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Lecture 1: Harmonic crystals
Christopher Mudry
Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland.
(Dated: September 6, 2001)
Abstract
Classical equations of motion for a finite chain of atoms are solved within the harmonic approximation.
The thermodynamic limit is constructed in terms of a one-dimensional (classical) field theory. Quantization
of the finite harmonic chain is undertaken. The thermodynamic limit is constructed in terms of a one-
dimensional quantum field theory describing phonons in a one-dimensional lattice.
Electronic address: [email protected]; URL: http://people.web.psi.ch/mudry
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I. CLASSICAL ONE-DIMENSIONAL CRYSTAL
A good reference for this section is chapter 12 of [1].
A. Discrete limit
For simplicity, I will consider a one dimensional world made ofN point-like objects (atoms) of
mass m and interacting through a potential V. I assume first that the potential V depends only on
the coordinates n , n 1 N, of the N atoms:
V V 1 N (1.1)
Furthermore, I assume that V has a non-degenerate minimum at
n na n 1 N (1.2)
where a is the lattice constant. For example, one could imagine that
V 1 Na
2
2
N 1
n 1
1 cos2
an 1 n
a
2
2
m2N
n 1
1 cos2
an
boundary terms (1.3)
For small deviations n about minimum (1.2), it is natural to expand the potential energy accord-ing to
V 1 1 N N V 1 N
N 1
n 1
2n 1 n
2 1
2m2
N
n 1
n2
boundary terms (1.4)
The dimensionfull constant is the elastic or spring constant. It measures the strength of the
linear restoring force between nearest neighbor atoms. The characteristic frequency measures
the strength of an external force that pins atoms to their equilibrium positions (1.2). To put it
differently, m2 is the curvature of the potential well that pins an atom to its equilibrium position.
Terms that have been neglected in are of several kinds. Only terms of quadratic order in the
nearest neighbor relative displacement n 1 n have been accounted for, and all interactions
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beyond the nearest neighbor range have been dropped. I have also omitted to specify boundary
terms. They are specified once boundary conditions have been imposed. In the limit N ,
the choice of boundary conditions should be immaterial since the bulk potential energy should
be of order L Na, whereas the energy contribution arising from boundary terms should be of
order L0 1. To minimize boundary effects in a finite system, one imposes periodic boundary
conditions
n N n n 1 N (1.5)
An open chain of atoms turns into a ring after imposing periodic boundary conditions. Further-
more, imposing periodic boundary conditions endows the potential with new symmetries within
the harmonic approximation defined by1
Vharmonic 1 1 N N :N
n 1
2n 1 n
2 1
2m2
N
n 1
n2
(1.7)
First, the shift of labelling
n n m n 1 N m (1.8)
leaves Eq. (1.7) invariant. Second, translational invariance is recovered in the absence of the
pinning potential:
0 Vharmonic 1 N Vharmonic 1 xa N xa x (1.9)
The kinetic energy of an open chain of atoms is simply given by
T 1 1 N N1
2m
N
n 1
dndt
21
2m
N
n 1
n2
(1.10)
As for the potential energy, the choice of boundary conditions only affects the kinetic energy by
terms of order L0. It is again natural to choose periodic boundary conditions if one is interested in
extensive properties of the system.
1
Without loss of generality, I have set the classical minimum of the potential energy to zero:
V 1 N 0 (1.6)
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The classical Lagrangian in the harmonic approximation and with periodic boundary condi-
tions is defined by subtracting from the kinetic energy (1.10) the potential energy (1.7)
:N
n 1
1
2m n
2
n 1 n2
m2 n2
(1.11)
The classical equations of motion follow from Lagrange equations
d
dt
n
n n 1 N (1.12)
They are
m n n 1 n 1 2n m2n n 1 N (1.13)
with the complex traveling wave solutions
n t ei kn t 2 2
m1 cos k 2 (1.14)
Imposing periodic boundary conditions allows to identify the normal modes. These are countable
many traveling waves with the frequency-wave number relation
l 2
m1 cos kl 2 kl
2
Nl l 1 N (1.15)
The most general real solution of Lagrange equations (1.13) obeying periodic boundary conditions
is
n t
N
l 1 Al e i kl n l t Al e i kl n lt n 1 N (1.16)
Here, the complex valued expansion coefficient Al is arbitrary.
