Quantum Mechanics 3: the quantum mechanics of many-particle systems W.J.P. Beenakker Academic year 2017 – 2018 Contents of the common part of the course : 1) Occupation-number representation 2) Quantum statistics (up to § 2.5.3) Module 1 (high-energy physics): 2) Quantum statistics (rest) 3) Relativistic 1-particle quantum mechanics 4) Quantization of the electromagnetic field 5) Many-particle interpretation of the relativistic quantum mechanics The following books have been used : F. Schwabl, “Advanced Quantum Mechanics”, third edition (Springer, 2005); David J. Griffiths, “Introduction to Quantum Mechanics”, second edition (Prentice Hall, Pearson Education Ltd, 2005); Eugen Merzbacher, “Quantum Mechanics”, third edition (John Wiley & Sons, 2003); B.H. Bransden and C.J. Joachain, “Quantum Mechanics”, second edition (Prentice Hall, Pearson Education Ltd, 2000).
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Quantum Mechanics 3:
the quantum mechanics of many-particle systems
W.J.P. Beenakker
Academic year 2017 – 2018
Contents of the common part of the course:
1) Occupation-number representation
2) Quantum statistics (up to § 2.5.3)
Module 1 (high-energy physics):
2) Quantum statistics (rest)
3) Relativistic 1-particle quantum mechanics
4) Quantization of the electromagnetic field
5) Many-particle interpretation of the relativistic quantum mechanics
The following books have been used:
F. Schwabl, “Advanced Quantum Mechanics”, third edition (Springer, 2005);
David J. Griffiths, “Introduction to Quantum Mechanics”, second edition
(Prentice Hall, Pearson Education Ltd, 2005);
Eugen Merzbacher, “Quantum Mechanics”, third edition (John Wiley & Sons, 2003);
B.H. Bransden and C.J. Joachain, “Quantum Mechanics”, second edition
with H ψE(q1, · · · , qN) = E ψE(q1, · · · , qN) . (4)
The Hamilton operator describing a non-interacting many-particle system comprises of
pure 1-particle Hamilton operators, i.e. Hamilton operators that depend exclusively on
observables belonging to individual single particles. Denoting the 1-particle Hamilton
operator of particle j by Hj , the N -particle Hamilton operator reads
H =N∑
j=1
Hj , with[Hj , Hk
]= 0 for all j , k = 1 , · · · , N . (5)
This implies that the Hamilton operators of the individual particles are compatible observ-
ables, as expected for non-interacting particles. Suppose now that the 1-particle energy-
eigenvalue equation
Hj ψλj(qj) = Eλj
ψλj(qj) (6)
gives rise to an orthonormal set ψλj(qj) of energy eigenfunctions belonging to the energy
eigenvalues Eλj, which are labeled by a complete set of quantum numbers. Examples of
such complete sets of quantum numbers are for instance λj = nj for a linear harmonic
3
oscillator or λj = (nj , ℓj , mℓj , msj ) for a 1-electron atom. For the orthonormal set of
energy eigenstates of the complete non-interacting N -particle system we can identify three
scenarios.
A) The particles are distinguishable. The orthonormal set of N -particle energy eigen-
states comprises of states of the form
ψE(q1, · · · , qN) = ψλ1(q1)ψλ2(q2) · · · ψλN(qN) , with E =
N∑
j=1
Eλj. (7)
Such so-called product functions describe an uncorrelated system for which the prop-
erties of a specific particle can be measured without being influenced by the other
particles.
The complete set of all product functions spans the space of all possible
N-particle states involving distinguishable particles.
B) The particles are indistinguishable bosons. The orthonormal set of N -particle en-
ergy eigenstates comprises of totally symmetric states of the form
ψS(q1, · · · , qN) =1
NS
∑
diff.perm.
ψλ1(qP (1))ψλ2(qP (2)
) · · · ψλN(q
P (N)) ,
with NS
=√
number of different permutations of λ1, · · · , λN . (8)
The possible energy eigenvalues are the same as in equation (7).
The space of bosonic N-particle states is spanned by a reduced set of linear
combinations of product functions. As expected, this describes a correlated
system for which the measurement of the properties of a specific particle is
influenced by the other particles.
C) The particles are indistinguishable fermions. The orthonormal set of N -particle en-
ergy eigenstates comprises of totally antisymmetric states of the form
ψA(q1, · · · , qN) =1√N !
∑
perm.
(−1)P ψλ1(qP (1))ψλ2(qP (2)
) · · · ψλN(q
P (N))
=1√N !
∣∣∣∣∣∣∣∣∣∣∣
ψλ1(q1) ψλ2(q1) · · · ψλN(q1)
ψλ1(q2) ψλ2(q2) · · · ψλN(q2)
......
. . ....
ψλ1(qN) ψλ2(qN) · · · ψλN(qN)
∣∣∣∣∣∣∣∣∣∣∣
, (9)
where the determinant is known as the Slater determinant. The possible energy
eigenvalues are again the same as in equation (7), bearing in mind that λ1, · · · , λN
4
have to be all different. If two complete sets of quantum numbers λj and λk coin-
cide, then two of the columns in the Slater determinant are identical and the totally
antisymmetric eigenfunction ψA vanishes. This is known as the
Pauli exclusion principle for identical fermions: no two identical fermions can be in
the same fully specified 1-particle quantum state.
Also the space of fermionic N-particle states is spanned by a reduced set
of linear combinations of product functions. This too describes a correlated
system for which the measurement of the properties of a specific particle is
influenced by the other particles.
1.2 Occupation-number representation
Our aim: we want to construct the space of quantum states (Fock space) for a
many-particle system consisting of an arbitrary number of unspecified identical
particles. This Fock space makes no statements about the physical scenario that
the considered particles are in, such as being subject to interactions, external
influences, etc.. It simply is the complex vector space (Hilbert space, to be more
precise) that includes all possible many-particle states, on which the quantum
mechanical many-particle theory should be formulated. The actual construction
of Fock space involves finding a complete set of basis states for the decomposition
of arbitrary many-particle state functions. The properties of these basis states
will fix the properties of the Fock space.
The general rules for the construction of Fock space are:
• states should not change when interchanging particles;
• in order to guarantee the superposition principle, Fock space should not change when
changing the representation of the basis states.
1.2.1 Construction of Fock space
Consider identical particles of an unspecified type and assume q to be a corresponding
complete set of commuting 1-particle observables. Take the fully specified eigenvalues of
these observables to be exclusively discrete: qj labeled by j = 1 , 2 , · · · . The correspond-
ing 1-particle basis of normalized eigenstates of q is indicated by |qj〉, j = 1 , 2 , · · · .Subsequently we span the space of many-particle states by means of (special) linear com-
binations of product functions constructed from these 1-particle basis states, in analogy
to the 1-particle energy eigenfunctions that were used in § 1.1 to span the space of non-
interacting many-particle states. Since we are dealing with identical particles, a legitimate
5
many-particle state can make no statement about the identity of a particle in a specific
1-particle eigenstate. Such a many-particle state can at best make statements about the
number of particles nj that reside in a given fully specified 1-particle eigenstate belong-
ing to the eigenvalue qj . These numbers are called occupation numbers and can take the
values 0 , 1 , · · · (if allowed).
Postulate (replacing the symmetrization postulate): the collective set of hermitian
number operators n1 , n2 , · · · , which count the number of identical particles in each of the
1-particle quantum states |q1〉 , |q2〉 , · · · , form a complete set of commuting many-particle
observables. The employed complete set of 1-particle observables q can be chosen freely.
The hidden postulate aspect is that the state space for interacting particles can be
constructed from non-interacting building blocks that are based on 1-particle ob-
servables. Since the complete set of 1-particle observables can be chosen freely,
the superposition principle is automatically incorporated without any restric-
tions. This will guarantee that no mixed symmetry will occur in Fock space.
Hence, the corresponding set of normalized eigenstates |n1, n2, · · ·〉 will span Fock space
completely:
0-particle state : |Ψ(0)〉 ≡ |0, 0, · · ·〉 ≡ vacuum state ,
where the last pair state only has relevance for bosonic systems. With the help of the
corresponding completeness relation, a 2-particle observable B can be written as
B =∑
j,j′,k,k′
|Ψ(2)jk 〉〈Ψ
(2)jk |B |Ψ(2)
j′k′〉〈Ψ(2)j′k′| .
Each individual term occurring in this sum brings the particle pair from a state |Ψ(2)j′k′〉
to a state |Ψ(2)jk 〉 , with the matrix element 〈Ψ(2)
jk |B |Ψ(2)j′k′〉 as corresponding weight factor.
Using this 2-particle observable a proper many-particle observable can be constructed:
B(2)tot =
1
2
∑
α,β 6=α
Bαβ ,
where Bαβ is the 2-particle observable belonging to particles α and β 6= α . The factor
of 1/2 is introduced here to avoid double counting. The corresponding action of this
observable in Fock space simply reads
B(2)tot =
1
2
∑
j,j′,k,k′
a†j a†k 〈Ψ
(2)jk |B |Ψ(2)
j′k′〉 ak′ aj′ =1
2
∑
j,j′,k,k′
〈Ψ(2)jk |B |Ψ(2)
j′k′〉 a†j a†k ak′ aj′ . (56)
This method of writing all creation operators on the left and all annihilation operators
on the right in a many-particle operator is usually referred to as normal ordering. We
speak of an additive 2-particle quantity if the corresponding many-particle observable can
be represented in the form (56). This type of expression has the same form for any
discrete 1-particle representation, bearing in mind that just like we have seen in § 1.4.1each creation/annihilation operator in the expression is linked to a corresponding annihi-
lation/creation operator that is hidden in one of the basis states in the matrix element.
Switching to a continuous representation yields accordingly
In addition we assume the temperature to be low enough for the system to be effectively
in the ground state, i.e. effectively we are dealing with a T = 0 system.2
Without interactions among the particles: the total kinetic energy operator of the
non-interacting identical-particle system is diagonal in the momentum representation:
Ttot =∑
~k
~2~k 2
2ma†~k a~k =
1
2
∑
~k 6=~0
~2~k 2
2m
(a†~k a~k + a†−~k
a−~k
), (94)
with kinetic-energy eigenvalues
E(0) =∑
~k
~2~k 2
2mn(0)~k
. (95)
2The relevant details regarding temperature dependence and regarding the quantization aspects of the
container can be found in chapter 2.
33
The second expression for Ttot is given for practical purposes only (see below). It exploits
the symmetry of the momentum summation under inversion of the momenta. By adding
the total momentum operator
~Ptot =∑
~k
~~k a†~k a~k =1
2
∑
~k 6=~0
~~k(a†~k a~k − a†−~k
a−~k
), (96)
we can readily read off the particle interpretation belonging to the creation and annihila-
tion operators used. Particles with energy ~2~k 2/(2m) and momentum ~~k are created by
a†~k , annihilated by a~k and counted by n~k = a†~k a~k . The occupation number n(0)~k
indicates
how many particles can be found in the given momentum eigenstate in the absence of mu-
tual interactions. The ground state of the non-interacting N -particle system automatically
has n(0)~0
= N and n(0)~k 6=~0
= 0.
Including a weak repulsive interaction among the particles: in analogy with ex-
ercise 4, the inclusion of weak pair interactions that depend exclusively on the distance
between the two particles will lead to an additive many-particle interaction term of the
form
V =1
2
∑
~k,~k ′, ~q
U(q) a†~k a†~k ′a~k ′+~q a~k−~q , (97)
where
U(q) ≡ U(|~q |) =1
V
∫
V
d~r U(r) exp(−i~q ·~r ) (98)
is the Fourier transform of the spatial pair interaction per unit of volume. As argued in
exercise 4, the pair momentum of the interacting particles remains conserved under the
interaction (owing to translational symmetry) and U(q) merely depends on the absolute
value of ~q (owing to rotational symmetry).
What do we expect for the energy eigenstates in the ineracting case?
• First of all the state |n~0 = N, n~k 6=~0 = 0〉 will no longer be the ground state of the
interacting many-particle system. By applying V we observe
V |n~0 = N, n~k 6=~0 = 0〉 =1
2
∑
~q
U(q) a†~q a†−~q a~0 a~0 |n~0 = N, n~k 6=~0 = 0〉
=1
2N(N−1)U(0)|n~0 = N, n~k 6=~0 = 0〉
+1
2
√
N(N−1)∑
~q 6=~0
U(q)|n~0 = N−2, n~q = n−~q = 1, other n~k = 0〉 .
Therefore we expect the true ground state to receive explicit (small) contributions
from pairs of particles that occupy excited 1-particle eigenstates with opposite mo-
mentum. In that way the net momentum ~Ptot= ~0 indeed remains unaffected.
34
• In the non-interacting case the low-energy N -particle excitations involve just a few
particles in excited 1-particle states with O(h/L) momenta. Since all particles then
come with O(L) de Broglie wavelengths, we expect the quantum mechanical influ-
ence of the particles to extend across the entire system. Upon including repulsive
effects, we expect a single excitation to involve more than just bringing a single par-
ticle into motion and hence to require more energy than in the non-interacting case.
Approximations for weakly repulsive dilute Bose gases (Bogolyubov, 1947):
if the pair interaction is sufficiently weak and repulsive, we expect for the
low-energy N-particle states that still almost all particles occupy the 1-particle
ground state, i.e. N− 〈n~0〉 ≪ N . This allows us to apply two approximation
steps to simplify the many-particle interaction term, which hold as long as the
gas is sufficiently dilute to avoid too many particles from ending up in excited
1-particle states.
Approximation 1: for obtaining the low-energy N -particle states the effect from inter-
actions among particles in excited 1-particle states can be neglected. This boils down to
only considering interaction terms with two or more creation and annihilation operators
that belong to the 1-particle ground state:
V ≈ 1
2U(0)
n2~0− n~0
︷ ︸︸ ︷
a†~0 a†~0a~0 a~0 + U(0)
n~0︷︸︸︷
a†~0 a~0∑
~k 6=~0
n~k︷︸︸︷
a†~k a~k +1
2
n~0︷︸︸︷
a†~0 a~0∑
~q 6=~0
U(q)(
a†~q a~q + a†−~q a−~q
)
+1
2
∑
~q 6=~0
U(q)(
a~0 a~0 a†~q a
†−~q + a†~0 a
†~0a~q a−~q
)
.
• The first term corresponds to the configuration ~k = ~k ′ = ~q = ~0 , where all creation
and annihilation operators refer to the 1-particle ground state.
• The remaining terms cover situations where only two out of three momenta vanish.
The second term corresponds to the configurations ~k = ~q = ~0 and ~k ′ = ~q = ~0 ,
the third term to ~k ′ = ~k − ~q = ~0 and ~k = ~k ′ + ~q = ~0 , and the last term to~k − ~q = ~k ′ + ~q = ~0 and ~k = ~k ′ = ~0.
Subsequently the number operator n~0 can be replaced everywhere by N − ∑~q 6=~0
n~q and the
total number of particles N can be taken as fixed and very large:
V ≈ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
U(q)(
N a†~q a~q + N a†−~q a−~q + a~0 a~0 a†~q a
†−~q + a†~0 a
†~0a−~q a~q
)
.
35
Approximation 2, traditional approach: in Bogolyubov’s approach it is used that
the operators a~0 a~0 and a†~0 a†~0
can be effectively replaced by N when applied to the
lowest-energy N -particle states. In principle this could involve extra phase factors e2iφ0
and e−2iφ0 , however, these can be absorbed into a redefinition of the remaining cre-
ation and annihilation operators. This approach suggests that we are dealing with an
approximately classical situation, where the fact that the operators a~0 and a†~0 do not
commute only affect the considered N -particle states in a negligible way (as if they were
coherent states with |λ| ≫ 1). This assumption is plausible, bearing in mind that√N −n ≈
√N if N ≫ n. As a result, magnitude-wise the action of a~0 and a†~0 on
states with N−〈n~0〉 ≪ N will be effectively the same. In this way the following effective
approximation is obtained for the total Hamilton operator Htot = Ttot + V applicable to
the lowest lying energy eigenstates:
Htot ≈ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
(~2q2
2m+N U(q)
)(
a†~q a~q + a†−~q a−~q
)
+1
2
∑
~q 6=~0
N U(q)(
a†~q a†−~q + a−~q a~q
)
. (99)
Owing to this second approximation the total number of particles is no longer conserved
under the interactions. The justification for such an approach is purely thermodynamic
by nature (see Ch. 2): “the physical properties of a system with a very large number of
particles do not change when adding/removing a particle”. Since approximations are un-
avoidable for the description of complex many-particle systems, non-additive quantities of
this type are ubiquitous in condensed-matter and low-temperature physics. Note, however,
that the total momentum of the many-particle system remains conserved under the inter-
action, since each term in H adds as much momentum as it subtracts.
Approximation 2, but this time conserving particle number (based on the bachelor
thesis by Leon Groenewegen): in order to avoid that particle number is not conserved
during the second approximation step we can again exploit approximation 1 and write
N ≈ a~0 a†~0. Without loss of accuracy this allows to rewrite the total Hamilton operator
Htot = Ttot + V as
Htot ≈ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
(~2q2
2mN+ U(q)
)(
a~0 a†~0a†~q a~q + a~0 a
†~0a†−~q a−~q
)
+1
2
∑
~q 6=~0
U(q)(
a~0 a~0 a†~q a
†−~q + a†~0 a
†~0a−~q a~q
)
36
≡ N(N−1)
2U(0) +
1
2
∑
~q 6=~0
(~2q2
2m+N U(q)
)(
b†~q b~q + b†−~q b−~q
)
+1
2
∑
~q 6=~0
N U(q)(
b†~q b†−~q + b−~q b~q
)
, (100)
where the operators
b†~q 6=~0
=a~0 a
†~q√N
=a†~q a~0√N
and b~q 6=~0 =a†~0 a~q√N
=a~q a
†~0√N
(101)
have a much clearer physical interpretation. By means of b†~q a particle is excited from
the 1-particle ground state to the 1-particle state with momentum ~~q 6= ~0, whereas b~q
describes the de-excitation of an excited particle to the 1-particle ground state. The total
number of particles is not affected in this way! However, the total Hamilton operator
is identical to equation (99), with the non-additive character of Htot simply following
from the fact that the interaction can both excite pairs of particles from the 1-particle
ground state and make them fall back to the ground state. As a result of the macroscopic
occupation of the 1-particle ground state, the operators b†~q and b~q approximately behave
as ordinary creation and annihilation operators:
[b~q , b~q ′
]= 0 and
[b~q , b
†~q ′
]=
1
N
[a†~0 a~q , a
†~q ′ a~0
]=[a~q , a
†~q ′
] n~0N
−a†~q ′ a~q
N≈[a~q , a
†~q ′
].
More details can be found in the bachelor thesis by Leon Groenewegen.
In analogy with § 1.6.2 it is opportune to switch now to a quasi-particle de-
scription that turns the approximated interacting system into a non-interacting
quasi-particle system, with a corresponding total Hamilton operator that is both
additive and diagonal (i.e. decoupled).
In this quasi-particle description we expect that we have to combine the operator pairs b†~qand b−~q as well as b~q and b†−~q , since
• both operators within such a pair describe the same change in momentum and there-
fore affect the total momentum of the many-particle system in the same way;
• the quasi-particle number operators will then generate the correct terms b†~q b~q , b−~q b†−~q ,
b†~q b†−~q and b−~q b~q .