To revert to the Hamilton-Jacobi formalism of classical mechanics, one introduces the canonical
momentum n conjugate to n through
n t :
n
imN
l 1
lA
le i kl n lt A
le i kl n l t n 1 N (1.17)
and construct the Hamiltonian
N
n 1
1
2
n2
m n 1 n
2m2 n
2 (1.18)
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from the Lagrangian (1.11) through a Legendre transformation. Hamilton-Jacobi equations of
motion are then
n
nn n
nn n 1 N (1.19)
where stands for the Poisson brackets.2
In the long wave number limit kl 1, the dispersion relation reduces to
2l
m k2l
2
k4l (1.21)
The pinning potential characterized by the potential wall curvature has opened up a gap in the
spectrum of normal modes. No solutions to Lagrange equations (1.13) can be found below the
characteristic frequency . By switching off the pinning potential, 0, the dispersion relation
simplifies to
2l
mk2l k
4l (1.22)
The proportionality constant m between frequency and wave number is interpreted as the
velocity of propagation of a sound wave in the one-dimensional harmonic chain.
B. Thermodynamic limit
The thermodynamic limit N emerges naturally if one is interested in the response of solids
to external perturbations as can be induced, say, by compressions. Of course, the characteristic
wave lengths of typical perturbations in daily life are much larger than the atomic separation.
Hence, the elastic response from a solid to a macroscopic perturbation is dominated by normalmodes with arbitrarily small wave numbers k 0. It is then much more economical not to account
for the discrete nature of the solid as is done in the Lagrangian (1.11). To this end, Eq. (1.11) is
first rewritten as
N
n 1
a1
2
m
an
2
an 1 n
a
2m
a2 n
2
:N
n 1
aLn (1.23)
2 The Poisson bracket f g of two functions f and g of the canonical variables n and n is defined by
f g :N
n 1
f
n
g
n
f
n
g
n(1.20)
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Interpret
:m
a :
n 1 na
Y : a and Ln (1.24)
as the mass per unit length, the elongation per unit length, the Youngs modulus,3 and the local
Lagrangian per unit length, respectively. Now write
L
0 dx
1
2
t
2
Y
x
2
2
2
:L
0dx (1.26)
whereby the following substitutions have been performed:
The discrete sum n has been replaced by the integral dx a over the semi-open interval
0 L .
The relative displacement n at time t has been replaced by the value of the real function at space-time coordinates x t obeying periodic boundary conditions in space:
x L t x t x 0 L t (1.27)
The time derivative of the relative displacement n at time t has been replaced by the value
of the time derivative t at space-time coordinate x t .
The discrete difference n 1 n at time t has been replaced by the lattice constant times
the value of the space derivative x at space-time coordinate x t .
The integrand in Eq. (1.26) is called the Lagrangian density. It is a real valued function of
space-time. From it, one obtains the continuum limit of Lagrange equations (1.12) according
to
t x t
t y tx
x t
x y t
x t
y t(1.28)
3 For an elastic rode obeying Hookes law, the extension of the rode per unit length is proportional to the exerted
force F with the Youngs modulus Y as the proportionality constant:
F Y (1.25)
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Here, the symbol x t is to be interpreted as the infinitesimal functional change of at
the given space-time coordinates x t induced by Taylor expansion,
x x t t x t
x x
t t
2 x2 t
2
(1.29)
One must keep in mind that , x , and t , are independent variables. Moreover,
one must use the rule
x t
y t x y
L
0dx x t
y t1 y 0 L (1.30)
that extends the rule
mn
m nN
m 1
mn
1 n 1 N (1.31)
to the continuum. Otherwise, all the usual rules of differentiation apply to .
Equations of motion (1.13) become the one-dimensional sound wave equation
2t v22x
2 0 v :Y
(1.32)
after replacing the finite difference
n 1 n 1 2n n 1 n n n 1 (1.33)
by a2 times the value of the second order space derivative 2x at space-time coordinates
x t .
The Hamiltonian in the continuum limit follows from Eq. (1.26) with the help of a (functional)
Legendre transform or directly from the continuum limit of Eq. (1.18),
L
0dx
1
2
2
Y
x
2
22
:
L
0 dx (1.34)
where the field is the canonically conjugate to :
x t :L
0dy
y t
t x t t x t (1.35)
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C. From sums to integrals
As we have seen, probing the one dimensional crystal on length scales much larger than the
lattice spacing a blurs our vision to the point where the crystal appears as an elastic continuum.
Viewed without an atomic microscope, the relative displacements n, n 1 N, become a
field x where x can be any real number provided N is sufficiently large.
The basic mathematical rules for this blurring or coarse graining is that for functions f thatvary slowly on the lattice scale,
n
f nadx
af x (1.36)
In particular,
f ma n
m nf na n
am n
af na f y dx x y f x (1.37)
justifies the identification
m na
x y (1.38)
Equation (1.38) tells us that the divergent quantity x 0 in real space should be thought of as
the reciprocal of the lattice spacing, i.e., the number of normal modes in reciprocal space per unit
volume 2 N in wave number space:
1
a
kl 1 kl
a
1
2 Nkl :
2
Nl (1.39)
How does one go from a discrete Fourier sum to a Fourier integral? Start from
N
l 1
eikl m n Nm n kl :2
Nl (1.40)
Multiply both sides of this equation by the reciprocal of the system size L Na:
1
L
N
l 1
eikl m nm n
a(1.41)
Since the right hand side should be identified with x y in the thermodynamic limit N ,
the left hand side should be identified with
1
L
N
l 1
eikla
m n a2 a
0
dk
2eik x y
dk
2eik x y (1.42)
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whereby
kl
ak m n a x y (1.43)
To see this, recall first that the periodic boundary conditions tell us that l 1 N could have
equally well be chosen to run between N 2 1 and N 2 if N is even or N 1 2 and
N 1 2 if N is odd. Hence, it is permissible to adopt the more symmetrical rule
1
L
N
l 1
f kl a
a
dk
2f k (1.44)
to convert a finite summation over wave numbers into an integral over the Brillouin zone (recip-
rocal space) a a as the thermodynamic limit N L a is taken. Now, if f x is a
slowly varying function on the lattice scale a, its Fourier transform f k will be essentially van-
ishing for k 1 a. In this case, the limits a can safely be replaced by the limits on
the right hand side of Eq. (1.44). We then arrive to the desired integral representation of the delta
function in real space,
x y
dk
2eik x y (1.45)
Observe that factors of 2 appear in an assymetrical way in integrals over real x and reciprocal
k spaces. Although this is purely a matter of convention when defining the Fourier transform,
there is a physical reasoning behind this choice. Indeed, Eq. (1.44) implies that dk 2 has the
physical meaning of the number of normal modes in reciprocal space with wave number between
k and k dk per unit volume L in real space. Correspondingly, the divergent quantity 2 k 0
in reciprocal space has the physical meaning of being the divergent volume L of the system
as is inferred from
x
dk
2eikx 2 k
dx eikx (1.46)
II. QUANTUM MECHANICAL ONE-DIMENSIONAL CRYSTAL
A. Reminiscences about the harmonic oscillator
I now turn to the task of giving a quantum mechanical description for a non-dissipative one-
dimensional harmonic crystal. One possible route consists in the construction of a Hilbert space
and of operators acting on it whose expectation values can be related to measurable properties of
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the crystal.4 In this setting, the time evolution of physical quantities can be calculated either in the
Schrodinger or in the Heisenberg picture. I will begin by reviewing these two approaches in the
context of a single harmonic oscillator. The extension to the harmonic crystal will then follow in
a very natural way.
The classical Hamiltonian that describes a single particle of unit mass m 1 confined to a
quadratic well with curvature 2 is
:1
2p2 2x2 (2.1)
Hamilton-Jacobi equations of motion are
dx t
dtx
pp t
d p t
dtp
x2x t (2.2)
Solutions to the classical equations of motion are
x t A cos t B sin t
p t A sin t B cos t (2.3)
The energy E of the particle is a constant of the motion that depends on the choice of initial
conditions through the two real valued constants A and B:
E1
2A2 B2 2 (2.4)
In the Schrodinger picture of quantum mechanics, the position x of the particle and its canonical
conjugate p become operators x and p that act on the Hilbert space of twice differentiable and
square integrable functions : and obey the commutation relation
x p : x p p x i (2.5)
The time evolution (or dynamics in short) of the system is encoded by Schrodinger equation
i t x t H x t (2.6)
4 Another route to quantization is by means of the path integral representation of quantum mechanics as is shown in
appendix B.