How to find the correct quasi-particle description will be addressed in the next intermezzo.
37
1.6.5 Intermezzo: the Bogolyubov transformation for bosons
Consider a many-particle system consisting of identical bosons that can occupy two fully
specified 1-particle quantum states |q1〉 and |q2〉 , such as the momentum states |~q 〉 and
|−~q 〉 in § 1.6.4. The corresponding creation and annihilation operators are given by a†1 , a†2
and a1 , a2 . The corresponding Fock space is spanned by the basis states |n1, n2〉 as given
in equation (28), where n1,2 represents the number of identical particles in each of the two
1-particle quantum states. Next we consider a non-additive Hamilton operator of the form
H = E (a†1 a1 + a†2 a2) + ∆(a†1 a†2 + a2 a1) (E > 0 and ∆ ∈ IR) . (102)
Evidently, the total number of particles is not conserved by such an operator, as the
occupation number of each of the two quantum states is raised by one or lowered by one
in the ∆ terms. However, by means of a so-called Bogolyubov transformation it can be
cast into an additive form up to a constant term.3 Such a transformation has the generic
form
c1 ≡ u1 a1 + v1 a†2 , c2 ≡ u2 a2 + v2 a
†1 (u1,2 and v1,2 ∈ IR) . (103)
The real constants u1 , u2 , v1 and v2 will be chosen in such a way that the operators c†1,2 and
c1,2 satisfy the same bosonic commutator algebra as a†1,2 and a1,2 . This will allow us to
formulate a new particle interpretation, where c†1,2 and c1,2 describe the creation and an-
nihilation of quasi particles. In order to guarantee that both a†1,2 , a1,2 and c†1,2 , c1,2
satisfy bosonic commutation relations, the following conditions should hold:
u1v2 − v1u2 = 0 and u21 − v21 = u22 − v22 = 1
⇒ u1 = + u2 , v1 = + v2 and u21 − v21 = 1
or u1 = −u2 , v1 = − v2 and u21 − v21 = 1 .
(104)
Proof: the commutation relations[c1, c1
]=[c2, c2
]=[c1, c
†2
]= 0 follow directly from the
bosonic commutation relations for a†1,2 and a1,2 . The indicated conditions for u1,2 and
v1,2 then simply follow from the fact that the other commutators
[c1, c2
] (103)====
[u1 a1 + v1 a
†2 , u2 a2 + v2 a
†1
] (26)==== (u1v2 − v1u2)1 ,
[c1, c
†1
] (103)====
[u1 a1 + v1 a
†2 , u1 a
†1 + v1 a2
] (26)==== (u21 − v21)1 ,
[c2, c
†2
] (103)====
[u2 a2 + v2 a
†1 , u2 a
†2 + v2 a1
] (26)==== (u22 − v22)1 ,
have to satisfy the usual bosonic commutation relations.
3If the states |q1〉 and |q2〉 carry opposite quantum numbers, such as the momentum in the states |~q 〉and |−~q 〉 in § 1.6.4, then both H and the Bogolyubov transformation conserve these quantum numbers.
38
The sign in this transformation can be chosen freely. Usually one chooses the plus sign,
resulting in the following generic form for a bosonic Bogolyubov transformation:
c1 ≡ u1 a1 + v1 a†2 , c2 ≡ u1 a2 + v1 a
†1 (105)
with inverse
a1 = u1 c1 − v1 c†2 , a2 = u1 c2 − v1 c
†1 . (106)
In the literature one usually opts for a parametrization of u1 and v1 in terms of a real
parameter η according to u1 = cosh η and v1 = sinh η , which automatically satisfies the
condition u21 − v21 = 1.
Bringing the non-additive Hamilton operator of equation (102) in additive form.
To this end we consider two combinations of quasi-particle number operators. First of all
c†1 c1 − c†2 c2(105)==== (u1 a
†1 + v1 a2)(u1 a1 + v1 a
†2) − (u1 a
†2 + v1 a1)(u1 a2 + v1 a
†1)
(26)==== u21 (a
†1 a1 − a†2 a2) − v21 (a1 a
†1 − a2 a
†2)
(26)==== (u21 − v21)(a
†1 a1 − a†2 a2)
(104)==== a†1 a1 − a†2 a2 . (107)
This expression tells us that certain quantum numbers are conserved under the transition
from particles to quasi particles, provided that these quantum numbers take on opposite
values in the states |q1〉 and |q2〉 (see the footnote on p. 38). In § 1.6.6 we will use this
property to conserve the momentum quantum numbers and thereby the total momentum
while performing the transformation, as anticipated in § 1.6.4. Secondly we have
This implies that for |∆| < E the non-additive Hamilton operator H in equation (102)
can be rewritten as
H = E (a†1 a1 + a†2 a2 + 1) + ∆(a†1 a†2 + a2 a1) − E 1
=√E2−∆2 (c†1 c1 + c†2 c2 + 1) − E 1 . (109)
Proof: based on equation (108) we are looking for a factor C such that C(u21 + v21) = E
and 2Cu1v1 = ∆. This implies that E and C should have the same sign and that
E2 −∆2 = C2(u21 − v21)2 (104)==== C2 E,C > 0
======⇒ |∆| < E and C =√E2−∆2 .
As promised at the start of this intermezzo, the Hamilton operator has been cast into a
form that consists exclusively of a unit operator and number operators for quasi particles.
In exercise 7 it will be shown that the corresponding ground state (i.e. the state without
quasi-particle excitations) is comprised of coherently created pairs of particles. In § 1.7.2the fermionic version of all this will be derived by employing a similar procedure.
39
1.6.6 Superfluidity for weakly repulsive spin-0 bosons (part 2)
The approximation (100) for the total Hamilton operator at the end of § 1.6.4 is exactly of
the form discussed in the previous intermezzo. Therefore, an appropriate set of Bogolyubov
transformations can be performed to recast the Hamilton operator in the additive form
H ≈ N(N−1)
2U(0) − 1
2
∑
~q 6=~0
(~2q2
2m+N U(q)− ǫ~q
)
+1
2
∑
~q 6=~0
ǫ~q (c†~q c~q + c†−~q c−~q)
=N(N−1)
2U(0) − 1
2
∑
~q 6=~0
(~2q2
2m+N U(q)− ǫ~q
)
+∑
~q 6=~0
ǫ~q c†~q c~q , (110)
with corresponding quasi-particle excitation spectrum
ǫ~q = ǫ−~q =~2q2
2m
√
1 +4mN U(q)
~2q2. (111)
For each pair of momenta ~q and − ~q with ~q 6= ~0 a bosonic Bogolyubov transformation
should be used, based on the following energy parameters in equations (102) and (109):
E → 1
2
(~2q2
2m+N U(q)
)
, ∆ → 1
2NU(q) .
For the total momentum operator we find with the help of approximation 2 on p. 37 that
~Ptot(96)
====1
2
∑
~q 6=~0
~~q(a†~q a~q − a†−~q a−~q
)≈ 1
2
∑
~q 6=~0
~~qa~0 a
†~0
N
(a†~q a~q − a†−~q a−~q
)(112)
(101)====
1
2
∑
~q 6=~0
~~q(b†~q b~q − b†−~q b−~q
) (107)====
1
2
∑
~q 6=~0
~~q(c†~q c~q − c†−~q c−~q
)=∑
~q 6=~0
~~q c†~q c~q .
Now we can read off the new particle interpretation. Quasi particles with energy ǫ~q and
momentum ~~q are created by c†~q , annihilated by c~q and counted by c†~q c~q . The ground
state of the new many-particle system still contains no quanta with momentum ~~q 6= ~0 .
However, both the composition of the ground state in terms of the original particles and
the shape (dispersion relation) of the elementary excitation spectrum have changed as a
result of the interaction.
40
1) Approximated excitation spectrum for weakly repulsive spin-0 bosons:
• For large excitation energies ~2q2 ≫ mN |U(q)| the quasi-particle excitation spec-
trum ǫ~q ≈ ~2q2/(2m)+NU(q) hardly differs from the non-interacting spectrum. As
such, the quasi particles have the same properties as the original bosons.
• For small excitation energies ~2q2 ≪ mN U(q) the quasi-particle excitation spec-
trum ǫ~q ≈ ~q√
N U(q)/m ≈ ~q√
N U(0)/m changes substantially. The quasi
particles describe massless quanta, i.e. quantized sound waves inside the considered
medium with speed of propagation
cs = limq→0
ǫ~q~q
=
√
N U(0)m
for U(0) =1
V
∫
V
d~r U(r) . (113)
The low-energy quasi-particle interpretation of the
interacting system therefore differs fundamentally
from the original particle interpretation of the non-
interacting system. Moreover, we have gone from a
free-particle system that does not allow for the possi-
bility of superfluidity to an interacting system that can
display superfluidity if u < uc ≈ cs .
ǫ~q
|~q |
linear
quadratic
2) Approximated ground state for weakly repulsive spin-0 bosons: another thing that has
changed substantially is the composition of the ground state of the interacting many-
particle system in terms of the original particles (see exercise 7).
• In the non-interacting case all particles occupy the 1-particle ground state with mo-
mentum ~0 and energy 0 (i.e. n(0)~0
= N and n(0)~k 6=~0
= 0). This condensate has spatial
correlations on all distance scales in view of the corresponding macroscopic de Broglie
wavelength.
• The latter still holds in the interacting case. However, the particle excitations for~k 6= ~0 are replaced by quasi-particle excitations with the same momentum. In the
new ground state none of these quasi-particle excitations are occupied (i.e. n~k=~0 ≈ N
and for the quasi particles n~k 6=~0 = 0). This situation without excited quasi particles
differs markedly from the situation without excited particles. Written in terms of
the original particle interpretation the new condensate actually contains particles
that are not in the ground state. To be more precise, it contains coherently excited
particle pairs with opposite momenta. The increase in kinetic energy is in that case
compensated by the decrease in repulsive interaction energy.
41
Remark: if the average spatial pair interaction U(0) would have been attractive, i.e. if
U(0) < 0, then the ground state would have been unstable. This can be read off directly
from the spectrum (111), which would become complex at very low energies. In that case
it would be energetically favourable for the system to have a large number of low-energy
particle pairs outside of the 1-particle ground state, which invalidates the assumptions of
the previous approach. The fermionic version of such a pairing effect and the ensuing
“pair-bonding” instability play a crucial role in the quantum mechanical description of
superconductivity. A first exposure to this phenomenon can be found in exercise 9.
1.6.7 The wonderful world of superfluid 4He: the two-fluid model
|~q | [A−1
]
ǫ~q ∗ k−1B
[K]
phononsrotons
The many-particle system worked out above is
often used to model the low-energy excitations
in liquid 4He, which becomes superfluid be-
low Tλ = 2.18K (P.L. Kapitsa, J.F. Allen and
A.D. Misener, 1937). At this point we have to
add a critical note. The pair interactions cannot
be truly weak, as required by the approximation
method, as we are dealing with a fluid rather than
a gas. As a consequence, a second branch of ex-
citations enters the spectrum at higher energies,
lowering the critical velocity for superfluidity.
fraction helium I
fraction helium IIAt the absolute zero of temperature 4He is com-
pletely superfluid, i.e. no low-energy excitations
are excited thermally. At a temperature around
0.9K a noticable influence of thermally induced
excitations sets in. On the temperature inter-
val 0 < T < Tλ there are effectively two fluids.
On the one hand there is the excitation-free con-
densate. This fluid, referred to as helium II, is
superfluid and carries no thermal energy. On the
other hand there is the collection of thermally induced quasi-particle excitations. This
fluid, referred to as helium I, carries the thermal energy and gives rise to friction. For
T > Tλ the influence of the helium II component is negligible. By means of this so-called
two-fluid model a few surprising phenomena can be understood.
How to recognize Tλ (P.L. Kapitsa, 1937): a remarkable superfluid phenomenon is that
the boiling of liquid helium abruptly subsides the moment that the temperature drops be-
low Tλ . Helium I will flow away from any spot where the fluid is locally warmer, thereby
carrying away thermal energy, whereas non-thermal helium II will flow towards that spot in
42
order to compensate for the drop in mass density. This superfluid helium II is characterized
by an infinitely efficient heat conductance, making it effectively impossible to have a tem-
perature gradient in the fluid. Already for temperatures just below Tλ the heat transport
becomes a millionfold more efficient and gas bubbles will have no time to form.
Friction for T < Tλ : objects moving through 4He at temperatures below Tλ experience
friction exclusively from the helium I component and not from the helium II component.
This allows us to experimentally determine the helium I and helium II fluid fractions (see
the picture on the previous page). For temperatures below about 0.9K the fluid behaves
almost entirely as a superfluid, giving rise to a persistent current once the fluid is set in
motion (e.g. at higher temperature).
Frictionless flow through porous media: as a result of friction the helium I component can
not flow through very narrow capillary channels. However, the superfluid helium II compo-
nent can do this without the need for a pressure difference between the two ends of such a
channel. So, 4He can flow through porous surfaces if T < Tλ . This flow will be completely
frictionless. Such a situation where a selective superfluid flow occurs is called a super leak.
Fisher and Pickett, Nature 444, 2006The fontain effect (J.F. Allen and H. Jones, 1938):
consider an experimental set-up consisting of two
containers with 4He that are connected by means
of a porous plug. Both containers are cooled down
to the same temperature below Tλ . If the tem-
perature in one of the containers is raised slightly,
e.g. by shining a pocket torch on it, then the num-
ber of thermally induced excitations will increase
in that container. As a result, the helium I com-
ponent increases at the expense of the helium II
component. In order to compensate for the differ-
ence in helium II concentration, helium II from the
other container will pass through the porous plug.
However, a compensatory reverse flow of helium I
to the other container, where the helium I concen-
tration is lower, is hampered by friction. Conse-
quently, an increased fluid concentration will ac-
cumulate in the heated container (heat pump). By providing the heated container with a
capillary safety-relief outlet, a spectacular helium fountain (fontain effect) is produced.
43
Creeping helium film: 4He has as an additional
property that the mutual van der Waals bond-
ing is weaker than the van der Waals bonding
to other atoms. Because of this a 30 nm thick
2-dimensional helium film (Rollin film) attaches
itself to the entire wall of a closed helium con-
tainer. If part of this helium film could flow or
drip to a lower level within the container (see
picture), then a superfluid helium II flow will
start that terminates only when the helium level
reaches its energetic optimum everywhere in the
container. During this process the helium liquid is seemingly defying gravity.
In case you want to see some video evidence for the bold statements that were made about
the weird and wacky world of superfluid 4He, then you are advised to have a look at
http://www.youtube.com/watch?v=2Z6UJbwxBZI or to do your own YouTube search.
1.7 Examples and applications: fermionic systems
1.7.1 Fermi sea and hole theory
Consider a fermionic many-particle system consisting of a very large, constant number N
of electrons with mass m. The electrons are contained inside a large cube with edges L
and periodic boundary conditions, giving rise to a discrete momentum spectrum:
In diagonal form the density matrix is given by the projection operator
ρdiagpure =
(
1 0
0 0
)
: 100% polarization with quantization axis parallel to ~P .
(154)
This corresponds to maximal order (read: maximal quantum information), with all
particles polarized in the direction of the polarization vector ~P .
• For 0 < |~P | < 1 the ensemble is partially polarized, represented by the inequality12< Tr(ρ2) = 1
2(1 + ~P 2) < 1.
57
• We speak of an unpolarized ensemble if |~P | = 0. In that case
ρ~P=~0
= 12I2 (155)
and Tr(ρ2) takes on its minimal value Tr(ρ2) = 12. In this situation as many parti-
cles are in the spin “↑” state as in the spin “↓” state. As such we are dealing with
an equal admixture of two totally polarized subensembles, one with spin parallel to
the quantization axis and one with spin antiparallel to the quantization axis. Note,
though, that the direction of this quantization axis can be chosen freely! An unpolar-
ized ensemble is in fact an example of a completely random ensemble with maximal
disorder (see § 2.4).
• An ensemble (beam) with degree of polarization |~P | can be viewed as being com-
posed of a totally polarized part and an unpolarized part. In diagonal form this
reads
ρdiag(151)==== |~P |
(
1 0
0 0
)
+1− |~P |
2
(
1 0
0 1
)
.
Freedom to choose: the density matrix (155) of a completely random ensemble does not
depend on the choice of representation for the considered space, in contrast to the density
matrix (154) for a pure ensemble. Moreover, a given mixed ensemble can be decomposed in
different ways in terms of pure ensembles . For instance, a mixture with 20% of the par-
ticles polarized in the positive x-direction, 20% in the negative x-direction, 30% in the
positive z-direction and 30% in the negative z-direction results in a net unpolarized en-
semble, since 0.2 (ρ~P =~ex
+ ρ~P =−~ex
) + 0.3 (ρ~P =~ez
+ ρ~P =−~ez
) = 12I2 .
2.3 The equation of motion for the density operator
As a next step towards quantum statistics we determine the equation of motion for the
density operator in the Schrodinger picture. Consider to this end a statistical mixture of
pure states that is characterized at t = t0 by the density operator
ρ(t0) =
N∑
α=1
Wα |α(t0)〉〈α(t0)| . (156)
Assume the weights Wα of the statistical mixture not to change over time. Then the
density operator evolves according to
|α(t)〉 = U(t, t0)|α(t0)〉 ⇒ ρ(t) =
N∑
α=1
Wα |α(t)〉〈α(t)| = U(t, t0) ρ(t0)U†(t, t0) . (157)
As shown in the lecture course Quantum Mechanics 2, the evolution operator U(t, t0)
satisfies the differential equation
i~∂
∂tU(t, t0) = H(t)U(t, t0) , (158)
58
where H(t) is the Hamilton operator belonging to the type of system described by the
ensemble. From this we obtain the following equation of motion for the density operator:
i~d
dtρ(t) =
[H(t), ρ(t)
], (159)
which is better known as the Liouville equation. This is the quantum mechanical analogue
of the equation of motion for the phase-space probability density in classical statistical me-
chanics, which can be formulated in terms of Poisson brackets as ∂ρcl/∂t = −ρcl ,Hcl .For this reason the name “density operator” was given to ρ .
Note: ρ(t) does not possess the typical time evolution that we would expect for a quantum
mechanical operator. Since ρ(t) is defined in terms of state functions, it is time indepen-
dent in the Heisenberg picture and time dependent in the Schrodinger picture. This is
precisely the opposite of the behaviour of a normal quantum mechanical operator.
Time evolution of the ensemble average of the dynamical variable A: the en-
semble average [A] as defined in § 2.1 satisfies the evolution equation
d
dt[A ]
(137)====
d
dtTr(ρA) = Tr
(ρ∂A
∂t
)+ Tr
( dρ
dtA)
(159)==== Tr
(ρ∂A
∂t
)− i
~Tr(HρA− ρHA
)= Tr
(ρ∂A
∂t
)− i
~Tr(ρAH − ρHA
)
(137)====
[ ∂A
∂t
]− i
~[AH − HA ] =
d
dt[A ] , (160)
where ∂A/∂t refers to the explicit time dependence of A . In the penultimate step we have
used that the trace is invariant under cyclic permutations, i.e. Tr(ABC) = Tr(CAB) =
Tr(BCA). The evolution equation (160) has the same form as the evolution equation for
the expectation value of a dynamical variable in ordinary QM. However, this time we have
averaged twice in view of the double statistics!