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where the quantum Hamiltonian H is given by
H1
2p2 2 x2 (2.7)
The time evolution of the wave function x t is unique once initial conditions x t 0 are
given. Solving the time-independent eigenvalue problem
Hn x nn x (2.8)
is tantamount to solving the time-dependent Schrodinger equation through the Ansatz
x t n
cnn x eint (2.9)
The expansion coefficients cn are time-independent and uniquely determined by the initial
condition, say x t 0 . As is well known, the energy eigenvalues n are given by
n n
1
2 n (2.10)
The energy eigenfunctions n x are Hermite polynomials multiplying a Gaussian:
0 x
1 4
e12x2
1 x4
3 1 4xe
12x2
2 x
4
1 4
2
x2 1 e12x2
...
n x1
2nn!
n 1 2
1 4 x
d
dx
n
e12x2 (2.11)
The Heisenberg picture of quantum mechanics is better suited than the Schrodinger picture to a
generalization to quantum field theory. In the Heisenberg picture and contrary to the Schrodinger
picture, operators are explicitly time-dependent. For any operator O, the solution to the operator
equation of motion5
i d
O tdt
O t H (2.12)
5 The assumption that the system is non-dissipative has been used here in that H does not depend explicitly on time:
t H 0.
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that replaces Schrodinger equation is
O t e iHt
O t 0 e iHt
(2.13)
By definition, the algebra obeyed by operators in the Schrodinger picture holds true in the Heisen-
berg picture provided operators are taken at equal-time. For example,
x t : e iHt
x t 0 e iHt
p t : e iHt
p t 0 e iHt
(2.14)
obey by construction the equal-time commutator
x t p t i t (2.15)
Finding the commutator of x t and p t at unequal times t t requires solving the dynamics of
the system, i.e., Eq. (2.12) with O substituted for x and p, respectively:
dx t
dtp t
d p t
dt 2
x t (2.16)
In other words, the Heisenberg operators x t and p t satisfy the same equation of motion as the
classical variables they replace:
d2 x t
dt22 x t 0 p t
dx t
dt(2.17)
The solution (2.3) can thus be borrowed with the caveat that A and B should be replaced by time-
independent operators A and B.
At this stage, it is more productive to depart from following a strategy dictated by the real valued
classical solution (2.3). The key observation is that the quantum Hamiltonian for the harmonic
oscillator takes the very simple form6
H a t a t1
2(2.18)
if the pair of canonically conjugate Hermitean operators x t and p t is traded for the pair a t
and a t of operators defined by
x t 2 a t a
t
p t2
ia t ia t (2.19)
6 Observe that I am anticipating that H does not depend explicitly on time.
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Once the equal-time commutator a t a t is known the Heisenberg equations are easily derived
from
ida t
dta t H
ida t
dta t H (2.20)
With the help of
a t
2x t i
p t
a t
2x t i
p t
(2.21)
one verifies that
x t p t i x t x t p t p t 0 a t a t 1 a t a t a t a t 0
(2.22)
The change of Hermitean operator valued variables to non-Hermitean operator valued variables
now proves advantageous in that the equations of motion for a t and a t decouple according to
da t
dti a t a t a t 0 e it
da t
dti a t a t a t 0 e it (2.23)
Below, I will write a for a t 0 and similarly for a. The time evolution of x t , p t , and H is
now explicitly given by
x t2
a e it a e it
p t i
2a e it a e it
H aa1
2(2.24)
As must be by the absence of dissipation, H is explicitly time-independent: t H 0. The Hilbert
space can now be constructed explicitly with purely algebraic methods. The Hilbert space is
defined by all possible linear combinations of the eigenstates
n :a
n
n!0 H n n n n (2.25)
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Here, the ground state or vacuum 0 is defined by the condition
a 0 0 (2.26)
One verifies that 0 x in Eq. (2.11) uniquely satisfies Eq. (2.26) by using the real space represen-
tation of the operator a.
B. Discrete limit
In the spirit of the Heisenberg picture for the harmonic oscillator and guided by the Fourier
expansions in Eqs. (1.16) and (1.17), I begin by defining the operators
n t :1
N
N
l 1
2lal e
i kl n l t al e
i kl n lt
n t : i1
N
N
l 1
l
2
al ei kl n l t a
l e
i kl n lt n 1 N (2.27)
where the frequency l and the integer label l are related by Eq. (1.15), i.e., (remember that m 1)
l 2 1 cos kl 2 kl :2
Nl l 1 N (2.28)
and the operator valued expansion coefficients al and al obey the harmonic oscillator algebra
al
al
l l al al al
al
0 l l 1 N (2.29)
The normalization factor 1 N is needed to cancel the factor of N present in the Fourier series
N
l 1
eikl m n Nm n (2.30)
that shows up when one verifies that the equal-time commutators
m t n t i m n m t n t m t n t 0 m n 1 N (2.31)
hold for all times. I am now ready to consistently define the Hamiltonian H for the quantum
one-dimensional harmonic crystal [compare with Eq. (1.18)]
H :N
n 1
1
2n t
2 n 1 t n t2 2 n t
2(2.32)
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With the help of the algebra (2.29), one verifies that H is explicitly time-independent and given by
HN
l 1
l al al
1
2(2.33)
Next task is to construct the Hilbert space for the one-dimensional quantum crystal by algebraic
methods. Assume that there exists a unique state 0 , the ground state or vacuum, defined by
al
0 0 l 1 N (2.34)
If so, the state
n1 n2 nN :N
l 1
1
nl!a
l
nl0 (2.35)
is normalized to one and is an eigenstate of H with energy eigenvalue
n1 nN :N
l 1
l nl1
2(2.36)
The ground state energy is of order N and given by
0 0 :1
2
N
l 1
l (2.37)
Excited states have at least one nl 0. They are called phonons. The eigenstate n1 n2 nN is
said to have n1 phonons in the first mode, n2 phonons in the second mode, and so on. Phonons can
be thought of as identical elementary particles since they possess a definite energy and momentum.
Because the phonon occupation number
nl n1 nl nN al al n1 nl nN (2.38)
is an arbitrary positive integer, phonons obey Bose-Einstein statistics. Upon switching on a suit-
able interaction [say by including cubic and quartic terms in the expansion (1.4)], phonons scatter
off one other just as other -ons (mesons, photons, gluons, and so on) known to physics. Although
we are en route towards constructing quantum fields x t out ofn t we have encountered par-
ticles. The duality between field and particle is the essence of quantum field theory.
The vector space spanned by the states labelled by the phonon occupation numbers
n1 nNN in Eq. (2.35) is the Hilbert space of the one-dimensional quantum crystal. The
mathematical structure of this Hilbert space is a symmetric tensor product ofN copies of the har-
monic oscillator Hilbert space. In physics, this symmetric tensor product is called a Fock space
when the emphasis is on the phonon as an elementary particle.
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C. Thermodynamic limit
Taking the thermodynamic limit N is a direct application to subsection II B of the rules
established in the context of the classical description of subsections I B and I C. Hence, with the
identifications7
N
n 1
a dx
1
Na
N
l 1
dk
2
l k 2v2
a21 cos ka 2 v2k2 2 ifka 1,
kln kx
al1
Naa k
n t a x t
n t a x t (2.39)
the canonically conjugate pairs of operators n t and n t are replaced by the quantum fields
x t :dk
2 2 ka k e i kx k t a k e i kx k t
x t : idk
2
k
2a k e i kx k t a k e i kx k t (2.40)
respectively.
8
Their equal-time commutators follow from the harmonic oscillator algebra
a k a k 2 k k a k a k a k a k 0 (2.41)
They are
x t y t i x y x t y t x t y t 0 (2.42)
7 Limits of integrations in real and reciprocal spaces are left unspecified in order to distinguish whether L Na is
held fixed or not in the thermodynamic limit N , i.e., whether the continuum limit a 0 is simultaneously
taken or not.8 The substitution rules al
1
Naa k , n t a x t , and n t a x t , are needed to cancel the
volume factor Na in Nl 1 Nadk2 .
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The Hamiltonian is
H dx1
2t x t
2v2 x x t
2 2 x t 2
dk
2
1
2 k a k a k a k a k (2.43)
The excitation spectrum is obtained by making use of the commutator between a k and a k .