2.4 Quantum mechanical ensembles in thermal equilibrium
As we have seen, there are marked differences between pure ensembles (with
maximal order) and completely random ensembles (with maximale disorder).
We are now going to investigate the generic differences a bit closer.
Let |k〉 be an orthonormal set of eigenstates of ρ corresponding to the eigenvalues ρk .As such, these eigenstates together span the (reduced) D-dimensional space on which the
pure state functions of the subensembles are defined. This dimensionality D indicates the
maximum number of independent quantum states that can be identified in the reduced
59
space on which we have chosen to consider the quantum mechanical systems. For instance,
the density matrix in the spin-1/2 spin space has dimensionality D = 2. With respect to
this basis we have
ρpure =
0. . . Ø
01
0Ø
. . .0
vs ρrandom =1
D
11 Ø
. . .Ø 1
1
.
Examples of these extreme forms of density matrices have been given in equations (154)
and (155) in § 2.2. In case of a pure ensemble the density matrix is simply the projection
matrix on the corresponding pure state vector, which depends crucially on the chosen
representation of the D-dimensional space. In case of a completely random ensemble, the
D orthonormal basis states each receive the same statistical weight 1/D to guarantee
that Tr(ρ) = 1. In that case each state is equally probable and the density matrix is
proportional to the D-dimensional identity matrix, which does not depend at all on the
chosen representation.
In order to quantify the differences we introduce the quantity
σ ≡ −Tr(ρ ln ρ)compl.==== −
D∑
k,k′=1
〈k|ρ|k′〉〈k′| ln ρ |k〉 = −D∑
k=1
ρk ln(ρk) ≡ S
kB
, (161)
where ρk represents the probability to find the system in the pure basis state |k〉 . This
quantity is minimal for pure states and maximal for a completely random ensemble :
σpure = 0 vs σrandom = −D∑
k=1
1
Dln(1/D) = ln(D) . (162)
That σrandom is maximal will be proven in § 2.4.3. In accordance with classical thermo-
dynamics, it can be deduced from σ = S/kB and σrandom = ln(D) that the quantity
S should be interpreted as the quantum mechanical entropy4 and kB as the well-known
Boltzmann constant. You could even say that this definition of the entropy is superior
to the classical one. In classical mechanics there is no such thing as counting states.
At best one could work with phase-space volumes that have to be made dimensionless
4This definition of entropy is also used in information theory in the form of the so-called Shannon
entropy −∑k Pk ln(Pk), which is an inverse measure for the amount of information that is encoded in
the probability distribution P1, · · · , PN . Each type of probability distribution in fact corresponds to a
specific type of ensemble. In the lecture course Statistical Mechanics a relation will be established between
the thermodynamical and quantum mechanical definitions of the entropy of a canonical ensemble.
60
by means of an arbitrary normalization factor. For this reason the classical entropy can
only be defined up to an additive constant. In QM, however, there is a natural unit of
where the 1-particle spinvector χs,ms(σ) as usual refers to a spin-s particle with spin
component ms~ along the z-axis. As a result, the degree of degeneracy of the energy
eigenvalues Eν is increased by a factor 2s+1.
70
νz
νy
νx
1
2
3
4
1 2 3 4 51
23
45
0
Next we make use of the fact that the box has macroscopic dimensions. As a result, the
1-particle energy levels are very closely spaced and it makes good sense to switch to a
continuous density of states D(Ekin) that counts the number of 1-particle quantum states
per unit of kinetic energy. In the continuum limit the number of 1-particle quantum states
with kinetic energy on the interval (Ekin , Ekin + dEkin) is simply given by D(Ekin)dEkin .
In order to determine D(Ekin) it will prove handy to introduce the quantized wave vector
~k = π~ν/L (νx,y,z = 1 , 2 , · · ·) , (186)
to determine the corresponding density of states D(k), and finally to insert the expression
Ekin(k) = ~2k2/(2m) for the kinetic energy in terms of the wave vector. As indicated
in the picture given above, a unit cube in ~ν-space contains exactly one spatial quantum
state and therefore 2s+1 fully specified quantum states (including spin). In view of the
positivity conditions νx,y,z > 0, all possible ~ν-values are in one octant of ~ν-space.
νx
νyν = 4
νx
νyν = 11
71
For ν-values that are not too small, the number of states N(k) with wave vector smaller
than k = πν/L is in good approximation related to the corresponding volume in ~ν-space
(as illustrated in the preceding picture for a 2-dimensional lattice):
N(k) =
k∫
0
dk′D(k′) ≈ (2s+ 1)1
8
( 4
3πν3)
(186)====
2s+ 1
6π2V k3 . (187)
By taking a derivative we obtain
D(k) ≈ d
dkN(k) =
2s+ 1
2π2V k2 . (188)
This expression is actually valid for an arbitrary spin-s particle in a box. Only
when we switch to D(Ekin) differences will occur between for instance relativis-
tic and non-relativistic scenarios, since in those cases the dispersion relation
between Ekin and k will be different (see exercise 17).
In the non-relativistic case considered here the dispersion relation reads k =√
2mEkin/~2
and therefore
D(Ekin) =( dk
dEkin
)
D(k) =2s+ 1
4π2
( 2m
~2
)3/2
V E1/2kin . (189)
The approximation that we just used is obviously not correct for very low kinetic 1-particle
energies. For instance, for k < π/L there are no kinetic energy levels possible at all. In
fact, in the low-energy regime the discrete character of the energy levels manifests itself.
This is caused by the fact that the box has finite dimensions that cannot be considered
macroscopic for de Broglie wavelengths λ(k) = 2π/k = O(L). As we will see later, such low
kinetic energies play a relevant role in bosonic gases at very low temperatures (see § 2.7).They do not play an important role in a Fermi gas in view of the small number of different
low-energy states N(k)(187)==== O[V/λ3(k)] (see later).
Geometrical aspects: the number of quantum states per unit of kinetic energy and per
unit of volume of the box is given in the continuum limit by D(Ekin)/V , which does not
depend on the volume! In the continuum limit the precise shape of the box in general
does not matter. If we indicate the surface area of the macroscopic box by S , then the
influence of the edge of the box on D(Ekin) is suppressed by a relative factor S/(kV )
(see exercise 15). This simply underlines the quantum mechanical expectation that the
presence of the edge of the box is felt up to order λ(k) distances away. The corresponding
suppression factor Sλ(k)/V is small if the kinetic energy is sufficiently high, as is true (on
average) for a Fermi gas that has a sufficiently large number of particles.
72
Fermi gas in a constant attractive potential: as we will see in § 2.5.3 and § 2.5.4,in certain physical scenarios the Fermi gas is bound by a constant attractive potential
V = −V0 < 0 inside the box. This has no bearing on the kinetic energy levels nor on the
corresponding energy eigenfunctions that were derived previously. However, the 1-particle
energy levels will shift according to
Eν
V0 6=0−−−−→ Eν − V0 , (190)
where Eν is the kinetic contribution to the total 1-particle energy. By means of the
definition
E ≡ Ekin − V0 ≥ −V0 (191)
the 1-particle density of states of the bound system is expressible in terms of the 1-particle
density of states (189) for free particles inside the box:
DV0(E) = D(Ekin) = D(E + V0) . (192)
Fermi gas in a constant homogeneous magnetic field: assume the Fermi gas to be
subjected to a constant homogeneous magnetic field in the z-direction, ~B = B ~ez . In spin
space this gives rise to an additional interaction
H spinB
= C Sz (C ∈ IR , C ∝ B) . (193)
Since this operator acts exclusively on spin space and the Hamilton operator (182) ex-
clusively on position space, the 1-particle energy eigenfunctions are still given by equa-
tion (185). However, the (2s+1)-fold spin degeneracy of the kinetic 1-particle energy
levels will be lifted according to
Eν
B 6=0−−−−→ Eν +ms~C ≡ Eν,ms , (194)
where Eν is the kinetic contribution to the total 1-particle energy Eν,ms for a particle
with spin component ms~ along the z-axis. For each value of the (magnetic) spin quantum
number ms this is equivalent to a constant potential shift V = ms~C. By defining
Ems ≡ Ekin +ms~C ≥ ms~C , (195)
the density of states for each of the 2s+1 individual spin states is expressible in terms of
the 1-particle density of states (189) for free particles inside the box:
Dms(Ems) =1
2s+ 1D(Ekin) =
1
2s+ 1D(Ems−ms~C ) . (196)
73
2.5.1 Ground state of a Fermi gas
In § 2.4.2 it was shown that a gas at temperature T = 0 will be in the ground state
(completely degenerate gas). For gases consisting of bosons or distinguishable particles
this automatically implies that all particles will populate the lowest 1-particle energy
eigenstate. For a Fermi gas this is not a possibility, bearing in mind that Pauli’s ex-
clusion principle tells us that at most one fermion can occupy a fully specified quantum
state. In that case the N particles will occupy the kinetic 1-particle energy eigenstates
one by one from the bottom up, with the highest occupied level given by the Fermi energy
EF ≡ ~2k2F/(2m). States with Ekin > EF will not be occupied (vacant) for a Fermi gas at
T = 0 and can only by excited by increasing the temperature or by including pair inter-
actions into the analysis. The collective set of states with Ekin ≤ EF form the so-called
Fermi sea. The value of the Fermi energy can be extracted readily from equation (187),
since
N = N(kF ) =
kF∫
0
dk D(k)(187)====
2s+ 1
6π2V k3F (197)
implies that
kF =( 6π2
2s+ 1
)1/3
ρ1/3N and EF =
~2k2F2m
=~2
2m
( 6π2
2s+ 1
)2/3
ρ2/3N (198)
in terms of the particle density ρN = N/V . Since N is large, the precise degree of
occupation of the (degenerate) highest energy level is immaterial. Moreover, the fact
that only a limited amount of particles will occupy the lowest kinetic energy eigenstates
also implies that it is a good approximation to replace the density of states by the con-
tinuous function in equation (189). Note that EF ∝ ρ2/3N /m, which tells us that the
Fermi energy increases with increasing particle density and that it will intrinsically be a
factor 103 smaller for protons and neutrons than for electrons. The total kinetic energy
of the ground state of the Fermi gas amounts to
ET=0kin,tot =
kF∫
0
dk~2k2
2mD(k)
(188)====
2s+ 1
10π2
~2
2mV k5F
(197),(198)=======
3
5NEF . (199)
As a consequence, the average kinetic energy per particle is pretty substantial in spite of
the extremely low temperature:
Ekin =ET=0
kin,tot
N=
3
5EF . (200)
74
Quite often a kind of effective (classical) temperature is assigned to a T = 0 Fermi gas.
This is done by invoking the classical principle of equipartition of energy:
Tcl ≡ ET=0kin,tot
3NkB/2
(199)====
2
5
EF
kB
≡ 2
5TF ∝ ρ
2/3N
m, (201)
where TF is called the Fermi temperature. For a given constant number of particles such
a completely degenerate Fermi gas actually exerts a finite pressure (degeneracy pressure)
P = −(
∂ET=0kin,tot
∂V
)
N
(198),(199)=======
2
3
ET=0kin,tot
V
(201)==== kBρNTcl ∝ ρ
5/3N
m, (202)
which obeys a kind of effective (classical) ideal gas law. The higher the particle density of
the Fermi gas becomes, the more the particles lose their individuality and the higher the
effective temperature will be, making it more difficult to excite the fermions. For finite
temperatures T ≪ TF the Fermi gas will actually behave as if we were dealing with a
completely degenerate Fermi gas (see § 2.7).
Calculational tool: to get a feel for the particle-specific aspects of the relation between
EF , TF and ρN , a few characteristic values are listed in the table below. In this table ρrelN
and T relF mark the particle density and Fermi temperature for which EF = mc2, so that
ρN ≪ ρrelN
and TF ≪ T relF are required for the non-relativistic approach to be valid:
EF = mc2(ρN/ρ
relN
)2/3and TF = T rel
F
(ρN/ρ
relN
)2/3. (203)
particle mc2 in MeV T relF in K ρrel
Nin m−3
electron 0.51100 0.5930× 1010 1.6589× 1036
proton 938.27 1.0888× 1013 1.0269× 1046
neutron 939.57 1.0903× 1013 1.0312× 1046
Ground state of a Fermi gas in a constant attractive potential: assume the Fermi
gas to be bound by a constant attractive potential V = −V0 < 0 inside the box. All quan-
tities related to the kinetic part of the energy remain unaffected, such as EF , ET=0kin,tot , Ekin
and the thermal gas pressure P . However, the total energy of the completely degenerate
Fermi gas will undergo a shift:
ET=0tot =
3
5NEF −NV0 ⇒ Eb ≡ V0 − Ekin = V0 − 3
5EF , (204)
with Eb the average binding energy per particle.
75
2.5.2 Fermi gas with periodic boundary conditions
In view of the applications in chapter 1 and the discussion of relativistic quantum theories,
we make a small excursion to non-interacting periodic systems with periodicity length L.
The previously-used rigid-wall boundary condition will now be replaced by a set of spatial
periodicity conditions for the wave function of the system:
The spatial 1-particle energy eigenfunctions are now periodic plane waves (see App.A.1):
ψ~k(~r ) =1√L3
exp(i[xkx + yky + zkz ]
)=
1√L3
exp(i~k · ~r ) ,
Ek =~2
2m(k2x + k2y + k2z) =
~2~k 2
2m≡ ~2k2
2m, (206)
in terms of the quantized wave vectors
~k =2π
L~ν , with νx,y,z = 0 ,±1 ,±2 , · · · . (207)
The remaining steps closely follow the previous discussion. Each unit cube in ~k-space
now contains precisely (2s+ 1) (L/2π)3 = (2s+ 1) V/(8π3) fully specified quantum states
(including spin). In the continuum limit the number of states N(k) with wave vector
smaller than k is related to the corresponding volume in ~k-space:
N(k) =
k∫
0
dk′D(k′) ≈ (2s+ 1)V
8π3
( 4
3πk3)
=2s+ 1
6π2V k3 . (208)
This expression is identical to the result (187) for a particle inside a cube with edges L
and rigid walls. As such, also the 1-particle density of states is identical for both types of
systems. The most prominent difference between both types of systems is the absence of
finite-size effects in the periodic case. This difference vanishes in the limit L → ∞ . In
that case we are effectively dealing with a system without explicit periodicity or enclosure.
In such scenarios we usually resort to using a periodic system with periodicity length L
and subsequently take the continuum limit L→ ∞ (see App.A.2).
Ground state: in the ground state a Fermi gas with periodic boundary conditions occu-
pies all 1-particle ~k-states inside a Fermi sphere with radius
kF =( 6π2
2s+ 1
)1/3
ρ1/3N (209)
for which N(kF ) = N . On the Fermi surface of the Fermi sphere the energy and momen-
tum are given by the Fermi energy EF and the Fermi momentum pF :
EF =~2k2F2m
and pF = ~kF . (210)
76
2.5.3 Fermi-gas model for conduction electrons in a metal
As a first example of a Fermi gas we consider conduction electrons in a piece of metal. In a
metal the electrons of the metal atoms can be divided into two categories. The bulk of all
the electrons occupy very narrow (nearly atomic) energy bands, being effectively bound to
the atomic nuclei to form metal ions. On the other hand there are the conduction electrons,
which occupy the partially filled highest energy band (conduction band). In an individual
(separate) metal atom these electrons would be the valence electrons situated in the outer-
most atomic shell. In first approximation these conduction electrons can roam freely inside
the metal lattice (nearly-free electron model). This phenomenon occurs when the average
distance between the metal ions is smaller than the average extension of the atomic orbitals
for these conduction electrons. In this oversimplified, semi-classical picture the conduction
electrons are kind of squeezed out of their atomic confinement by the metal lattice, as
sketched in the picture displayed below where the hatched area represents the conduction
band. For characteristic metal densities of ρNat= 1028–1029metal atoms/m3 the effective
volume per metal atom corresponds to the volume of a sphere of radius rs =(3/4πρNat
)1/3,
which is no more than 2– 6 times the Bohr radius a0 = 0.529× 10−10m.
separate atom
energy levels
V (x)
. . .
. . .
macroscopic chain
energy bands (closely spaced energy levels)
V (x)
“edge” “edge”
In first approximation the conduction electrons form a Fermi gas (electron gas), with the
edge of the metal delimiting the macroscopic box (Sommerfeld, 1928).6 In this very crude
approximation the explicit electron–electron as well as electron–ion interactions are rep-
resented by an averaged constant (attractive) potential V = −V0 < 0.
6As a whole, the combination of metal ions and electron gas is neutral. Such a combination of an
electron gas and a complementary collection of positively charged particles is also referred to as a plasma.
77
V
x0 L
EF conduction band
−V0
metal
As we have seen in § 2.5.1 this adds a constant additive contribution −NV0 to the total
energy of the electron gas. This constant term is not essential for the thermal response of
the system. However, it is instrumental in binding the conduction electrons to the metal.
Inside the metal lattice the conduction electrons can be considered approximately free,
with scattering being suppressed due to the fact that scattering into occupied 1-particle
states is forbidden (Pauli-blocked) by the exclusion principle. This eliminates the large
majority of all electron–electron and electron–ion collisions. Pauli-allowed collisions, with
a sufficiently large energy transfer, actually provides the mechanism of weak energy contact
that allows the gas of conduction electrons to achieve thermodynamical equilibrium inside
the metal. This will play a crucial role in understanding the thermal properties of metals,
as we will see shortly.
The heat capacity of metals at room temperature.
Consider the heat capacity(∂Etot/∂T
)
V,Nof an ideal gas consisting of a constant num-
ber N of conduction electrons inside a fixed volume V . In classical statistical mechanics,
equipartition of energy tells us that each free particle of an ideal gas contributes 3kB/2
to the heat capacity. For N conduction electrons this would amount to a total classical
heat capacity of at least 3NkB/2. This is nowhere near the experimentally measured heat
capacities of metals, which turn out to have much smaller values.
Question: can we understand this based on what we have learned so far?
As mentioned above, in first approximation the conduction electrons form a Fermi gas
with the edge of the metal delimiting the macroscopic box. The particle density of the
conduction-electron gas equals the density of metal atoms multiplied by the number of con-
duction electrons Zv per metal atom. For typical conductors such as copper (Cu) or silver
(Ag) we have Zv = 1 and both densities coincide. A conduction-electron gas at T = 0 has
an average kinetic energy per conduction electron of Ekin = 3EF/5, which translates into a
78
classical gas at temperature Tcl = 2EF/(5kB) = 2TF/5. For instance, silver has a particle
density of ρN = 5.8× 1028 particles/m3 for both metal ions and conduction electrons. By
means of the table below equation (203) we find the Fermi energy to be EF ≈ 5.5 eV,
which corresponds to an effective classical temperature of Tcl ≈ 2.5×104K. By increasing
the thermal energy of the system, conduction electrons can start to occupy excited states
with kinetic 1-particle energy Ekin > EF . At room temperature, i.e. Tk ≈ 300K, the
available thermal energy per conduction electron amounts to 3kBTk/2 ≈ 0.039 eV ≪ EF .
As a consequence, only a small fraction of the conduction electrons is liable to be excited,
i.e. conduction electrons with kinetic energy Ekin = EF −O(kBTk). In the figure displayed
below this is illustrated by plotting the degree of occupation n(Ekin) of the fully specified
quantum states with given kinetic energy Ekin .