It is given by
H E0 :dk
2 k a k a k (2.44)
and is observed to vanish for the vacuum 0 . The operation of subtracting from the Hamiltoni-
an the ground state energy E0 is called normal ordering. It amounts to placing all annihilation
operators a k to the right of the creation operators a k . The ground state energy
E0 : 0 H 0
dk
2
1
2 k 2 k 0
Volume in real spacedk
2
1
2 k
modes
1
2modes (2.45)
can be ill-defined for two distinct reasons. First, if N with a held fixed, there exists an upper
cut-off to the integral over reciprocal space at the Brillouin zone boundaries a and E0 is only
infrared divergent due to the fact that 2 k 0 is the diverging volume L Na in real space.
Second, even ifL Na is kept finite while both the infrared N and ultraviolet a 0 limits
are taken, the absence of an upper cut-off in the k integral can cause the zero point energy density
E0 L to diverge as well. Divergences of E0 or E0 L are only of practical relevance if one can
control experimentally k or the density of states modes and thereby measure changes in E0 or
E0 L. For example, this can be achieved in a resonant cavity whose size is variable. If so, changes
of E0 with the cavity size can be measured. These changes in the zero point energy are known
as the Casimir energy. Another possibility to measure E0 indirectly occurs when some internal
parameters entering the microscopic Hamiltonian can be tuned so as to lower E0 to the point
where E0 becomes negative thereby signalling an instability associated to spontaneous symmetry
breaking (the vacuum 0 is not non-degenerate anymore when E0 0). Finally, divergences of
E0 L matter greatly if the energy-momentum tensors of matter fields are dynamical variables as
is the case in cosmological models.
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D. Higher dimensional generalizations
Generalizations to higher dimensions are straightforward. The coordinates x 1 and k 1
in real and reciprocal one-dimensional spaces need only be replaced by the vectors r d and
k d, in real and reciprocal d-dimensional spaces, respectively.
APPENDIX A: THE HARMONIC OSCILLATOR ALGEBRA AND ITS COHERENT STATES
1. Boson algebra
The quantum Hamiltonian for the harmonic oscillator is
H aa1
2(A1)
when represented in terms of the lowering (annihilation) and raising (creation) operators a and a,
respectively, that obey the boson algebra
a a 1 a a a a 0 (A2)
A complete, orthogonal, and normalized basis of H is given by
na n
n!0 H n n
1
2n (A3)
where the ground state (vacuum) 0 is annihilated by a:
a 0 0 (A4)
As it should be ifH is to be Hermitean, annihilation a and creation a operators are thus adjoint to
each other and represented by
a n n n 1 a n n 1 n 1
m a n nm 1 n m a n n 1m 1 n (A5)
The single-particle Hilbert space1
of twice differentiable and square integrable functions onthe real line for the harmonic oscillator can be reinterpretedas the Fock space for the annihila-
tion and creation operators a and a, respectively, since the number operator
N : aa (A6)
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commutes with the Hamiltonian and the Fock space is, by definition, the direct sum of the
energy eigenspaces:
1 :
n 0
n (A7)
One possible resolution of the identity 11 on 1 is
11
n 0
n n (A8)
More informations on the harmonic oscillator can be found in chapter V of Ref. [2].
2. Coherent states
Define the uncountable set of harmonic oscillator coherent states, in short bosonic coherent
states, by
cs : e a 0 :
n 0
n
n!n (A9)
The adjoint set is
cs : 0 ea :
n 0
n n
n! (A10)
Properties of boson coherent states are:
Coherent state cs is a right eigenstate with eigenvalue of the annihilation operator a,9
a cs a e a 0
n 0
n
n!a n
n 1
n
n!n n 1
n 1
n 1
n 1 !n 1
cs (A11)
9 Non-Hermitean operators need not have same left and right eigenstates.
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Coherent state cs is a left eigenstate with eigenvalue of the creation operator a,
a cs cs cs a
cs (A12)
The action of creation operator a on coherent state cs is differentiation with respect to
,
a cs a e a
0
n 0
n
n!a n
n 0
n
n!n 1 n 1
n 0
d
d
n 1
n 1 !n 1
d
d cs (A13)
The action of creation operator a on coherent state cs is differentiation with respect to ,
a csd
d cs cs a
d
dcs (A14)
The overlap cs cs between two coherent states is exp ,
cs cs
m n 0
m mm!
nn!
n
m n m n m n
n 0
n
n!
e (A15)
There exists a resolution of the identity in terms of boson coherent states,
11 dz dz2i
e z z z cscs z
:1
dRez
dImz e z z z cscs z (A16)
Proof:
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Write
O :dz dz
2ie z z z cs cs z (A17)
By construction, O belongs to the algebra of operators generated by a and a.
Step 1: With the help of Eqs. (A11) and (A14),
a z cs cs z a z cs cs z z cs cs z a
z z cs cs z z csd
dzcs z
z does not depend on z so that z can be factorized zd
dzz cs cs z (A18)
Hence, after making use of integration by parts,
a Odz dz
2ie z z z
d
dzz cs cs z
0 (A19)
Step 2: By taking the adjoint of Eq. (A19), a
O 0.
Step 3:
0 O 0dz dz
2ie z z 0 z cs cs z 0
m n m n m ndz dz
2ie z z
1 (A20)
Step 4: Any linear operator from to belongs to the algebra generated by a and a. Since
O commutes with both a and a
by steps 1 and 2, O commutes with all linear operators from
to . By Schurs lemma, O must be proportional to the identity operator. By Step 3, the
proportionality factor is 1.
For any operator A : ,
Tr A :
n 0
n A n
By the resolution of identity Eq. (A16)
dz dz
2i
e z z
n 0
n z cscs z A n
dz dz
2ie z z cs z A
n 0
n n z cs
By the resolution of identity Eq. (A8)
dz dz
2ie z z cs z A z cs (A21)
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Any operator A : is some linear combination of products of as and as. Normal
ordering of A, which is denoted : A :, is the operation of moving all creation operators to the
left of annihilation operators as if all operators were to commute. For example,
A aaaa aaa : A : aaaa aaa A 2aa a (A22)
The matrix element of any normal ordered operator : A a a : between any two coherent
states cs z and z cs follows from Eqs. (A11), (A12), and (A15),
cs z : A a a : z cs cs z : A z z : z cs e
z z : A z z : (A23)
Here, : A z z : is the complex valued function obtained from the normal ordered operator
: A a a : by substituting a for the complex number z and a for the complex number z .
Define the continuous family of unitary operators
D : ea
a (A24)
From Glauber formula10
D e 2
2 e a
e a (A26)
which implies that
D 0 e 2
2 e a
e a 0
e 2
2 e a 0
e 2
2 cs (A27)
Hence, D is the unitary transformation that rotates the vacuum 0 into the coherent state
cs, up to a proportionality constant.
More informations on bosonic coherent states can be found in complement GV of Ref. [2].
10
Let A and B be two operators that both commute with their commutator A B . Then,
eA eB eA B e12 A B (A25)
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APPENDIX B: PATH INTEGRAL REPRESENTATION OF THE ANHARMONIC OSCILLA-
TOR
Define the anharmonic oscillator of order n 2 3 4 by
HnH0
HnH0 : a
a1
2Hn :
2n
m 3
m a a
m
(B1)
Of the real parameters m, m 3 4 2n, it is only required that 2n 0. This insures that there
exists a vacuum 0 annihilated by a. With the help of the boson algebra (A2) it is possible to move
all annihilation operators to the right of the creation operators in the interaction Hn. Having done
that, Hn is normal ordered, i.e., Hn : Hn :. Evidently, Hn : Hn : cannot be written anymore as a
polynomial in x a a of degree 2n. For example, a a 3 aaa 3aaa 2a H c .