Ekin
n(Ekin)
0
0.5
1 T = 0
0<T≪ TF
EF
The distributions for T = 0 and 0 < T ≪ TF only differ marginally. Later on in this
chapter it will be proven that at room temperature only a fraction of order Tk/TF of
the conduction electrons will participate in the heat conductivity of a metal. This heat
conductivity is actually largely determined by the thermal motion of the ions in the metal
lattice (phonons) and not so much by the freely moving collective of conduction electrons!
This is the reason why you will not be burning your fingers on a piece of metal at room
temperature, in spite of the fact that the metal possesses certain characteristic properties
of a classical gas at temperature Tcl = O(104K).
Note: the above-given Fermi-gas model does not apply to all properties of a metal. For
instance, in order to adequately describe the electrical conductivity of metals, the inter-
actions between the conduction electrons and the lattice of metal ions are indispensable.
Among other things, these interactions give rise to the band structure in metals and the
Cooper-pair binding in low-temperature superconductors.
79
2.5.4 Fermi-gas model for heavy nuclei
Hahn –Ravenhall –Hofstadter (1956)
nucleon distribution in arbitrary units
r [fm]
Ca
V
Co
InSb
Au
Bi
Up to now it has only been possible to de-
scribe the collective nuclear forces inside
atomic nuclei by means of simplified mod-
els. These models parametrize our present
experimental and theoretical knowledge.
On the one hand, the radius of the charge
distribution inside the nucleus, i.e. the pro-
ton distribution, can be inferred from scat-
tering experiments and spectroscopy. For
example: the scattering of electrons off nu-
clei and the Rontgen spectra of muonic
atoms. In the latter case an atomic elec-
tron is replaced by a muon, which is 200
times heavier and therefore has 200 times
smaller Bohr orbits. On the other hand, information about the radius of the complete
nucleon distribution, involving both protons and neutrons, can be inferred from the scat-
tering of α-particles (helium nuclei) off nuclei (see figure). In that case a relatively sharp
energy threshold is observed, marking the minimum energy that the α-particles need in or-
der to overcome the nuclear Coulomb-repulsion and become sensitive to the nuclear forces.
By means of these experiments we have learned that the nucleon density is almost constant
inside a heavy nucleus:
ρN ≈ 0.17× 1045 nucleons/m3 ⇒ mass density ≈ 2.8× 1017 kg/m3 . (211)
ρ/ρN
nuclear radius
surface thickness
r [fm]
This density only has a mild dependence
on the type of nucleus and falls off rel-
atively quickly in the vicinity of the nu-
clear surface. Therefore it makes sense to
assign to each nucleus a spherical volume
with effective radius R , which is defined
in such a way that it contains all nucleons
when assuming a constant nucleon den-
sity ρN . This nuclear radius roughly sat-
isfies the relation
R ≈ r0A1/3 , with r0 ≡
( 3
4πρN
)1/3
≈ 1.12× 10−15m = 1.12 fm . (212)
The quantity A is the mass number of the nucleus, which is the sum of the number of
protons Z (charge number) and the number of neutrons A−Z .
80
Thomas – Fermi model: these observations can be combined into a simple Fermi-gas
model (Thomas–Fermi model) for heavy atomic nuclei. In this model the nucleus is com-
prised of two independent Fermi gases of protons and neutrons that are bound to a sphere
of radius R by a constant attractive potential V = −V0 < 0 inside the sphere. This po-
tential well replaces the average short-distance interactions with neighbouring nucleons.
The distribution of the two types of nucleons inside the nucleus are in first approximation
taken to be uniform, i.e. finite-size effects are neglected. At first sight it may seem non-
sensical to “leave out” the actual strong interactions among the nucleons. However, a few
properties of nuclei can be explained by Pauli’s exclusion principle alone. By studying the
Fermi-gas model we want to figure out which properties fall into this category and which
properties require detailed knowledge about the dynamics of the nuclear forces. If we wish,
the model can be refined by additionally taking into account the Coulomb repulsion be-
tween protons as well as an approximation for the nuclear forces between two nucleons.
This obviously makes the model less simple, since the interactions are very sensitive to
the actual distance between the two nucleons that participate in the interaction and there-
fore to their 2-particle spin state. After all, spatial symmetrization/antisymmetrization
decreases/increases the average distance between the particles.
V
r0 R
E(n)F
−V0
neutrons
V
r0 R′
E(p)F
∆VCoul/Z
per protonprotons
including Coulomb repulsion
In the ground state both Fermi gases fill all energy levels up to the Fermi energy, causing the
ground-state energy of a heavy nucleus to simply amount to 35
[ZE
(p)F +(A−Z)E(n)
F
]−AV0
in this very simple model. So, if we do not account for Coulomb repulsion between the
protons the total ground-state energy becomes minimal for Z = A/2 (see exercise 16). The
pure Fermi-gas model therefore predicts ground-state nuclei that consist of equal amounts
of protons and neutrons:
ρ(p)N = ρ
(n)N =
ρN
2
(203),(211)======= 8.25× 10−3 ρrelN . (213)
81
The empirical fact that the ratio Z/(A − Z) decreases for increasing A-values can only
be understood if we do account for the Coulomb repulsion between the protons (see exer-
cise 16), making it energetically favourable to have slightly more neutrons than protons in
the nucleus. Since the proton mass (Mp) and neutron mass (Mn) are almost identical, both
Fermi gases will have roughly the same Fermi energy in the absence of Coulomb repulsion:
EF(198)====
~2
2Mp
(
3π2 ρN
2
)2/3 (203),(213)
−−−−−−→ EF ≈ 38MeV , (214)
with corresponding Fermi temperature TF = EF/kB
(203)==== O(1011K). The average kinetic
energy per nucleon then reads
Ekin =3
5EF ≈ 23MeV (215)
and the average binding energy per nucleon
Eb = V0 − Ekin . (216)
By scattering low-energy neutrons off complex nuclei, the depth of the potential well V0 is
measured to be about 40MeV. The Fermi-gas model therefore predicts an average binding
energy per nucleon of 40MeV− 23MeV = 17MeV.
Eb
[MeV]
A
fusion fission
very stable nuclei
asymmetric interactions
surface effects
The experimental data on the binding energy per nucleon (see figure) follows directly from
the difference between the observed mass of the heavy nucleus and the sum of the rest
masses of the individual nucleons:
AEb =[ZMp + (A− Z)Mn − M
]c2 ,
Mpc2 = 938.27MeV and Mnc
2 = 939.57MeV . (217)
82
The corresponding experimental results for the binding energy per nucleon can be cast into
a rather insightful form, known as the semi-empirical mass formula (Bethe –Weizsacker
formula, 1935):
Eb = aV − aSA−1/3 − aC
Z2
A4/3− aa
( 2Z−AA
)2
+ pair interactions ,
aV = 15.8MeV , aS = 18.3MeV , aC = 0.71MeV and aa = 23.2MeV . (218)
The so-called volume coefficient aV is almost completely explained by the exclusion prin-
ciple. Furthermore, it will be shown in the exercise class that the Fermi-gas model will be
less successful in predicting the asymmetry coefficient aa , which quantifies the influence
of Z 6= A/2 on the binding energy. That should not come as a big surprise as we expect
nucleon-sensitive interactions such as Coulomb and isopsin interactions to play a role in
that quantity. By contrast, the binding-energy effects caused by a uniform charge distri-
bution of protons inside the nucleus are found to agree extremely well with the observed
Coulomb coefficient aC.
Additional remarks: the fact that the total binding energy of a heavy nucleus is approx-
imately proportional to the mass number A is called saturation. In fact the first signs
of saturation already set in for relatively light nuclei with A ≥ 12. The physical picture
behind this is that the nucleons of heavy nuclei only interact with a small, fixed number
of nearest neighbours in view of the short-range nature of the nuclear forces, which fall
off exponentially on characteristic distance scales of roughly 1.4 fm. For a closest packing
of nucleons the number of nearest neighbours indeed equals 12. This is in sharp contrast
with long-range interactions, such as Coulomb interactions, which involve the interactions
between all pairs of particles. In that case we expect the binding energy to be propor-
tional to the number of pairs 12A(A− 1), as can be read off from the Coulomb-repulsion
term ∝ Z2 in the semi-empirical mass formula. This nuclear saturation effect legitimizes
the point of view that each nucleon in the inner parts of a large nucleus experiences an
effective (averaged) nuclear binding potential that does not depend on the total number of
nucleons. The saturation effect also predicts the nuclear surface to have a dual influence
on the binding energy (see the surface term ∝ A−1/3 = r0/R in the semi-empirical mass
formula). First of all, there are finite-size effects that affect the density of states of the
Fermi gas, as worked out in exercise 15(ii). Secondly, the particle density has a sharp
drop-off in the vicinity of the nuclear surface, causing the nucleons near the edge of the
nucleus to have fewer nearest neighbours and consequently less binding energy. The latter
aspect gives rise to surface tension, resembling the kind of surface tension that is observed
for liquid droplets. For this reason the Fermi-gas model including finite-size effects is also
referred to as the liquid-drop model.
83
2.5.5 Stars in the final stage of stellar evolution
A star can be regarded as a many-particle system for which the gravitational pressure
is in equilibrium with the thermal and radiation pressure. During the stellar evolution
various nuclear-fusion stages can be identified, with the fusion fuel being consumed step-
by-step (and shell-by-shell) from the inside out. In the final stage of stellar evolution the
fusion fuel gets exhausted in the stellar core, where lowly reactive nuclei such as 12C- and16O-nuclei have amassed. If the fusion reactions effectively cease in the stellar core, then
so does the radiation pressure. With one component of the compensatory pressure gone,
the star will start to implode in an attempt to find a new equilibrium. While the star
implodes the outer layers of stellar matter are ejected, leaving behind a stellar core with
a thin atmosphere. Depending on its mass, this stellar core can go through different phases.
Phase 1: Fermi-gas model for white dwarfs.
During the implosion the thermal (kinetic) pressure increases in the stellar core, where the
matter is being compressed. Assume, for convenience, that the stellar core is comprised
of one characteristic type of atom with charge number Z and mass number A = 2Z . If
the stellar matter is compressed to the point that the distance between the atomic nuclei
becomes smaller than the Bohr radius aZ = a0/Z of a 1-electron atom consisting of a
nucleus with charge Ze and one electron, then the atoms will be compressed into full
ionization (pressure ionization). In first approximation the electrons that are “squeezed
out of the atoms” will form an electron gas inside a spherical volume.7 For atoms with
A = 12 –16 this phenomenon occurs when the mass density inside the stellar core exceeds
the 107 kg/m3 barrier. A star that reaches a new stable equilibrium during this electron-
gas phase is called a white dwarf.
Let’s assume the electron density of the electron gas to be uniform, which can be replaced
by a more realistic density profile in a refined version of the model (see exercise 18). This
constant density can be translated by means of the plasma relation
ρ(e)N = Zρ
(nuclei)N =
ρM
AMpZ (219)
in terms of the uniform mass density ρM of the stellar core. In this expression AMp is the
mass per characteristic atomic nucleus and Z indicates how many electrons are released
per atom. For typical white-dwarf mass densities ρM = 107–1011 kg/m3, the electron den-
sity is given by ρ(e)N = 1033–1037 electrons/m3 and the corresponding Fermi temperature by
TF = 108–1010K (remembering that A = 2Z ). The typical thermodynamic temperature
T <107K of a white dwarf is (much) smaller than the Fermi temperature of the electron
gas, which therefore effectively will be in the ground state.
7Just as in § 2.5.3 the electrons and ions (nuclei) together form a plasma.
84
r
r+drSince the kinetic energy of the electrons substantially
exceeds the thermal energy of the much heavier nu-
clei, we have to look for a stable equilibrium con-
dition based exclusively on the kinetic energy of the
electron gas at T = 0 and the gravitational potential
of the star. The gravitational potential is obtained
straightforwardly by building up the star in spherical layers
Egrav = −R∫
0
dr(4πr2ρM
)( 4
3πr3ρM
)GN
r= − 16
3π2ρ2MGN
R∫
0
dr r4
= −( 4
3πR3ρM
)2 3GN
5R= − 3GNM
2
5R. (220)
Here M and R are the mass and radius of the imploded star, whereas GN is Newton’s
gravitational constant. The total energy of the star is now approximately given by
ET(199)
=======3
5NEF − 3GNM
2
5R
(198)==== N
3~2
10me
(3π2ρ(e)N )2/3 − 3GNM
2
5R
(219)=======
( M
AMpZ) 3~2
10me
( 3π2ZρM
AMp
)2/3
− 3GNM2
5R
ρM =M/V=======
[3
10
( 9π
4
)2/3 ~2
me
( Z
AMp
)5/3]M5/3
R2− 3GNM
2
5R≡ a
M5/3
R2− 3GNM
2
5R.
The corresponding equilibrium condition reads dET/dR = 0, which yields
− 2aM5/3
R3+
3GNM2
5R2= 0 ⇒ MR3 =
( 10a
3GN
)3 A=2Z==== 7.30× 1050 kgm3 . (221)
In this way we have established a relation between the mass and radius (∼ 104 km) of
a white dwarf. This relation runs counterintuitive to our gut feeling that a larger star
should necessarily be a heavier star. However, this counterintuitive relation is confirmed
by astronomical observations. In fact we are dealing here with a pure quantum mechani-
cal phenomenon caused by degenerate quantum matter that manifests itself in a directly
measurable way in an object of astronomical dimensions!
Question: Is the non-relativistic approach actually viable here?
For a mass density of ρM = 107 kg/m3 we have ρ(e)N = 3 × 1033 electrons/m3 ≪ ρrel
N, as
can be extracted from the table below equation (203). At “low” mass densities the non-
relativistic approach is indeed perfectly acceptable. However, at “high” mass densities
relativistic corrections are unavoidable. For a mass density of ρM = 1011 kg/m3 the other
85
extreme is reached, since ρ(e)N = 3 × 1037 electrons/m3 ≫ ρrelN . In that case the Fermi gas
should be treated (ultra) relativistically. The ultra-relativistic treatment, which implies
neglecting the electron mass with respect to the kinetic energy, gives rise to a completely
different type of equilibrium condition (see exercise 18): stable white dwarfs have an up-
per limit MC
on the mass. This upper limit of roughly 1.4 solar masses is called the
Chandrasekhar limit.
radius versus mass for celestial bodies
Phase 2: Fermi-gas model for neutron stars.
Heavy stars can pass phase 1 without reaching a stable equilibrium. Such a star will
continue to implode. For increasing mass density electrons can be “captured” by the
protons through the inverse β-decay process
e− + nucleus(A,Z) → νe + nucleus(A,Z−1) , (222)
with the approximately massless neutrinos escaping (evaporating) and thereby cooling the
star. The physical mechanism behind this inverse β-decay process looks as follows. On
the right-hand side of the equation the nuclei have less total binding energy in view of
the disturbed balance between the number of protons and neutrons. For sufficiently high
mass densities this energy increase will be compensated by an energy reduction caused
by lowering the number of electrons in the electron gas. At a mass density of roughly
86
ρM = 1015 kg/m3 the kinetic pressure of the neutrons will become more important than
the pressure of the electron gas. At even higher densities in the range of 1017–1018 kg/m3
this will eventually result in a Fermi gas that is predominantly comprised of neutrons
(neutron gas). A star that reaches stability during this neutron-gas phase is therefore
called a neutron star. The main differences with the Fermi gas of phase 1 are summarized
by two simple substitutions: Z → A and me → Mn ≈ Mp . Apart from that every-
thing else stays the same. As a result, the a-parameter in equation (221) will receive
an extra factor (me/Mp)(A/Z)5/3 = O(10−3). This intrinsic factor of 1000 reduction for
the characteristic radii of neutron stars is indeed confirmed by astronomical observations
(R ∼ 10 km). Again the ultra-relativistic treatment gives rise to an upper limit on the
mass of a neutron star. This upper limit has to be taken with a pinch of salt, though. For
mass densities of the order of the nuclear mass density 2.8 × 1017 kg/m3 we also expect
the dynamics of the strong nuclear force to play an important role.
2.6 Systems consisting of non-interacting particles
As a second application of ensembles in thermal equilibrium, we try to derive the statistical
properties of systems consisting of a large number of non-interacting particles. We assume
the corresponding 1-particle energy spectrum to be discrete, as is for instance the case
for systems that are contained inside an enclosure or trapped in a potential well. Three
fundamentally different scenarios can be identified.
(A) The particles are distinguishable, for example by means of their positions. This will
give rise to the so-called Maxwell–Boltzmann statistics.
(B) The particles are indistinguishable identical bosons. This will give rise to the so-called
Bose–Einstein statistics.
(C) The particles are indistinguishable identical fermions. This will give rise to the so-
called Fermi–Dirac statistics.
For deriving the statistical distributions in scenarios (B) and (C) we will use the grand-
canonical ensemble approach of § 2.4.4. The Maxwell–Boltzmann distribution can then be
obtained by taking the classical limit. Since the particles are assumed to be non-interacting,
we choose to work with an occupation-number representation based on a complete set of
1-particle observables that includes the 1-particle Hamilton operator. The total Hamilton
operator and total number operator of the many-particle system have a simple additive
form in this representation:
N =∑
j
nj and Htot(N) =∑
j
Ej nj , (223)
with E1 ≤ E2 ≤ E3 ≤ · · · the ordered fully specified 1-particle energy spectrum. The
87
statistical properties of the ensemble can be specified by the average occupation numbers
nk = [nk] , since
N =∑
j
nj and Etot =∑
j
Ej nj . (224)
For the grand-canonical density operator ρ = exp(−βHtot(N)−αN
)/Z(α, T ) this implies
ρ a†k = exp(−βEk − α) a†k ρ . (225)
Proof: consider the operator O(λ) ≡ exp(λA) a†k exp(−λA), with the observable in the
exponent given by A ≡ −βHtot(N) − αN(223)==== −∑j (βEj + α)nj . With the help of
[A, a†k
] (16)==== − (βEk + α) a†k we can derive for arbitrary real λ that
d
dλO(λ) = exp(λA)
[A, a†k
]exp(−λA) = − (βEk + α)O(λ)
⇒ O(λ) = exp(−λ [βEk + α]
)O(0) : exp(λA) a†k exp(−λA) = exp
(−λ [βEk + α]
)a†k .
The operator identity in equation (225) is then obtained by simply inserting λ = 1 and
subsequently right-multiplying the result by ρ = exp(A)/Z(α, T ).
The desired statistical distribution for identical particles then becomes
nk(17),(137)==== Tr(ρ a†k ak)
(225)==== exp(−βEk − α) Tr(a†k ρ ak) = exp(−βEk − α) Tr(ρ ak a
†k)
(26),(27)==== exp(−βEk − α) Tr
(ρ[1± nk]
) (139)==== (1± nk) exp(−βEk − α)
⇒ nk =1
exp(βEk + α)∓ 1, (226)
where the minus (plus) sign refers to bosons (fermions). In the limit α→ ∞ the identity of
the particles does not matter anymore and the quantum mechanical distributions smoothly
approach the well-known classical Maxwell–Boltzmann distribution nk = exp(−βEk−α).In this asymptotic limit 1-particle quantum effects dominate the statistics as a result of
the suppression factor exp(−αN) in the grand canonical density matrix.
All this can be summarized by the following distribution for the average occupation number
belonging to a fully specified 1-particle energy level with energy E :
n(E) =1
exp(βE + α) + γ, with γMB = 0 , γBE = −1 and γFD = 1 . (227)
Note: the temperature and fugacity parameters β and α are not independent of each
other in view of the conditions (181) for the ensemble averages Etot and N .
88
0
1
2
3
4
5
6
7
8
-2 -1 0 1 2 3
n(E)
βE+α
Bose–Einstein
Maxwell–Boltzmann
Fermi–Dirac
quantum mechanical domain
classical
domain
The quantum mechanical distributions
The characteristic properties of these distributions can be read off directly from the plots
displayed above.
• The three distributions approach each other for large values of βE + α . For instance
this applies to the high-energy limit or to situations for which α is large. In such situ-
ations the identity of the particles does not matter anymore and quantum mechanical
statistics effectively merges with classical Maxwell–Boltzmann statistics.
• As anticipated n(E) ∈ [0, 1] for the Fermi–Dirac distribution.
• For the Bose–Einstein distribution we have to demand that βE + α > 0 for each
energy E, in order to guarantee that n(E) ≥ 0. Hence, α is bounded from below
by α > −βEground , with Eground the 1-particle ground-state energy.
The Fermi energy: in the low-temperature limit T → 0 ( β → ∞ ) the Fermi–Dirac
distribution approaches the distribution
n(E)β→∞−−→
1 if E < EF
0 if E > EF
,
for which all energy states are occupied up to the maximum energy EF = − limβ→∞
α/β .
For a Fermi gas this maximum energy of course coincides with the definition for the Fermi
energy that was introduced in § 2.5.
89
Freeze out of bosonic degrees of freedom: the fact that βE + α ≥ βEground + α > 0
for the Bose–Einstein distribution has as direct consequence that
n(E)
n(Eground)=
exp(βEground + α)− 1
exp(βE + α)− 1≤ exp
(−β [E−Eground]
).
If an energy gap ∆E is present between the 1-particle ground state and the first excited
state, then thermal excitations are heavily suppressed for temperatures kBT ≪ ∆E . Such
a scenario for instance applies to a harmonic oscillator with characteristic frequency ω for
temperatures kBT ≪ ~ω (see exercise 14). In that case we speak of an effective freeze
out of the corresponding bosonic degree of freedom since the excited states containing one
or more bosonic energy quanta ~ω will be scarcely populated. The effective freeze out of
such a degree of freedom goes hand in hand with a reduced thermal response (e.g. heat
conduction) of the system. Applied to complex systems such as molecular gases or crystal
lattices, this phenomenon can occur step-by-step: for decreasing temperature more and
more vibrational modes of the complex system will freeze out.
Bose –Einstein condensation: if βE + α ↓ 0 in the Bose–Einstein distribution, then
the corresponding average occupation number diverges. This sharp rise in the degree of
occupation is unique for the bosonic distribution and is responsible for the physical phe-
nomenon of Bose–Einstein condensation. The fact that bosons have an increased proba-
bility to occupy the same quantum state can result in a noticeable (macroscopic) fraction
of the particles occupying the ground state already at finite temperatures. Whether this
phenomenon actually occurs or not depends on the properties of the system considered.
2.7 Ideal gases in three dimensions
An important example of non-interacting many-particle systems in thermal equilibrium is
provided by the ideal gases. These systems consist of non-interacting particles of mass m
and spin s that are contained inside a macroscopically large enclosure with fixed volume
V . We assume the enclosure to contain a very large, approximately constant number N
of free identical particles. The 1-particle energy spectrum pertaining to such an enclosed
system is derived in § 2.5. In view of the macroscopic volume and the very large number
of particles, we can in most situations8 resort to taking the continuum limit (189) for the
1-particle density of states D(Ekin):
D(Ekin) =
2s+ 1
4π2
( 2m
~2
)3/2
V E1/2kin if Ekin ≥ 0
0 if Ekin < 0
. (228)
8This actually does not hold for ideal gases consisting of identical bosons at very low temperatures (see
the discussion of Bose–Einstein condensation in § 2.7.2).
90
The average number of gas particles with kinetic energy on the interval [Ekin , Ekin+dEkin]
is then given by n(Ekin, T ) dEkin , with
n(Ekin, T ) = n(Ekin)D(Ekin)(227)====
D(Ekin)
exp(βEkin + α) + γ. (229)
The quantity α is determined by the constant particle density ρN = N/V :
ρN =1
V
∞∫
−∞
dEkin n(Ekin, T )(228),(229)=======
2s+ 1
4π2
( 2m
~2
)3/2∞∫
0
dEkinE
1/2kin
exp(βEkin + α) + γ
x=βEkin , (166)=========
2s+ 1
4π2
( 2mkBT
~2
)3/2∞∫
0
dxx1/2
exp(x+ α) + γ, (230)
where the integral is a strictly decreasing function of α . This last observation will play
an important role in the phenomenon of Bose–Einstein condensation discussed in § 2.7.2.By means of the thermal de Broglie wavelength
In this expression N0 represents the average number of bosons in the 1-particle ground
state. This ground-state occupation number can be neglected with respect to N, unless
α = O(1/N) ≈ 0. As such, the density of particles in the 1-particle ground state starts
to play a prominent role if the temperature is lowered to the critical temperature T0 for
which α ≈ 0. By taking the ratio with respect to ρN , equation (241) can be rewritten in
terms of the critical temperature:
N0
N= 1 − (2s+1)f1/2(α)
ρNλ3T= 1 − f1/2(α)
f1/2(0)
λ3T0
λ3T= 1 − f1/2(α)
f1/2(0)
( T
T0
)3/2
. (242)
The last term on the right-hand side represents the fraction of bosons that occupy excited
1-particle states.
T < T0 : below the critical temperature a noticeable fraction of all bosons should be in
the ground state, since α cannot become smaller than 0 and therefore at most a fraction
(T/T0)3/2 < 1 of all bosons can occupy excited 1-particle states. In that case we say
that the ground state is occupied macroscopically. According to equation (241), getting a
fraction N0/N = O(1) of all bosons to occupy the ground state requires α = O(1/N) ≈ 0.
To very good approximation the fraction of bosons in the ground state is therefore given by
N0
N≈ 1−
( T
T0
)3/2
. (243)
This observable tendency of the identical bosons to occupy the same quantum state is called
Bose–Einstein condensation (BEC). This condensation takes place in momentum space
and should not be confused with the more traditional meaning of the word condensation as
a transition from a gaseous phase to a liquid phase. To increase the amount of condensation
requires lower temperatures or higher densities .10 The latter statement follows from the
fact that according to equation (239) T0 ∝ ρ2/3N . The condensate of bosons in the ground
state forms a degenerate Bose gas, which will not exert pressure nor contribute to the heat
capacity. At low temperatures only the fraction of bosons outside of the condensate will
10Note: it is far from easy to realize Bose–Einstein condensation in the lab. At high densities any
interaction among the bosons tends to completely swamp the symmetrization effects. The density should
therefore be kept low, causing T0 to approach absolute zero (microkelvins or less).
94
0
0.5
1
1.5
2
0 0.5 1 1.5 2 2.5 3
CV
ρNkB
T/T0
∝ (T/T0)3/2
classical
limit
contribute to the pressure and heat capacity of the Bose gas, giving rise to an additional
factor (T/T0)3/2 in the temperature dependence of these quantities (as the plot for the
heat capacity reveals). More details about this are given in the lecture course on Statistical
Mechanics. In § 2.7.4 we will briefly discuss why it took another 70 years to actually realize
the first gaseous Bose–Einstein condensates in the lab.
T > T0 : in that case α has to differ noticeably from 0 to avoid that the ratio N0/N in
equation (242) becomes negative. Automatically only a negligible fraction of the bosons
occupy the ground state.
Occupation of the first excited state for T <T0 : in order to say something sensible
about this we consider the 1-particle energy spectrum (184) for a particle in a cubic box.
For the energy difference between the ground state with ν =√3 and the first excited state
with ν =√6 we can derive that
β (Eex−Eground) =3~2π2
2mL2kBT
V =L3
==== N−2/3 T0T
3h2ρ2/3N
8mkBT0
(231)==== N−2/3 T0
T
3π
4(ρNλ
3T0)2/3
(239)==== N−2/3 T0
T
3π
4
(2.612 (2s+1)
)2/3= O(N−2/3T0/T ) .
The maximum average occupancy of the first excited energy level is obtained for the
minimum value for α , i.e. for α ↓ − βEground :
n(Eex) ≤ 1
exp(β [Eex−Eground]
)− 1
=1
exp(O[N−2/3T0/T ]
)− 1
≤ O(N2/3) ≪ N .
So, solely the 1-particle ground state can achieve macroscopic occupancy in a 3-dimensional
Bose gas! The adjacent first excited energy level can contain many particles, but never-
theless will not be able to reach macroscopic occupancy.
95
The phase transition at T = T0 : as explained on page 92, the quantum mechanical
domain of an ideal gas is reached if ρNλ3T = O(1). In that case the wave packets of the
individual particles will overlap and therefore the quantum mechanical indistinguishability
of the identical particles becomes relevant. In the vicinity of the critical temperature T0
this is indeed the case. The remarkable aspect of T0 is that the gas ondergoes a rapid
transition (phase transition) at that temperature.
For an arbitrary temperature T < T0 an ideal Bose gas is comprised of a fixed,
saturated number of particles in excited states. The distribution of these par-
ticles has a canonical form: it is described by a Bose–Einstein distribution
with α = 0. The remaining particles occupy the ground state, which thereby
achieves macroscopic occupancy. The presence of the condensate is a direct
consequence of the underlying particle conservation law.
Proof: for T < T0 we know that α vanishes, which implies that the excited states are
maximally occupied according to a canonical distribution. From equation (243) we know
that the number of particles that do not occupy the ground state is given by N (T/T0)3/2.
When increasing the number of particles by a factor f while keeping the temperature
constant, we get
N → fN(239)
===⇒ T0 → f 2/3T0 and N (T/T0)3/2 → N (T/T0)
3/2.
The number of particles outside the condensate indeed remains constant. We could actually
have read this off directly from equation (239), since
N (T/T0)3/2 (231),(239)
======= 2.612 (2s+1)V/λ3T
is indeed independent of the particle density and merely counts the number of times the
characteristic thermal volume λ3T fits into the system volume V .
The fact that we have added additional particles merely shows up in the occupancy of
the ground state! One could say that this condensate “does not occupy volume”, as
it does not exert pressure, and therefore can be adjusted freely to account for the cor-
rect total particle density. On this ground we anticipate that the Bose–Einstein distri-
bution will have a O(N) statistical uncertainty in the occupancy of the ground state,
as will be shown in exercise 19. Moreover, the presence of the condensate manifests
itself in long-range correlations (long-range order) in view of the corresponding macro-
scopic de Broglie wavelength. The high-occupancy condensate behaves approximately as
a (pseudo classical) matter wave that can be described quantum mechanically by means
of a complex function (order parameter) ψ0 =√N0 exp(iφ0).
96
Quantum mechanical indistinguishability and Bose–Einstein condensation:
the reason behind the absence of condensation for distinguishable particles resides in the
number of possible ways to excite particles from the ground state. Assume N0 spin-0
particles to occupy the ground state. Then there are 3N0 different first excited states if
the particles are distinguishable and merely 3 if the particles are indistinguishable.
N0 Eground
0 Eex (3×)
ground state
N0−1 Eground
1 Eex (3×)
excited state
For large enough N0 it becomes entropically more favourable for distinguishable particles
to be excited because of the kB ln(3N0) gain in entropy. For indistinguishable particles the
gain in entropy is substantially smaller and independent of N0 , i.e. kB ln(3), which lifts
the entropic obstruction for having a large number of particles in the ground state.
Bose–Einstein condensation versus D(Ekin) for low energies: the previous indis-
tinguishability argument is actually not able to guarantee the occurrence of a condensate.
If the continuous density of states is not small for low-energy 1-particle states, then there
are nevertheless ample of ways to excite a particle with a minimum gain of energy. This
can keep the system from actually achieving a macroscopic occupation of the ground state.
In the above-given derivation for a 3-dimensional Bose gas it was crucial for the occurrence
of Bose–Einstein condensation that the strictly decreasing integral f1/2(α) remained finite
when α reached its minimal value. In exercise 20 it will be investigated how the formation
of a condensate depends on the dimensionality of the system and on the energy–momentum
relation of the particles. The results of this investigation can be summarized as follows:
Bose–Einstein condensation is indeed not possible if D(Ekin) does not vanish
for Ekin = Eground : in that case∫∞−∞ dEkinD(Ekin)/
[exp(βEkin+α)−1
]is not
finite for α ↓ − βEground and the continuous density of states D(Ekin) does not
require being corrected.
The role of the particle conservation law: in the absence of particle conservation
there is no reason for a condensate to form. A photon gas in thermal equilibrium with an
atomic heat bath is an example of this (see § 4.3.2). As a result of being absorbed by
the atoms, the photons actually do not show up as particles in the atomic heat bath and
therefore they exclusively contribute to the thermodynamic balance of energy. In that case
the entropy of the heat bath actually does not depend on the number of photons.
On the other hand, a canonical ensemble with a fixed number of particles is able to give rise
to Bose–Einstein condensation, as is for instance the case for low-energy photons in the
97
curved-mirror microresonator of exercise 21. In that case the fixed number of particles acts
as a boundary condition on the occupation numbers of the 1-particle energy levels, which
resembles the action of keeping the ensemble average N constant in a grand-canonical
ensemble.
2.7.3 Fermi gases at T 6= 0 (no exam material)
For ideal gases that consist of identical fermions there can be no condensation phenomenon,
since maximally 2s + 1 fermions can occupy the ground state as a consequence of the
exclusion principle. Therefore there is no need to correct the continuous density of states
D(Ekin). We simply obtain
ρNλ3T/(2s+1) = g1/2(α) , g1/2(α) ≡ 2√
π
∞∫
0
dxx1/2
exp(x+ α) + 1. (244)
The low-temperature domain (degenerate Fermi gases): for Fermi gases we speak
of low temperatures if ρNλ3T is large, which implies that α has to be strongly negative in
order to satisfy equation (244). In that case the following integral approximation can be
used for a function G(x) that grows slower than exponentially for large x:
∞∫
0
dxG(x)
exp(x+ α) + 1≈(
1 +O[α−2 ])
−α∫
0
dx G(x) , (245)
provided that the integral on the right-hand side exists. This approximation follows from
the fact that the function n(x) ≡ [exp(x+α)+1]−1 effectively behaves like a step function
for large −α , with a deviation that is symmetric around x = −α (zie plotted curve). This
deviation is only significant for a limited range of x-values: x+ α = (E−µ)/kBT = O(1),
which corresponds to kinetic energies that only differ from the Fermi energy by O(kBT ).
x
n(x)
0
0.5
1step function approximation
actual distribution
−α
x+ α = (E−µ)/kBT = O(1)
98
For a degenerate Fermi gas equation (244) leads to the following approximate relation
between α, ρN and T :
ρNλ3T/(2s+1) ≈ 4
3√π(−α)3/2
(
1 +O[α−2 ])
(231)===⇒ −α ≈ ~2
2mkBT
( 6π2ρN
2s+ 1
)2/3 (198)====
EF
kBT
(201)====
TFT
. (246)
This relation is valid up to O(α−2) = O(T 2/T 2F ). In EF and TF we immediately recognize
the Fermi energy and Fermi temperature as derived for T = 0 Fermi gases in equations
(198) and (201). In accordance with the examples discussed in § 2.5 the domain of degen-
erate Fermi gases corresponds to −α ≈ TF/T ≫ 1, which can be reached by lowering the
temperature or increasing the density and (related to that) TF . Only a O(T/TF ) fraction
of the particles will participate in the thermal response of a degenerate Fermi gas. This
can be read off from equation (233) by employing the integral approximation (245):
U ≈ ρNkBT25(−α)5/2
23(−α)3/2
(
1 +O[α−2 ])
(246)====
3
5ρNkBTF
(
1 +O[T 2/T 2F ])
⇒ CV =( ∂U
∂T
)
V,N= O(ρNkBT/TF ) . (247)
More details will be given in the lecture course on Statistical Mechanics. Contrary to Bose
gases, in a degenerate Fermi gas almost all particles will contribute to the kinetic energy
and pressure of the gas, regardless of the temperature.
Chemical potential: in statistical mechanics α is usually rewritten in terms of the so-
called chemical potential
µ ≡ − α
β
(166)==== −αkBT . (248)
For a degenerate Fermi gas we have µ ≈ EF , so that
n(E)(227)====
1
exp(β [E − µ]
)+ 1
≈ 1
exp(β [E −EF ]
)+ 1
. (249)
For a completely degenerate Fermi gas at T = 0 all states will be occupied up to the
Fermi energy:
n(E) =
1 if E < EF
0 if E > EF
,
in accordance with the definition given in § 2.5.
99
Example: Pauli paramagnetism of a degenerate electron gas.
Consider an ideal electron gas with constant volume V and constant number of electrons N .
Assume that T ≪ TF , which implies that the chemical potential µ of the gas satisfies
βµ = µ/kBT ≫ 1. Subject this system to a very weak constant homogeneous magnetic
field in the z-direction, ~B = B~ez . In spin space this gives rise to an extra interaction
H spinB
= − ~MS · ~B = −B ~MS · ~ez =2µ
BB
~Sz (0 < µ
BB ≪ kBT ≪ µ) ,
in terms of the Bohr-magneton µB. As explained on page 73, the magnetically induced
spin interaction leads to a splitting of the kinetic 1-particle energy levels, thereby lifting the
spin degeneracy. The average number of electrons with spin parallel (N+) and antiparallel
(N−) to the magnetic field are given by
N± =
∞∫
−∞
dE±D±(E±)
exp(β [E± − µ ]
)+ 1
(196)====
∞∫
±µBB
dE±
12D(E± ∓ µ
BB)
exp(β [E± − µ ]
)+ 1
,
with E± the 1-electron energy for states with spin component ± ~/2 along the z-axis and
with D(Ekin) the kinetic 1-particle density of states (228) for m = me and s = 1/2. By
means of the change of variable E± = Ekin ± µBB this can be simplified to
N± =V
4π2
( 2me
~2
)3/2∞∫
0
dEkinE
1/2kin
exp(β [Ekin − µ ± µ
BB ])+ 1
, (250)
with Ekin the kinetic 1-electron energy. This allows us to derive the following additive ex-
pressions for the (constant) electron density and the magnetization density, i.e. the average
magnetic moment per unit of volume in the direction of the magnetic field:
ρN =N
V=
N+ + N−V
,
MS
≡ 1
V B [ ~M(1)Stot
· ~B ] = − 2µB
~V[ ~S
(1)tot · ~ez ]
(53)==== − µ
B
V(N+ − N−) . (251)
In the limit µBB ≪ µ this can be simplified by means of the integral approximation (245):
ρN ≈ 1
6π2
( 2me
~2
)3/2(
[µ− µBB ]3/2 + [µ+ µ
BB ]3/2
)
≈ 1
3π2
( 2meµ
~2
)3/2
,
MS
≈ µB
6π2
( 2me
~2
)3/2(
[µ+ µBB ]3/2 − [µ− µ
BB ]3/2
)
≈ 1
6π2
( 2meµ
~2
)3/2( 3µ2BB
µ
)
.