In the representation in which H is normal ordered, the canonical partition function on the
Hilbert space in Eq. (A7) is defined to be
Z : Tr exp H Tr exp : H :
n 0
n exp : H : n (B2)
I will now give an alternative representation of the canonical partition function that relies on the
use of coherent states. I begin with the trace formula (A21):
Zd0d0
2ie 00 cs 0 exp : H : 0 cs (B3)
For M a large positive integer, write
exp : H : exp
M
M 1
j 0
: H :
1
M
M 1
j 0
: H :
M
2
(B4)
Insert the resolution of identity (A8) M 1 -times,
e :H: e
M:
H:1
j M 1
dj dj
2ie jj j cs cs j e
M:
H: (B5)
Equation (A23) together with Eq. (B4) gives
cs 0 e
M: H: M 1 cs e
0M 1
M:H 0 M 1 :
M
2
(B6)
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and
cs j e
M:H: j 1 cs e
jj 1
M:H j j 1 :
M
2
j M 1 M 2 1
(B7)
The operator valued function : H : of a and a has been replaced by a complex valued function
:H: of and , respectively. Altogether, a M-dimensional integral representation of the partition
function has been found,
Z exp 0M 1
j 0
dj dj2i
expM
j 0
j j j 1
M: H j j 1 :
M
2
(B8)
whereby
N : 0 N : 0 (B9)
It is customary to write, in the limit M , the functional path integral representation of the
partition function
Z exp 0 eSE (B10)
where the so-called Euclidean action SE is given by
SE
0
d : H : (B11)
and the complex valued fields and obey the periodic boundary conditions
(B12)
Hence, their Fourier transform are
1
l
l eil
1
l l eil
l :
2
l (B13)
The frequencies l are the so-called boson Matsubara frequencies.
Convergence of the (functional) integral representing the partition function is guaranteed by
the contribution 2n 2n to the interaction : Hn :. Thus, convergence of an integral
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is the counterpart in a path integral representation to the existence of a ground state in operator
language.
Quantum mechanics at zero temperature is recovered from the partition function after perform-
ing the analytical continuation (also called a Wick rotation)
it d idt it (B14)
under which
SE iS
i
dt t it t : H t t : (B15)
The path integral representation of the anharmonic oscillator relies solely on two properties of
bosonic coherent states: Equations (A8) and (A23). Raising, a, and lowering, a, operators are
not unique to bosons. As we shall see, one can also associate raising and lowering operators to
fermions. Raising and lowering operators are also well known to be involved in the theory of the
angular momentum. In general, raising and lowering operators appear whenever a finite (infinite)
set of operators obey a finite (infinite) dimensional Lie algebra. Coherent states are those states
that are eigenstates of lowering operators in the Lie algebra and they obey extensions of Eqs. (A8)
and (A23). Hence, it is possible to generalize the path integral representation of the partition
function for the anharmonic oscillator to Hamiltonians expressed in terms of operator obeying a
fermion, spin, or any type of Lie algebra. Due to the non-vanishing overlap of coherent states, a
first order imaginary-time derivative term always appears in the action. This term is called a Berry
phase when it yields a pure phase in an otherwise real valued Euclidean action as is the case,
say, when dealing with spin Hamiltonians.11 It is the first order time derivative term that encodes
11 By writing [compare with Eq. (2.21)]
1
2x ip
1
2x ip (B16)
we can derive the path integral representation of the (an)harmonic oscillator in terms of the coordinate and momen-
tum of the single particle of unit mass m 1, unit characteristic frequency 1, and with 1. The first order
partial derivative term becomes purely imaginary
0d i
0d xp (B17)
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quantum mechanics in the path integral representation of the partition function. A reference on
generalized coherent states is the book in Ref. [3].
APPENDIX C: HIGHER DIMENSIONAL GENERALIZATIONS
The path integral representation of the partition function for a single anharmonic oscillator is
a functional integral over the exponential of the Euclidean classical action in 0-dimensional space(B11). The path integral representation of the quantum field theory of a d-dimensional continuum
of coupled anharmonic oscillator is a functional integral over the exponential of the Euclidean
classical action in d-dimensional space of the form
SE
0dt ddr E
0d ddr r r : H r r : (C1)
The classical fields r , and r obey periodic boundary conditions in imaginary time ,
r r r r (C2)
At zero temperature, analytical continuation it of the action yields
S
dt ddr
dt ddr r it r : H r r : (C3)
The classical canonical field conjugate to r t is
r t :
t r t r t (C4)
Canonical quantization is obtained by replacing the classical fields r t and r t with quan-
tum fields r t and r t that obey the equal-time algebra
r t r t r r r t r t r t r t 0 (C5)
[1] H. Goldstein, Classical mechanics, (Addison-Wesley, New-York, 1980).
[2] C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum mechanics, (Hermann, Paris, 1977).
[3] A. M. Perelomov, Generalized coherent states and their applications, (Springer, Berlin, 1986).
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