In lowest-order approximation we have µ ≈ EF and
MS
≈ µ2B
3ρN
2EF
B EF = kBTF======⇒ χS(T ) ≡
( dMS
dB)
B=0= µ2
B
3ρN
2kBTF. (252)
100
The paramagnetic susceptibility χS(T ) of the electron gas is constant for T ≪ TF . This
low-temperature phenomenon is called Pauli paramagnetism. It stresses that a degenerate
Fermi gas does not obey Curie’s classical law χS(T ) ∝ T−1 for weak magnetic fields. This
is caused by the effective quantum mechanical interaction induced by the fermionic anti-
symmetrization procedure. In the classical high-temperature limit the magnetic allignment
of the magnetic moments is counteracted by thermal disorder. This leads to a magneti-
zation density that depends on the ratio of the magnetic interaction energy µBB and the
characteristic thermal energy kBT , resulting in a net 1/T temperature dependence of the
paramagnetic susceptibility. At low temperatures only a O(T/TF ) fraction of the electrons
take part in thermal excitations, effectively replacing the 1/T temperature dependence by
a temperature-independent factor 1/TF . The exclusion principle is more effective at coun-
teracting magnetic allignment than thermal disorder, since the characteristic temperature
TF of a degenerate Fermi gas can largely exceed the (ambient) temperature T .
2.7.4 Experimental realization of Bose–Einstein condensates
Superfluid helium: 4He-atoms consist of two protons, two neutrons and two electrons,
which implies that the corresponding total spin is integer. The 4He-atoms can thus be
regarded as bosons. They satisfy Bose–Einstein statistics and can therefore give rise to the
formation of a Bose–Einstein condensate at low temperatures. As argued in § 1.6.4 –1.6.6,this condensate plays an essential role in the phenomenon of superfluidity in liquid 4He
at temperatures below 2.2K. The rare 3He-isotope consists of two protons, one neutron
and two electrons, which implies that the corresponding total spin is half integer. The3He-atoms can thus be regarded as fermions. They satisfy Fermi–Dirac statistics and can
therefore not give rise to a similar type of condensate as 4He-atoms. Pairs of 3He-atoms,
however, can behave bosonically. As a result of pairing effects, also 3He can display su-
perfluidity, but only at temperatures below 2mK.
Ultracold dilute gases: for long the existence of superfluidity was seen as an indica-
tion that it should be possible to realize a gaseous Bose–Einstein condensate in the lab.
However, that was all that could be said, since the observed superfluidity involved flu-
ids rather than an ideal or weakly-interacting gas. It took another 70 years to realize a
Bose–Einstein condensate in a nearly ideal gas. The main problem was to refrigerate the
gas without creating a liquid or solid and without the atoms binding into molecules. To
achieve this dilute, neutral gases had to be used without contact with any walls in order
to avoid freezing. This presented two new challenges:
• a Bose–Einstein condensate in a low-density gas requires very low temperatures,
since T0 ∝ ρ2/3N ;
• the gas has to be refrigerated without physical contact with the outside world.
101
By means of two advanced cooling techniques, i.e. laser cooling and evaporative cooling
(see the lecture course Quantum Mechanics 2 for more details), Eric Cornell and Carl
Wieman were finally able to produce the first gaseous Bose–Einstein condensate in 1995,
consisting of about a thousand rubidium-87 atoms at a temperature of 170 nK. In the same
year a similar feat was realized for ultracold dilute gases of lithium-7 atoms (Randy Hulet)
and sodium-23 atoms (Wolfgang Ketterle). In the latter case, the number of condensate
particles had already increased a thousandfold. In 2001 Cornell, Wieman and Ketterle
were awarded the Nobel prize for this achievement.
Cornell & Wieman, JILA, 1995: transition from a “normal” gas to a condensate.
Velocity distribution of the gas particles for T > T0 , T < T0 and T ≪ T0 .
R.G. Hulet et al., Rice University, 2005
Application: isotope effects for lithium.
In this experiment ultracold gases were used
that consisted of fermionic 6Li-atoms and
bosonic 7Li-atoms. At very low tempera-
tures we expect that a magnetically trapped6Li-gas has a larger spatial extent than a7Li-gas for the same number of gas atoms,
since in the fermionic case the average ki-
netic energy is larger. As can be observed
in the figure on the right, this is indeed con-
firmed experimentally.
102
3 Relativistic 1-particle quantum mechanics
In this chapter we will attempt to cast non-relativistic quantum mechanics into
a relativistically viable form. To phrase it differently, we will try to find a
relativistic analogue of the Schrodinger equation.
Similar material can be found in Schwabl (Ch. 5–7 and Ch. 10,11), Merzbacher
(Ch. 24) and Bransden & Joachain (Ch. 15).
The desired equation should have exactly the same form for any inertial observer (relativity
principle) and should have solutions that respect the quantum mechanical postulates. As a
starting point for the construction of the relativistic wave equation we will use the standard
connection between plane waves of the type exp(i~k · ~r − iωt) and free particles with
energy E = ~ω and momentum ~p = ~~k (de Broglie hypothesis). The non-relativistic
QM followed from (slightly) modifying the classical Hamilton–Jacobi equation to obtain a
wave equation, which allowed to model particle–wave duality by describing the particles
as a superposition of plane waves. The correct energy and momentum of the particles
could be extracted from the plane waves by means of the position-space operators
E = ~ω → E = i~∂
∂t, ~p = ~~k → ~p = − i~ ~ , ~r → ~r = ~r .
Applied to the classical expression E = Hcl(~r, ~p ) = ~p 2/(2m) + V (~r ) for the energy of a
particle with mass m in a potential field V , the Schrodinger equation
i~∂
∂tψ(~r, t) = Hcl(~r, ~p )ψ(~r, t) =
(
− ~2
2m~2
+ V (~r ))
ψ(~r, t)
is obtained. However, in this equation ~r and t are not treated on equal footing, invalidating
the relativity principle. The plan is now to recast this idea into a relativistic form in flat
spacetime. To this end we will adopt the following plan of attack to construct a relativistic
QM for free particles.
(1) Particle–wave duality: construct a wave equation that incorporates an operator
version of the correct relativistic relation between energy and momentum. This
will imply that any solution to the wave equation should also satisfy the Klein–
Gordon equation, providing a clear link to plane-wave decompositions (see § 3.1)!
(2) Relativity principle and spin: make sure that the wave equation satisfies the rela-
tivity principle. By specifying this for spatial rotations, we will be able to determine
the spin of the particles belonging to the wave equation!
(3) QM: extract a continuity equation from the wave equation and verify whether this
allows a probability interpretation that conforms with the superposition principle.
(4) Check the non-relativistic limit and compare with experimental observations.
103
3.1 First attempt: the Klein–Gordon equation (1926)
First read appendix D, where the conventions and definitions of special relativity are sum-
marized. The classical energy of a free relativistic particle with rest mass m is given by
E =√
m2c4 + ~p 2c2 . (253)
As a first ansatz for a relativistic wave equation in position space, we could start from
this classical expression and simply substitute E → E = i~ ∂∂t
and ~p → ~p = − i~ ~ in
analogy with the non-relativistic case:
i~∂
∂tψ(x) =
√
m2c4 − ~2c2 ~2ψ(x) (x = position 4-vector) . (254)
The “square-root operator” has to be defined by means of a power series (see the lecture
course Quantum Mechanics 2), which effectively results in a differential operator of infi-
nite order. This hugely complicates the discussion. More importantly, in this equation
~r and t are treated asymmetrically, making it not directly suitable as frame-independent
expression . . . however, these drawbacks would be lifted if we would decide to apply the
differential operators twice on the left- and right-hand side of the equation.
The wave equation: as a first serious attempt it therefore makes sense to start from the
squared expression E2 = m2c4+~p 2c2 . Upon performing the substitution we obtain a wave
equation that does treat ~r and t on equal footing: the so-called Klein–Gordon equation
− ~2 ∂
2
∂t2ψ(x) =
(
m2c4 − ~2c2 ~
2 )
ψ(x) , (255)
which is a second order differential equation in time. In contrast to the non-relativistic
case, this implies that two boundary conditions are needed to fully determine the time
evolution of the wave function. It should be noted, though, that by squaring the energy–
momentum relation we have introduced a sign ambiguity for the energy. We will see that
this aspect will come back to haunt us while trying to interpret the wave function.
Implementation of the relativity principle: to guarantee that the form of the Klein–
Gordon equation is the same for each inertial observer, we first introduce a manifest co-
variant notation in terms of covariant and contravariant vectors:
(
~2c2∂µ∂
µ +m2c4)
ψ(x) = 0 ⇒(
+m2c2
~2
)
ψ(x) = 0 , (256)
in terms of the d’Alembertian
≡ ∂µ∂µ (D.11)
====1
c2∂2
∂t2− ~2
. (257)
104
1) Generic approach: as this will be a recurring theme in the subsequent discussions,
we first try to formulate the generic procedure to implement the relativity principle. To
this end we build in the possibility that the wave function ψ(x) has several components
(intrinsic degrees of freedom):
ψ(x) =
ψ1(x)...
ψN(x)
. (258)
The transformation characteristic of this (multi-component) wave function under Poincare
transformations is defined as
ψ′(x′) ≡ M(Λ)ψ(x) , (259)
where the matrix M(Λ) denotes how the Poincare transformations act on the space
spanned by the intrinsic degrees of freedom. As indicated earlier, this transformation
characteristic should follow from the demand of having the same wave equation for each
inertial observer:
if Dψ(x) = 0 for the linear differential operator D and wave function ψ(x) in
inertial system S, then it should follow automatically that also D′ψ′(x′) = 0
for the transformed differential operator D′ and wave function ψ′(x′) in inertial
system S ′.
The linear differential operator D is invariant under translations by a constant 4-vector aµ
since ∂/∂xµ = ∂/∂ (xµ + aµ). Therefore M(Λ) is independent of a as well as the coordi-
nates x and x′. Furthermore, M(Λ) is a linear operator (matrix) in the space of intrinsic
degrees of freedom, because the superposition principle should hold in both S and S ′.
Additionally we require the matrices M(Λ) to reflect the properties (group structure) of
the Lorentz transformations:
M(Λ1Λ2) = M(Λ1)M(Λ2) and M(Λ−11 ) = M−1(Λ1) (260)
for arbitrary Lorentz transformations Λ1 and Λ2 . This guarantees that in the absence of
a Lorentz transformation also nothing happens in the space of intrinsic degrees of freedom.
This fixes the normalization of the matrices M(Λ) in such a way that a (projective) matrix
representation of the Lorentz group is obtained.
2) Relativity principle and spin for the Klein–Gordon equation: next we specif-
ically consider the Klein–Gordon equation. We know that the d’Alembertian and speed
of light are scalar quantities, which do not change under the transition from one inertial
frame to the other. Consequently also the Klein–Gordon operator is invariant under a
change of inertial system:
D′KG =
(
′ +
m2c ′ 2
~2
)
=(
+m2c2
~2
)
= DKG .
105
This differential operator acts on each component of ψ(x) separately without mixing the
components, implying that each component should separately satisfy the Klein–Gordon
equation. As such, the transformation matrix M(Λ) in the Klein–Gordon theory is propor-
tional to the identity matrix. In order to guarantee the validity of the relativity principle it
hence suffices to demand that ψ(x) is a scalar field11, i.e. ψ′(x′) = ψ(x) = ψ(Λ−1[x′−a]
).
The Poincare transformations therefore act exclusively on x-space, leaving the intrinsic de-
grees of freedom of ψ(x) untouched. The particles described by ψ(x) are spin-0 particles,
as a spatial rotation merely affects the (spacetime) argument of the wave function without
mixing the intrinsic components. The generator of infinitesimal spatial rotations is in this
case exclusively given by the orbital angular momentum operator (see the lecture course
Quantum Mechanics 2).
The non-relativistic limit: before investigating the probabilistic interpretation of the
wave function, we first check whether the relativistic theory reproduces non-relativistic QM
at low velocities. To extract a non-relativistic wave function from the solution ψ(x) to the
Klein–Gordon equation, the rest-mass term has to be removed from the time evolution:
ψ(x) ≡ ψNR(x) exp(−imc2t/~) .
After multiplication by exp(imc2t/~), the Klein–Gordon equation (255) yields
(
− ~2 ∂
2
∂t2+ 2i~mc2
∂
∂t+ m2c4
)
ψNR(x) =(
m2c4 − ~2c2 ~
2)
ψNR(x)
⇒ i~∂
∂tψNR(x) − ~
2
2mc2∂2
∂t2ψNR(x) = − ~
2
2m~2ψNR(x) .
The wave function ψNR(x) oscillates (in time) due to the non-relativistic kinetic energy
ENR = E −mc2 ≪ mc2, so that
∣∣∣
~2
2mc2∂2
∂t2ψNR(x)
∣∣∣ ≪
∣∣∣~
∂
∂tψNR(x)
∣∣∣ .
In the non-relativistic limit the Klein–Gordon equation for ψ(x) changes into a Schrodinger
equation for ψNR(x).
Problems with the Klein–Gordon equation: last but not least we have a closer look
at the quantum mechanical interpretation of the solutions to the Klein–Gordon equation.
For those solutions we will try to find a continuity equation of the form ∂ρ/∂t+~·~j = 0, in
analogy with what can be done in the non-relativistic case for solutions to the Schrodinger
equation. This continuity equation will be interpreted as a manifestation of conservation
11A second possible solution is a so-called pseudoscalar field, for which ψ′(x′) = det(Λ)ψ(x).
106
of probability, with ρ being the quantum mechanical probability density and ~j the quan-
tum flux. For solutions to the Klein–Gordon equation we can derive in a trivial way that
0(256)==== ψ∗(x)ψ(x)− ψ(x)ψ∗(x)
(257)==== ∂µ
[ψ∗(x)∂µψ(x)− ψ(x)∂µψ∗(x)
]
(D.11)===⇒ ∂
∂tρ(x) + ~ ·~j(x) = 0 , (261)
with
ρ(x) ≡ i~
2mc2
[
ψ∗(x)∂
∂tψ(x)− ψ(x)
∂
∂tψ∗(x)
]
,
~j(x) ≡ − i~
2m
[
ψ∗(x)~ψ(x)− ψ(x)~ψ∗(x)]
. (262)
The prefactor of ρ(x) is chosen here in such a way that in the non-relativistic limit we
have ρ(x) ≈ |ψNR(x)|2 . For a proper quantum mechanical interpretation we would like to
interpret ρ(x) as relativistic probability density. To check this, we consider the plane-wave
solutions to the Klein–Gordon equation (256):
ψp(x) = exp(−ip ·x/~) , with p ·p = p2 = m2c2 ⇒ p0 =E
c= ±
√
m2c2 + ~p 2 .
(263)
E
0
mc2
−mc2
Based on these plane-wave solutions we find that
ρ(x) is not positive definite if p0 < 0 (see exer-
cise 22). As such, ρ(x) is not suitable as proba-
bility density. The next logical step is to simply re-
gard all negative-energy solutions as being unphysi-
cal and remove them from the theory. This approach
turns out to be not viable, since it will unavoidably
clash with the fundamental superposition principle
(see exercise 22). Apart from the problems with the
probabilistic interpretation, the negative-energy so-
lutions will result in a free-particle energy spectrum
that is unbounded from below (as sketched in the
figure on the right). This highly unstable scenario
would imply that an infinite amount of energy could
be extracted from the system if an external perturba-
tion would induce a transition from a positive-energy
eigenstate to a negative-energy eigenstate.
107
Two strategies present themselves at this point. On the one hand we could try to inter-
pret the negative-energy states in a different way. We will come back to this option in
chapter 5, when trying to link the Klein–Gordon theory to the quantum theory of the
electromagnetic field that is formulated in chapter 4. On the other hand we could follow
the rather logical strategy of trying to construct an alternative relativistic wave equation
that explicitly does not involve squaring the energy–momentum relation. After all, the
sign ambiguity for the energy was caused by this squaring action. In § 3.2 we will address
this latter (historical) approach, resulting in an entirely different wave equation that does
allow for a viable probability interpretation.
Static, radially symmetric solutions to the Klein–Gordon equation:
to round things off, we briefly discuss the physical applicability of the Klein–Gordon equa-
tion. To this end we consider time-independent (static) solutions φ(r) with radial sym-
metry. In that case the Klein–Gordon equation simplifies to
(~2
− m2c2
~2
)
φ(r) =1
r
( d2
dr2− m2c2
~2
)(rφ(r)
)= 0 .
This differential equation has exponential solutions, with the physically interesting branch
consisting of exponentially decreasing functions of the form
φYu(r) = Cexp(−r/a)
r, with a ≡ ~
mc> 0 . (264)
Solutions of this type are used to describe interactions that are transmitted by massive in-
termediary bosons, such as the weak nuclear forces (transmitted by the W and Z bosons)
and the interactions among nucleons (transmitted by pions), see the lecture course “Struc-
tuur der Materie: Subatomaire Fysica”. In the latter case φYu(r) represents an attractive
interaction potential of the Yukawa-type (i.e. with C < 0), where the constant length scale
a = ~/mc is a measure for the effective range of the potential.
If the mass would have been absent (i.e. for m = 0) we would have been dealing with the
well-known Laplace equation
~2φ(r) =
1
r
d2
dr2(rφ(r)
)= 0 ,
with solutions
φ(r) =A
r+ B .
For B=0 such a solution can be used in the description of infinite-range interactions that
are transmitted by massless intermediary bosons, such as the electromagnetic Coulomb-
interactions (transmitted by photons) and the strong nuclear forces (transmitted by gluons).
108
3.2 Second attempt: the Dirac equation (1928)
Based on the previous observations we now use the non-relativistic expression
i~∂
∂tψ(x) = Hψ(x) (265)
as starting point, with H independent of ∂/∂t. In this way the problematic differential
operator − ~2∂2/∂t2 is absent. The essential difference with the usual non-relativistic
Schrodinger equation resides in the requirement that space and time coordinates are to
be treated on equal footing. As a result, H should now be taken linear in ~ . Intrinsic
degrees of freedom are accounted for by writing the wave function as a column vector with
N components:
ψ(x) =
ψ1(x)...
ψN(x)
. (266)
For a free particle H will not depend on t or ~r . The simplest form of the Hamilton
operator then amounts to
H = c ~α · ~p + βmc2 = c3∑
j=1
αj pj + βmc2 . (267)
The operators ~α and β are independent of t, ~r , E , ~p and act as N×N matrices on
the N components of ψ(x), bearing in mind that H should be a linear operator in order
to guarantee the validity of the superposition principle. In this way we have obtained the
so-called Dirac equation for a free particle with rest mass m:
(
i~∂
∂t+ i~c ~α · ~ − βmc2
)
ψ(x) = 0 . (268)
In the Dirac equation the correct relativistic relation between energy and mo-
mentum should be contained. Or to phrase it differently, the matrices ~α and β
should be chosen in such a way that each component of a solution to the Dirac
equation should automatically be a solution to the Klein–Gordon equation.
This gives rise to the following set of matrix identities:
β2 = IN ,αj , αk
= 2δjk IN and
αj , β
= 0 (j , k = 1 , 2 , 3) . (269)
From the anticommutation relationsαj , β
= 0 we can readily deduce that ~α and β
should indeed be matrices. The symbol IN is used to denote an identity matrix of rank N .
109
Proof: 0(268)====
(
i~∂
∂t− i~c ~α · ~ + βmc2
)(
i~∂
∂t+ i~c ~α · ~ − βmc2
)
ψ(x)
=(
− ~2 ∂
2
∂t2+
~2c2
2
3∑
j,k=1
αj, αk
jk + i~mc33∑
j=1
αj, β
j − β2m2c4)
ψ(x) .
This yields the Klein–Gordon equation (255) if the matrix relations listed above are sat-
isfied. In order to guarantee conservation of probability, we additionally have to impose
that H = H†, which translates into the matrices ~α and β having to be hermitian:
αj = (αj)† and β = β† . (270)
Armed with this set of conditions we now try to find explicit solutions for ~α and β with
the lowest possible rank N .
1. The matrices ~α and β have eigenvalues ± 1.
Proof: the eigenvalues of ~α and β are real, since the matrices are hermitian. More-
over, from the matrix relations (269) it follows that β2 = (αj)2 = IN , causing the
square of the eigenvalues to be 1.
2. The matrices ~α and β are traceless.
Proof: from the matrix relations in equation (269) it can for instance be inferred that
Tr(αj) = Tr(β2αj)cyclic==== Tr(βαjβ)
αj ,β=0======= −Tr(αjβ2) = −Tr(αj) = 0.
3. Since the trace equals the sum of the eigenvalues, the results of steps 1 and 2 can
be combined into the requirement that the matrices ~α and β should have equal
numbers of +1 and −1 eigenvalues. The rank N thus has to be even.
4. Based on steps 1 to 3, the lowest possible value for the rank is N = 2. In that case we
are dealing with 2×2 matrices, spanned by the identity matrix I2 and the three Pauli
spin matrices ~σ . However, we are looking for four different, mutually anticommut-
ing matrices ~α and β . This is impossible for a representation of the Dirac equation
with N = 2 , since the identity matrix commutes with all 2×2 matrices.
From this we can draw the conclusion that the representation of the Dirac equation with
the lowest rank involves N = 4. The corresponding wave functions with four intrinsic
degrees of freedom are called Dirac spinors:
ψ(x) =
ψ1(x)...
ψ4(x)
. (271)
In fact there are infinitely many, physically equivalent representations for the matrices ~α
and β . After all, matrix relations are invariant under arbitrary unitary transformations.
110
In the remainder of this lecture course we will adopt the so-called
Dirac representation : β =
(
I2 Ø
Ø − I2
)
, ~α =
(
Ø ~σ
~σ Ø
)
, (272)
where the 4×4 matrices have been subdivided into 2×2 blocks. As will be proven in the
exercise class, these matrices indeed satisfy the matrix relations (269) and (270). The
corresponding proof will be based on the fundamental properties of the Pauli spin matrices
σ1 = σx , σ2 = σy and σ3 = σz . In this notation the commutator algebra for the Pauli
spin matrices reads[σj , σk
]= 2i
3∑
l=1
ǫjkl σl , (273)
in terms of the completely antisymmetric coefficient
ǫjkl =
+1 if (j, k, l) is an even permutation of (1, 2, 3)
−1 if (j, k, l) is an odd permutation of (1, 2, 3)
0 else
. (274)
3.2.1 The probability interpretation of the Dirac equation
As the Dirac equation is linear in both the spatial gradient and the time derivative, the
probability interpretation of the Dirac equation can be obtained in a straightforward way.
Define the hermitian conjugate row vector
ψ†(x) ≡(ψ∗1(x), · · · , ψ∗
4(x)). (275)
Then
∂
∂t
[ψ†(x)ψ(x)
]= ψ†(x)
∂ψ(x)
∂t+∂ψ†(x)
∂tψ(x)
(268)==== − c ψ†(x)
(~α · ~ +
imc
~β)ψ(x)
− c(~ψ†(x)
)· ~α†ψ(x) +
imc2
~ψ†(x)β†ψ(x)
(270)==== − ~ ·
[ψ†(x)c~α ψ(x)
],
from which the following continuity equation can be extracted:
∂
∂tρ(x) + ~ ·~j(x) = 0 , (276)
with
ρ(x) ≡ ψ†(x)ψ(x) =4∑
i=1
|ψi(x)|2 and ~j(x) ≡ ψ†(x)c~α ψ(x) . (277)
This looks really promising, since this time ρ(x) is guaranteed to be positive definite and
therefore allows an interpretation as probability density. In that interpretation the vector ~j
will play the role of quantum flux, which would imply that the matrix c~α can be interpreted
as a kind of velocity operator in Dirac spinor space. For a viable probability interpretation
it was essential that the matrices ~α and β (and as such also H) are hermitian.
111
3.2.2 Covariant formulation of the Dirac equation
We need one more ingredient before being able to use the Dirac equation as a relativistic
wave equation. We have to demand that ψ(x) transforms in such a way under Poincare
transformations that the Dirac equation has the same form for any inertial observer. As a
first preparatory step we rewrite the Dirac equation in a form that facilitates the transition
to other inertial frames. To this end we combine the four matrices β and ~α in the 4-vector-
like object
γµ : γ0 ≡ β and
γ1
γ2
γ3
≡ ~γ ≡ β ~α (278)
⇒ Dirac representation : γ0 =
(
I2 Ø
Ø − I2
)
, ~γ =
(
Ø ~σ
−~σ Ø
)
.
These are the so-called γ-matrices of Dirac. This notation is slightly misleading in the
sense that it gives the impression that γµ is a contravariant vector quantity. For that
to be true an appropriate contraction in spinor space is required, which also includes the
Dirac spinors in the vector quantity. The γ-matrices have the following properties (which
Similarly we find for the time reversal transformation that
ψ′(x′) = S(ΛT)ψ∗(x) , S(ΛT) = exp(iϕT)σ
13 (ϕT ∈ IR) . (296)
(III) Spacetime translations: finally we consider constant spacetime translations of the
origin, i.e. x′µ = xµ+aµ with aµ a constant 4-vector. Since the Dirac operator (i~∂/−mc)is invariant under such translations, there is no need for a compensatory transformation
in spinor space.
115
3.2.4 Solutions to the Dirac equation
The Dirac equation for a free particle involves a time-independent Hamilton operator
that does not depend on ~x, i.e.[H, ~p
]= 0. As such, solutions can be found that are
simultaneous eigenfunctions of H and ~p :
ψp(x) = u(~p ) exp(−ip · x/~) , (297)
where the eigenvalues E of E = i~ ∂/∂t and ~p of ~p = − i~~ are combined into the
contravariant momentum vector pµ with p0 = E/c. These stationary plane-wave solutions
to the Dirac equation have to satisfy the additional equation
(i~∂/ −mc
)ψp(x) = 0
(297)===⇒ ( p/−mc)u(~p ) = 0 . (298)
Solutions in the rest system (zero-momentum solutions): we start by trying to
find the solutions in the so-called rest system, for which ~p = ~0 . In that case u(~0 ) should
satisfy the eigenvalue equation (Eγ0−mc2)u(~0 ) = 0, where the matrix γ0 =
(
I2 ØØ − I2
)
has eigenvalues ± 1. The desired plane-wave solutions thus read
E = E+ = mc2 : eigenvectors u(1)(~0 ) =
1000
, u(2)(~0 ) =
0100
,
(299)
E = E− = −mc2 : eigenvectors u(3)(~0 ) =
0010
, u(4)(~0 ) =
0001
.
Negative energies and the Dirac sea: we observe again that negative-energy states
occur. This time this is caused by the fact that H is linear in ~p rather than quadratic.
The Dirac spinors seem to describe two types of spin-1/2 particles simultaneously, one
type having positive energies and one having negative energies. Again the free-particle
energy spectrum is unbounded from below. However, since we are dealing with fermions
here, induced transitions between positive- and negative-energy states can be avoided
(Pauli-blocked). To this end Dirac assumed that all 1-fermion negative-energy states are
occupied in the vacuum state of the universe. In this way the vacuum consists of an infinite
sea of particles with negative energies (Dirac sea) and no particles with positive energies.
The exclusion principle then forbids the unwanted transitions to these occupied negative-
energy states. In § 3.2.6 we will come back to the implications of this specific assumption
and we will also have a critical look at potential problems linked to the existence of such
a Dirac sea.
116
Finite-momentum solutions: for completeness we have a brief look at how the plane-
wave solutions change for finite momenta. These solutions can be obtained by transforming
u(~0 ) to the correct inertial frame by means of an appropriate Lorentz boost S(Λ). How-
ever, it is much simpler to make use of the fact that any solution to the Dirac equation
is automatically a solution to the Klein–Gordon equation, which causes the desired plane-
wave solutions to satisfy the condition p2 = m2c2. The corresponding eigenvectors can thus
be obtained straight from the solutions (299) in the rest system:
u(r)(~p ) = N (p/ +mc)u(r)(~0 ) , E =
E+ =√
m2c4 + ~p 2c2 for r = 1, 2
E− = −√
m2c4 + ~p 2c2 for r = 3, 4,
since (p/−mc)u(r)(~p ) = N (p2 −m2c2) u(r)(~0 ) = 0 for p2 = m2c2.
While determining the normalization factor N we should properly take into account that
the normalization of ψ changes under Lorentz boosts. This is a direct consequence of
conservation of probability: the probability to observe the system as being localized in
a spatial volume d~x around ~x in the rest frame S should be identical to the proba-
bility to observe the system as being localized in the transformed spatial volume d~x ′
around ~x ′ in the boosted inertial frame S ′. Phrased differently, we have to demand that
The field operator ~A(~r, t) contains the wave aspects that belong to the photon
states. This reflects particle–wave duality in the sense that each creation and
annihilation operator corresponds to a specific term in the plane-wave expansion
of the vector potential ~A(~r, t).
Let’s see how this differs from the approach that we adopted in the previous chapter.
• The positive-frequency terms: when attempting to set up a 1-particle QM we would
not have been dealing with the operator a~k,λ exp(i~k · ~r − iωkt), but rather with the
plane wave ∝ exp(i~k · ~r − iωkt) as the corresponding stationary solution to the rel-
ativistic wave equation. By acting with the operators E = i~ ∂/∂t and ~p = − i~ ~on this plane wave we would have reached the conclusion that we were dealing with
a state with energy ~ωk and momentum ~~k . However, to the quantized electromag-
netic theory a many-particle interpretation should be assigned, with the annihilation
operator a~k,λ exp(i~k · ~r − iωkt) acting on multi-photon states. Applied to a multi-
photon state with n~k,λ photons in the 1-photon state with quantum numbers ~k and
λ , this operator gives rise to a similar type of plane wave:
a~k,λ exp(i~k · ~r − iωkt) → √
n~k,λ exp(i~k · ~r − iωkt) ,
where the size of the proportionality factor depends on the occupation number of
the given 1-photon state. In this context particle –wave duality refers to the fact
that the annihilation of a photon with positive energy ~ωk and momentum ~~k will
always be accompanied by a plane-wave factor ∝ exp(i~k · ~r − iωkt).
134
• The negative-frequency terms: the main difference between the 1-particle and many-
particle theory follows from the operator a†~k,λ exp(iωkt − i~k · ~r ). In the 1-particle
theory this would have corresponded to a plane wave ∝ exp(iωkt − i~k · ~r ), whichwe would have interpreted as a state with negative energy −~ωk and momentum
−~~k in the previous chapter. In the many-particle theory, however, the signs of
these quantum numbers are effectively reversed, since we are dealing with pho-
ton creation rather than annihilation! The operator a†~k,λ exp(iωkt − i~k · ~r ) cor-
responds to the creation of a photon with positive energy ~ωk and momentum ~~k ,
which will always be accompanied by a plane-wave factor ∝ exp(iωkt− i~k · ~r ).
Second quantization: the prescription for quantizing the electromagnetic field that we
have just discussed is an explicit example of second quantization, as described in § 1.5 for
the Schrodinger field. Starting from a classical wave equation we switched directly to a cor-
responding many-particle theory formulated in the Heisenberg picture. The classical vector
potential ~A(~r, t) was a classical, three-component function defined in each spacetime point.
The quantized vector potential ~A(~r, t) is a field operator that acts on the state functions
in the Fock space for photons. In this field operator ~r and t are not quantum mechanical
variables. They are simply parameters that parametrize the plane waves. In this approach
~r and t are treated on equal footing, in contrast to the non-relativistic QM where t is
a parameter and ~r an eigenvalue belonging to the observable ~r . This aspect is precisely
what is needed for constructing the relativistic QM (see Ch. 5).
4.4 Interactions with non-relativistic quantum systems
In classical electrodynamics the influence of a classical electromagnetic field on a classical
particle with mass m and charge q follows from minimal substitution. In the Coulomb
gauge this implies that the momentum ~p of the classical particle should be replaced by
~p−q ~A(~r, t), where ~r is the coordinate of the particle. After this substitution ~p represents
the canonical momentum of the particle in the presence of the electromagnetic field.
The quantum mechanical Hamilton operator of the complete system is given by
H = H0 + He.m. + V (t) , with V (t) = Vint(t) + Vspin(t) , (358)
where He.m. is the Hamilton-operator (346) of the free electromagnetic field inside an
enclosure with volume V and periodic boundary conditions. The term H0 represents the
Hamilton operator of the unperturbed non-relativistic particle, which can be a free particle
or a particle that is bound by a potential. The interactions between the particle and the
135
electromagnetic field are given by
Vint(t) = − q
m~A(~r, t) · ~p +
q2
2m~A 2(~r, t) , (359)
Vspin(t) = − ~MS · ~B(~r, t) , ~MS = − |e|~2m
g~S
~. (360)
The first interaction is obtained by applying minimal substitution to the kinetic term
~p 2/(2m) and by subsequently inserting the Coulomb condition ~ · ~A(~r, t) = 0. The sec-
ond interaction is the extra interaction between the magnetic field and the spin-induced
intrinsic magnetic dipole moment of the particle that we have encountered in chapter 3.
The interaction V (t) acts both on the charged-particle states, through ~r and ~p , and on
the multi-photon states, through a~k,λ and a†~k,λ . The field operator ~A(~r, t) provides the
contact between the charged particle and the radiation field, with ~r being the position at
which the particle experiences the interaction and with the interaction corresponding to
the creation (emission) and annihilation (absorption) of photons. To phrase it differently:
the field operator ~A(~r, t) is the universal force carrier (mediator) of the electromagnetic
interactions. Stronger radiation fields will result in stronger emission and absorption ef-
fects in view of the annihilation and creation factors√nj and
√nj+1 in equation (351)
being larger in that case. The difference between these two factors will play an important
role in the quantum mechanical phenomenon of spontaneous photon emission.
Vint(t) in first-order perturbation theory: absorption/emission of one photon.
In order to assess the implications of the quantization of the electromagnetic field, we
consider electromagnetic processes that involve the absorption/emission of precisely one
photon with quantum numbers ~k and λ . To further simplify the problem we assume for
convenience that the interaction Vint(t) can be treated as a weak perturbation and that
the magnetic interaction Vspin(t) can be neglected with repsect to Vint(t) (as is true for the
dipole approximation in § 4.4.2). As a result, the spin of the particle can be effectively left
out in the analysis. Owing to the periodic time dependence of the electromagnetic field,
we can apply periodic perturbation theory (see the lecture course Quantum Mechanics 2).
At first order in perturbation theory, only the linear term in ~A contributes to 1-photon
processes and effectively
V (t) →∑
~k
2∑
λ=1
[
C†~k,λ
exp(−iωkt) + C~k,λ exp(iωkt)]
,
with C~k,λ
(344),(359)======= − q
m
√
~
2Vǫ0ωk
a†~k,λ exp(−i~k · ~r )~ǫλ(~ek) · ~p . (361)
136
After all, at first order in perturbation theory the number of photons changes by ± 1 due
to the linear term in ~A and by 0 , ± 2 due to the quadratic term.12 Take the system prior
to the interaction to be in the following unperturbed initial state:
|φ(0)i 〉 = |ψA〉| · · · , n~k,λ, · · ·〉 , (362)
|ψA〉 = unperturbed initial state of the charged particle with energy EA ,
| · · · , n~k,λ, · · ·〉 = photon state with n~k,λ photons of the type (~k, λ) .
Subsequently we wait a sufficiently long time and ask ourselves the question what the
The action of a radiation field |n~k,λ〉 on a charge carrier is in that case equivalent with
the classical action of a monochromatic plane wave.
138
We thus observe a clear distinction between the classical theories of radiation and particles
(such as electrons).
• The classical limit of the Schrodinger wave mechanics for an electron describes the
behaviour of that particle as governed by Newtonian mechanics, whereas a classical
gas of such particles is obtained at low densities.
• The classical limit of the quantum theory of radiation describes the behaviour of
very many photons at large densities by means of a classical electromagnetic wave
that satisfies the Maxwell equations.
This also explains why first the wave aspects of light and the particle aspects of electrons
were observed, causing both phenomena to appear fundamentally different.
4.4.2 Spontaneous photon emission (no exam material)
Using the density of states for photons from § 4.2.2, the transition rate for spontaneous
emission of a photon with polarization λ and ~ek within a solid angle dΩ around ~eΩ is
given by
W spont.em.A→B+γλ, dΩ
=2π
~dΩ
∫
d(~ωk) ρλ(~ωk,Ω) |(C~k,λ )fi|2 δ(EB − EA + ~ωk)
(365),(352)=======
q2ωAB
8π2~c3m2ǫ0
∣∣∣〈ψB|exp(−iωAB
~eΩ · ~r/c) ~p |ψA〉 · ~ǫλ(~eΩ)∣∣∣
2
dΩ ,
with the usual definition ωAB
≡ (EA −EB)/~
. (367)
Application 1: can a free particle with mass m 6= 0 and well-defined momentum ~pA ≡ ~~kA
spontaneously emit a photon? Answer: no!
Proof: the unperturbed spatial wave function of the particle prior to the interaction is
ψA(~r ) = (2π)−3/2 exp(i~r · ~kA) .
As unperturbed final state of the particle we consider an arbitrary eigenfunction of the
momentum operator belonging to the momentum eigenvalue ~pB ≡ ~~kB :
ψB(~r ) = (2π)−3/2 exp(i~r · ~kB) .
The relevant matrix element for the transition is then
〈ψB|exp(−i~k · ~r ) ~p |ψA〉 = ~~kA (2π)−3
∫
d~r exp(i~r · [~kA−~kB−~k ]
)= ~~kA δ(~kA−~kB−~k ) .
139
Based on equation (367) this implies that, for the transition to be possible, two conditions
should be satisfied simultaneously . . . conservation of energy as well as momentum:
EB =√
~p 2B c
2 +m2c4 = EA−~ωk =√
~p 2A c
2 +m2c4 − ~ck and ~pB = ~pA−~~k .
Both conditions can be merged to arrive at
~p 2B +m2c2
momentum======= (~pA − ~~k)2 +m2c2
energy==== ~p 2
A +m2c2 + ~2k2 − 2~k
√
~p 2A +m2c2
k 6=0===⇒
√
~p 2A +m2c2 = ~pA ·
~k
k≤ |~pA| ,
which is impossible if m 6= 0. This energy–momentum conservation argument holds just
as well for photon absorption and for 1-photon processes caused by Vspin(t).
Application 2: spontaneous transitions of 1-electron atoms in the dippole approximation.
For spontaneous-emission transitions of relatively light 1-electron atoms (such as hydro-
gen) the wavelength of the emitted photon is much larger than the typical size of the
emitting atom. For instance, the transition from the n = 2 to the n = 1 state in-
volves an energy difference ∆E = 3mec2 (Zα)2/8. This corresponds to a wavelength
λγ = 2π/k = 2π~c/∆E ≈ (1.2/Z2) × 10−7m, which is much larger than the typical
< ratoom >= O(10−10m) size of the charge distribution of the 1-electron atom, provided
that Z is not too large. In the electric dipole approximation we exploit this fact by replac-
ing the Fourier mode exp(−i~k · ~r ) by 1, since kr ≪ 1 across the full extent of the atom,
and by leaving out the magnetic interaction. The justification for the latter approximation
lies in the fact that the factor ~ǫλ(~ek) · ~p in Vint(t) gives rise to O(~/<ratoom>) effects,
whereas the factor ~~k ×~ǫλ(~ek) in Vspin(t) leads to much weaker O(~k) effects.
By means of the commutation relation
H0 =~p 2
2me+ V (~r ) ⇒
[H0, ~r
]= − i~
∂H0
∂ ~p= − i~
~p
me
the relevant matrix element for spontaneous photon emission in the dipole approximation
is given by
〈ψB|~p |ψA〉 =ime
~〈ψB|
[H0, ~r
]|ψA〉 = − iω
ABme 〈ψB|~r |ψA〉 ≡ − iω
ABme~rBA
.
From equation (367) it follows that in dipole approximation
W spont.em.A→B+γλ, dΩ
≈ e2ω3AB
8π2~c3ǫ0|~r
BA· ~ǫλ(~eΩ)|2 dΩ . (368)
140
Subsequently we sum over all photon polarizations, using the completeness relation
2∑
λ=1
ǫλ,j(~eΩ)ǫ∗λ, l(~eΩ)
~ǫλ(~eΩ) ∈ IR3
=======2∑
λ=1
ǫλ,j(~eΩ)ǫλ, l(~eΩ)(334)==== δjl − eΩ, j eΩ, l ,
and integrate over all directions ~eΩ while defining the z-axis along ~rBA
:
W spont.em.A→B+γ ≈ e2ω3
AB
8π2~c3ǫ0
∫
dΩ(|~r
BA|2 − |~r
BA· ~eΩ|2
)=
e2ω3AB
|~rBA
|28π2~c3ǫ0
2π∫
0
dφ
1∫
−1
dcos θ sin2 θ
=e2ω3
AB
3π~c3ǫ0|~r
BA|2 =
4
3αem
ω3AB
c2|~r
BA|2 ≈ W spont.em.
A→B+γ . (369)
In this expression αem is the electromagnetic fine-structure constant, which is defined in
the usual way as
αem ≡ e2
4πǫ0~c≈ 1/137 . (370)
The average lifetime τA and decay width ΓA of the state A can be derived from
τA =1
ΓA=
1∑
j
W spont.em.A→Bj +γ
, (371)
which is summed over all allowed final states Bj for electric dipole transitions. These
allowed final states have to satisfy the so-called dipole selection rules |∆ℓ| = 1 and
|∆mℓ| = 0 or 1, which is related to the spin of the emitted photon. Sometimes ~rBA
= ~0
for all states with EB < EA . For instance this is the case for the 2s state of the hy-
drogen atom, for which no dipole transition to the 1s ground state is possible as both
states have ℓ = 0. To determine the (longer) lifetime of such states one would have
to take along the next term in the multipole expansion. This implies that the magnetic
interaction cannot be discarded (consequence: magnetic dipole transitions) and that in
Vint(t) the Fourier mode exp(−i~k · ~r ) should be replaced by 1 − i~k · ~r (consequence:
electric quadrupole transitions). For the 2s state of the hydrogen atom even these tran-
sitions are not possible: the 1-photon transition 2s → 1s + γ is forbidden by angular
momentum conservation. The first allowed transition is the much weaker 2-photon transi-
tion 2s → 1s + γ + γ , which explains why the lifetime of the 2s state (122ms) is so much
larger than the one for the 2p state (1.6 ns).
4.4.3 The photon gas: quantum statistics for photons
As final example of the interaction between quantum systems and quantized electromag-
netic fields we consider the quantum statistics for photons. From equations (364)–(366) a
141
condition can be derived for thermal equilibrium between an atomic heat bath at temper-
ature T and electromagnetic radiation with quantum numbers ~k and λ (see exercise 31).
From this equilibrium condition it follows that the average occupation number of the cor-
responding 1-photon state at temperature T = 1/(kBβ) is given by
n~k,λ =1
exp(β~ωk)− 1, (372)
which we recognize as a Bose–Einstein distribution with α = 0. This is exactly the same
distribution as derived in exercise 14 for a canonical ensemble of systems that each consist
of a single linear harmonic oscillator. The fact that photons in general do not allow for
a grand canonical treatment has to do with the absence of a photon conservation law in
most quantum mechanical settings.13 As a result of absorption by the atoms in the heat
bath, the photons do not migrate to the heat bath as particles and exclusively contribute
to the thermodynamic energy balance. The entropy of the heat bath is in fact independent
of the number of photons. The absence of a photon conservation law also implies that no
Bose–Einstein condensation of low-energy photons can occur in such situations. It will
be entropically favourable for the atomic heat bath to simply absorb such photons: the
condensate only involves a single quantum state whereas the number of heat-bath states
increases upon energy absorption.
Let’s now enclose the electromagnetic field inside a black body with walls that can ab-
sorb/emit photons of all energies. In this idealized situation we are dealing with an ideal
photon gas. The energy distribution of the radiation field per unit of energy and volume
then reads
U(E)(372)====
1
V
E
exp(βE)− 1
∫
dΩ
2∑
λ=1
ρλ(E,Ω)(352)====
8π
h3c3E3
exp(βE)− 1. (373)
From this expression we can derive Planck’s law for the energy distribution per unit of
frequency ν = E/h and volume:
U(ν) = U(E)dE(ν)
dν
(373)====
8πhν3
c31
exp(hν/kBT )− 1, (374)
as well as the law of Stefan–Boltzmann for the total energy density:
Utot =
∞∫
0
dE U(E)(373), x=βE=======
k4BT4
π2~3c3
∞∫
0
dxx3
exp(x)− 1=
π2k4B15~3c3
T 4 . (375)
13We used to think that a photon gas with a conservation law for the total number of photons was not
realizable in the lab. However, in 2010 the situation changed. The photon gas inside the microresonator
described in exercise 21 effectively satisfies a photon conservation law and as such allows the formation of
a Bose–Einstein condensate.
142
Experimental realization: use a large cavity (furnace) with a tiny hole in it. If the walls
of the cavity are sufficiently rough and opaque, any light entering the cavity will be reflected
very many times before it will be able to escape the cavity through the tiny hole. The light
then has a high probability of being absorbed by the walls before it manages to escape. In
good approximation the tiny hole acts as a black body. By varying the temperature of the
cavity, Planck’s law of black-body radiation can be verified experimentally.
Another almost perfect example of Planck’s spectrum for black-body radiation is provided
by the present-day cosmic microwave background (CMB), which corresponds to a temper-
ature of T = 2.725K. The present-day CMB radiation has evolved from a snapshot of our
universe roughly 380,000 years after the Big Bang, when the universe became transparent
to radiation. Due to the expansion of the universe, the earlier (distant) universe was of
course warmer than the universe today. However, the corresponding radiation is redshifted
due to that same expansion, causing it to appear as having the same temperature as the
universe today!
143
5 Many-particle interpretation of the relativistic QM
Motivated by the quantization of the electromagnetic field, this chapter will be
devoted to assigning a many-particle interpretation to the solutions of the rela-
tivistic wave equations that were constructed in chapter 3.
More extended versions of the material covered in this chapter can be found in
Schwabl (Ch. 13) and Merzbacher (Ch. 24).
Negative-energy states and the 1-particle relativistic QM: in chapter 3 the at-
tempts to construct a relativistic 1-particle quantum theory always resulted in a set of
hard to interpret negative-energy states. In the bosonic case, such as in the Klein–Gordon
theory, this resulted in unsurmountable problems that basically disqualified the theory as
a viable 1-particle quantum theory. In the fermionic case, such as in the free Dirac theory,
Pauli’s exclusion principle provided a way out by assuming that all 1-fermion negative-
energy states are occupied in the vacuum state of the universe. These states in the Dirac
sea behaved dynamically as if we were dealing with objects with opposite charge. This
can be highlighted by looking at the relativistic 1-electron atom, which has bound states
with positive energy. If we would invert the nuclear charge (Z → −Z), only bound states
with negative energy occur. This is consistent with the picture that the positively charged
nucleus effectively repels the negative-energy electrons and that upon inverting the nuclear
charge the negative-energy electrons are effectively attracted. Another way to read off this
opposite charge is by selecting negative-energy states in the Dirac equation (308) by means
of the substitution E = ENR −mc2. In that case we obtain the same eigenvalue equation
as in § 3.2.5 (I) with ψU↔ ψ
D, ENR → −ENR , ~p → − ~p and q → − q . It was shown in
exercise 27 how a negative-energy state can indeed be turned into a positive-energy state
with opposite charge by performing charge conjugation.
Hidden inside the free Dirac theory apparently resides the description of a second type of
particle with opposite charge. By employing hole theory (see the discussion in § 1.7.1),these particles with opposite charge could be made manifest as holes in the Dirac sea.
However, hole theory will not be able to describe all observed phenomena involving the
oppositely charged particles.
A conflict with hole theory: the occurrence of the β+ decay process p→ n+e++νe in
the decay of proton-rich nuclei actually contradicts this hole-theory interpretation. Within
hole theory we would have liked to interpret the produced e+ as a hole in the Dirac sea.
However, the creation of a hole in the Dirac sea should automatically be accompanied by
the simultaneous creation of a positive-energy electron (see the discussion on electron–hole
excitations in § 1.7.1), which evidently is not the case in the observed β+ decay process!
144
Many-particle interpretation of the relativistic QM: all this seems to suggest that
it may be better to not set up the relativistic QM as a 1-particle theory, but rather as a
many-particle theory in analogy to the treatment of the electromagnetic field. This idea
does not contradict with the non-relativistic QM. After all, as noted earlier, photons are
massless and should therefore always be treated relativistically. These photons can (in
most situations) be created and annihilated without energy threshold. Massive particles,
however, do have an energy threshold as a direct consequence of their non-zero rest mass:
• for kinetic energies ≪ mc2 such systems behave non-relativistically, i.e. they corre-
spond to a fixed number of particles;
• for kinetic energies = O(mc2) the many-particle aspects manifest themselves .
The many-particle theory will be set up along the lines worked out for the electromagnetic
field: the field equation will be solved by performing a Fourier decomposition (plane-
wave expansion) and the Hamiltonian belonging to the field equation will be expressed
as an infinite set of independent linear harmonic oscillators. Finally, this Hamiltonian
will be quantized in the Heisenberg picture and the fundamental energy quanta will be
interpreted as particles. Particle–wave duality then follows automatically from the fact
that the creation/annihilation of a particle is always accompanied by a corresponding
plane-wave factor, as can be read off from equation (344) in the electromagnetic case.
From this point onward the 1-particle quantum mechanical approach makes way for a
many-particle formulation based on second quantization. This many-particle formulation
is referred to as relativistic quantum field theory and is usually described in terms of path
integrals or Lagrangian densities for continuous systems. In this lecture series we will
content ourselves with deriving the salient aspects of quantization while avoiding field-
theoretic bells and whistles as much as possible.
5.1 Quantization of the Klein–Gordon field
Again we start off by considering the Klein–Gordon equation for free spin-0 particles with
rest mass m and charge q . For the derivation of the many-particle formulation we ignore
the problems with the 1-particle quantum mechanical interpretation and treat the Klein–
Gordon equation as a relativistic classical wave equation.
The classical energy density: in order to highlight the analogy with the electromagnetic
case we will rescale the Klein–Gordon field according to ψ(x) =√2mc2 φ(x)/~ . This
rescaled Klein–Gordon field φ(x) satisfies the field equation
(
+m2c2
~2
)
φ(x) =( 1
c2∂2
∂t2− ~2
+m2c2
~2
)
φ(x) = 0 (376)
145
and has a corresponding classical energy density14 (without proof)
ρH(x) =
( ∂
∂tφ∗(x)
)( ∂
∂tφ(x)
)
+ c2(~φ∗(x)
)
·(~φ(x)
)
+m2c4
~2φ∗(x)φ(x) . (377)
In this form the similarity with the classical electromagnetic energy density is apparent.
The first two terms are the analogues of the electric and magnetic contributions to the
energy density. The last term is a rest-mass term, which is absent for the massless pho-
tons. The other differences between both energy densities reflect that photons have spin 1,
whereas the particles belonging to the Klein–Gordon theory have spin 0 (see § 3.1).Remark: in the relativistic quantum-field-theory approach all this is handled more sys-
tematically by working with Lagrangian densities for continuous systems, using φ(x) and
φ∗(x) as generalized coordinates. Everything else follows from this Lagrangian density.
The Klein–Gordon equation simply follows from the Euler–Lagrange equation (equation
of motion) that belongs to this Lagrangian density. Moreover, if we go from the Lagrangian
density to the corresponding Hamiltonian density of the system we obtain an expression
that coincides with the energy density given above.
Plane-wave expansion and quantization: in analogy to the electromagnetic case we
also solve the Klein–Gordon equation inside a finite cubic enclosure with periodic boundary
conditions:
φ(x) =∑
~p
~√2E~p
u~p (~x )[
a~p (t) + c~p (t)]
, u~p (~x ) =1√V
exp(i~p · ~x/~) . (378)
The positive- and negative-frequency Fourier coefficients a~p (t) and c~p (t) satisfy
to count the number of particles and antiparticles. The total Hamilton operator of the free
non-interacting many-particle system can be written compactly as
∫
V
d~x ψ†(x) i~∂
∂tψ(x)
(380),(395),(398)=========
∑
~p
∑
λ
E~p
(
a†~p ,λ(t) a~p ,λ(t) − b−~p ,λ(t) b†−~p ,λ(t)
)
=∑
~p
∑
λ
E~p
(
n(+)~p ,λ + n
(−)−~p ,λ − 1
)
=∑
~p
∑
λ
E~p
(
n(+)~p ,λ + n
(−)~p ,λ − 1
)
= H . (402)
The energy spectrum is indeed nicely bounded from below thanks to the anticommutation
relations for b†~p ,λ(t) and b~p ,λ(t). In the many-particle formulation, the probability density
of § 3.2.1 receives the meaning of total charge operator of the many-particle system, since
Qtot ≡ q
∫
V
d~x ψ†(x)ψ(x)(380),(395),(398)========= q
∑
~p
∑
λ
(
a†~p ,λ(t) a~p ,λ(t) + b−~p ,λ(t) b†−~p ,λ(t)
)
(400),(401)======= q
∑
~p
∑
λ
(
n(+)~p ,λ − n
(−)~p ,λ + 1
)
= Qtot (403)
indeed counts particles with charge q and antiparticles with charge −q . Finally, the
total momentum operator of the free non-interacting many-particle system is given by the
additive expression
~P =∑
~p
∑
λ
~p(
n(+)~p ,λ + n
(−)~p ,λ
)
. (404)
151
Vacuum energy: in the expression for the Hamilton operator given above the vacuum
has an infinite negative energy, but it does define a lowest-energy state (ground state).
As a result of the fermionic quantization conditions the vacuum has a zero-point energy
per vibration mode that is opposite to the corresponding zero-point energy for bosonic
quantization conditions. It is very well possible that nature has equal amounts of bosonic
and fermionic degrees of freedom, allowing for an automatic cancellation of the infinities
in the zero-point energies. This idea is for instance used in supersymmetry , which links
bosonic to fermionic degrees of freedom by means of a symmetry. Quantum mechanically
speaking this cancellation of infinities is not essential, but cosmologically it is since an
energy density curves spacetime.
Locality: the requirement of locality forces us to impose the full set of anticommutation
relations (and not just the one for the b-operators). In exercise 32 it will be shown that in
that case
ψi(~x, t), ψi′(~x
′, t)
=ψ†i (~x, t), ψ
†i′(~x
′, t)
= 0 , (405)
ψi(~x, t), ψ
†i′(~x
′, t)
= δii′ δ(~x− ~x ′) 1 (i, i′ = 1 , · · · , 4) ,
causing the relevant anticommutators of the Dirac fields to vanish in all required situa-
tions. This would not have been the case if the creation and annihilation operators would
have satisfied bosonic commutation relations. In this context, an important thing to note
is that physical observables should consist of even numbers of Dirac fields, since physical
observations should not change under 2π rotations of space while Dirac spinors flip sign
(see exercixe 25). Products of even numbers of Dirac fields give rise to commutators of
observables that can be written in terms of anticommutators of Dirac fields (see the trick
on p. 10), allowing locality to be realized by means of either anticommutation relations or
commutation relations.
Spin – statistics theorem: spin and statistics are related in the relativistic QM. The quan-
tization of relativistic wave equations belonging to arbitrary spins will result in the generic
statement that bosons should have integer spin and fermions half integer spin.
In order to explicitly prove this theorem one should first of all demand the
energy spectrum to be bounded from below, This will determine whether the
creation and annihilation operators b†~p ,λ and b~p ,λ satisfy commutation relations
or anticommutation relations. From the requirement of locality then follows that
a†~p ,λ and a~p ,λ should satisfy the same operator algebra and that both sets of
operators should also commute or anticommute mutually.
152
5.2.1 Extra: the 1-(quasi)particle approximation for electrons
Starting from the many-particle theory for electrons and positrons (anti-electrons) we will
now try to understand the success of the 1-particle Dirac theory. Define to this end the
electron vacuum |0e〉 , with ψ(x)|0e〉 ≡ 0 and 〈0e|0e〉 ≡ 1 . (406)
This implies that ∀~p ,λ
a~p ,λ |0e〉 = b†~p ,λ |0e〉 = 0, i.e. this vacuum contains no electrons.
However, all possible 1-positron states are occupied in this vacuum, causing it to contain an
infinite number of positrons. We are dealing here with a quasi-particle vacuum belonging
to the quasi-particle annihilation operator ψ(x) in the Heisenberg picture. In view of
the exclusion principle, this way of constructing the electron vacuum exclusively works for
fermions and not for bosons.
Both adding an electron to the electron vacuum and removing a positron from the electron
vacuum can be regarded as the creation of a kind of 1-electron state, where in the latter
case we are dealing with a negative effective energy but electron-like effective charge − |e|with respect to the vacuum. In this quasi-particle picture an arbitrary 1-electron state can
be defined as a
1-charge state |1e〉 , with |1e〉 ≡∫
V
d~x ψ†(x)ψe(x)|0e〉 . (407)
This 1-charge state is defined in the Heisenberg picture, just as the field operator ψ(x).
This automatically implies that the time dependence of ψ†(x) should be compensated
by the time dependence of the spinor coefficient ψe(x). The coefficient ψe(x) will be
identified with the spinor wave function in the 1-electron Dirac theory. In exercise 32 it
will be proven that the states ψ†(x)|0e〉 have the following orthonormality property: