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Feynman DiagramPart I - Interaction Picture
June 1, 2016
Contents
1 Note 1
2 Natural Units 1
3 Interaction Picture 23.1 Equation of Motion for Observables .
. . . . . . . . . . . . . . . 23.2 Equation of Motion for wave
function . . . . . . . . . . . . . . . 33.3 The evolution operator
in Interaction picture . . . . . . . . . . . 33.4 Perturbative
Solution for Û . . . . . . . . . . . . . . . . . . . . . 4
3.4.1 Time Ordering Operator T . . . . . . . . . . . . . . . . .
73.5 Adiabatic Switch on . . . . . . . . . . . . . . . . . . . . .
. . . . 8
4 Todo 9
A Proof of Gell-Mann and Low Theorem 9
Abstract
In this article, we introduces the basic Interaction Picture in
quan-tum mechanics, then introduce the idea of ”adiabatic turn on”,
which issupported by Gell-Mann and Low theorem (proved in
appendix).
1 Note
This note is intended for learning Feynman Diagram techniques in
many-bodyphysics. However, due to the limited time and the growing
size of this document,I have to split it into small parts. This
first part provides one foundationalmaterial for the beginning of
Feynman Diagrams, i.e. Interaction Picture ofquantum mechanics.
Further updates will be found on my blog [1] within thissummer
vocation.
The style of the article might be fastidious. Yet I have try to
maintainsome physical viewpoints. Also, I will provide readers with
reference to originalmaterials whenever possible.
1
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2 Natural Units
For convenience, I will use the unit system s.t. ~ = 1. An
example of such unitsystems is Plank units. 1 Usually, calculation
down in such unit systems arecarried out mathematically first, then
those normalized (being set to 1) units,such as ~ here, are
inserted back to the equations by dimensional analysis. Forexample,
in this article when one sees the imaginary unit i, one should
replaceit with i~ or sometimes i~ to get back to S.I. units.
3 Interaction Picture
The Interaction Picture 2 is a very important way to view
physical systems asinteracting systems growing from the underlying
ground state systems. In Inter-action Picture, both the wave
functions and the operators are time dependent.The Hamiltonian is
separated into the ground state part and the interactionpart:
H = H0 + V
where
• H0 : is taken to be an exactly solvable Hamiltonian. Note that
H0 shouldbe bilinear in C† and C for Wick’s theoren to be
applicable. 3
The operators and states are related to Schrödinger’s Picture
by
|ΨI〉 = eiH0t |Ψs〉 = eiH0te−iHt |ΨH〉 (3.0.1)Ô1 = e
iH0tÔSe−iH0t (3.0.2)
where subscripts denoting:
• I:Interaction Picture
• S: Schrödinger Picture
• H: Heisenberg Picture
Remark: we see that the ”inner product” value is unaltered by
this:
〈ΨI |ÔS |ΨS〉 = 〈ΨS |e−iH0t · (eiH0tÔe−iH0t) · eiH0t|ΨS〉 = 〈ΨS
|ÔS |ΨS〉
Physically, this leaves the probabilities (the only important
value) unchanged.The evolution of states is governed by equation of
motion. In Schrödinger
Picture, it is the Schrödinger’s equation. In Interaction
Picture, we will mainlydeal the evolution operator.
1Readers could refer to [4] for details.2This part follows much
of section 6 of [2]. Similar material could also be found on
section
2.1 of [3].3See page 79, Sec 2.4 of [3]
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3.1 Equation of Motion for Observables
Before we talk about evolution operator, we mention two useful
results. Thefirst is a equation of motion for observables in
Interaction Picture.
Since: ÔI(t) = eiĤ0tÔSe
−iĤ0t, we have:
i · ∂ÔI(t)∂t
= i[iĤ0e
iĤ0tÔSe−iĤ0t] + eiĤ0tÔS(−iĤ0)e−iĤ0t
]= −Ĥ0(eiĤ0tÔSe−iĤ0t) + eiĤ0tÔSe−iĤ0tĤ0= [ÔI(t), Ĥ0]
(3.1.1)
This looks very similar to the Heisenberg’s equation, which is
reasonable sincein both pictures the operators is
time-dependent.
3.2 Equation of Motion for wave function
Another important result is the following equation:
i∂
∂t|ΨI(t)〉 =i
∂
∂t(eiĤ0t |ΨS(t)〉) (3.2.1)
=i(iĤ0) · |ΨI(t)〉+ eiĤ0t ·∂
∂t|ΨS(t)〉 (3.2.2)
=− Ĥ0 |ΨI(t)〉+ eiĤ0t(H0 + V ) |ΨS(t)〉 (3.2.3)
=[−Ĥ0 + eiH0t(Ĥ0 + V )e−iĤ0t] |ΨI(t)〉 (3.2.4)=V̂I |ΨI(t)〉
(3.2.5)
However, in Interaction Picture, there is one operator which is
even moreimportant than the above equations. It is the evolution
operator to be definedbelow.
3.3 The evolution operator in Interaction picture
Note: From now on, we work mostly in Interaction Picture. Hence
the subscriptI will be neglected, when it is convenient, for
operators or states in InteractionPicture.
The evolution operator Û(t, t0) is defined s.t.
|Ψ(t)〉 = Û(t, t0) |Ψ(t0)〉 (3.3.1)
We observe that Û behaves just like the time-evolution operator
in SchrödingerPicture: it brings a state at time t0 to a later
time t. For finite time, it can begiven a formal expression as
follows:
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By equation (3.0.1) on the preceding page we have
|ΨI(t)〉 = eiH0t |ΨS(t)〉 (3.3.2)= eiH0t · eiH(t−t0) |ΨS(t0)〉 (by
Schrödinger equation)
(3.3.3)
= eiH0t · e−iH(t−t0) · eiH0t0 |ΨI(t0)〉 (by 3.0.1)(3.3.4)
so:
Û(t, t0) = eiH0te−iH(t−t0)eiH0t0 (3.3.5)
The evolution operator has following properties:
1. Û(t0, t0) = 1
Physically, this means that a state is not evolving.
2. Û(t2, t1)Û(t1, t0) = Û(t2, t0) :
Physically, an evolving state should obviously be continuous in
time, hencethis property. It can also be demonstrated
mathematically:
Û(t0, t1)Û(t1, t0) =eiH0t2e−iH(t2−t1)e−iH0t1 ·
eiH0t1e−iH(t1−t0)e−iH0t0
=eiH0t2e−iH(t2−t0)e−iH0t0
3. Û(t, t0) · Û(t0, t) = 1This is a result of 2.
4. Û(t, t0) is unitary:
Because U(t, t0) is a transformation of basis that brings all
basis at timet0 to time t.
Also, one can easily see that mathematically: Û(t, t0) · Û†(t,
t0) = 1
5. U−1(t, t0) = U†(t, t0) = U(t0, t)
This means that ”inverse” = ”dagger” for Û . This is result of
4.
3.4 Perturbative Solution for Û
The operator Û has another more useful form which is the basis
for perturbativeanalysis of complex interacting systems. To obtain
such form, we first derive an
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integral equation for Û By equation (3.2.5) on the previous
page:
i∂
∂t· Û(t, t0) |ΨI(t0)〉 = (V̂ )I Û(t, t0) |ΨI(t0)〉 (3.4.1)
→ i ∂∂tÛ(t, t0) = −iV̂IU(t, t0) (3.4.2)
→ Û(t, t0)− Û(t0, t0) = −i ·∫ tt0
V̂IU(t, t0)dt (3.4.3)
or Û(t, t0) = 1− i∫ tt0
V̂IU(t1, t0)dt1 (3.4.4)
The integral equation is solved by iterative methods. We first
start with a (5.7.6) ofSakurai P356= (2.18) ofMakan P68
primitive guess:
Û (1)(t, t0) = 1−∫ tt0
V̂ (t′)dt′
Here, Û (1) is the first order solution.Then one improves on
this with
Û (2)(t, t0) =1−∫ tt0
V̂ (t′)Û (1)(t′, t0)dt′ (3.4.5)
=1−∫ tt0
V̂ (t2)(1−∫ t2t0
V̂ (t1)dt1)dt2 (3.4.6)
This is called it the second order solution.And one improves
again:
U (3) =1−∫ tt0
dt3 · V̂ (t3) · Û (2)(t3, t0) (3.4.7)
=1−∫ tt0
dt3V̂ (t3) · [1−∫ t3t0
dt2V̂ (t2) · (1−∫ t2t0
dt1V̂ (t1))dt1] (3.4.8)
Inductively we get the expression:
Û(t, t0) =
∞∑n=0
(−i)n∫ tt0
dt1
∫ t1t0
dt2 · · ·∫ tn−1t0
dtnV̂ (t1) · · · V̂ (tn) (3.4.9)
We remark that we have already obtain U in the form of
U = Σ1(−i)nan
If we can find another value bn = n!an, then
U =∑i
(−i)n
n!bn = (Formally writing as) e
−ib
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This is related to other forms of Wick’s theorem, which we will
not discussuntil the subsequent parts of this series.
Now we change the expression for Û a little bit. Let us take
the case ofn = 2 as an example. The main part is the integral:
∫ tt0
dt2
∫ t2t0
dt1V̂I(t2)V̂I(t1) (3.4.10)
The integration is taken diagrammatically:
Figure 1: t2 > t1
If we relabel t1 to t2, t2 to t1, we get:∫ tt0
dt1
∫ t1t0
dt2 · V̂I(t1)V̂I(t2) (3.4.11)
the same? 3.4.10 and 3.4.11.Then the integration is changed
into:
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Figure 2: t1 > t2
We observe that doing the two integration is equivalent to doing
a full inte-gral over the region marked by (t0 6 t1 6 t and t0 6 t2
6 t),
Figure 3: Whole region
while restricting the value of integration in the proper region
and puttingthe operators in correct order.
For example:equation (3.4.10) on page 5 is:
∫ tt0dt2 ·
∫ t2t0dt1θ(t1 − t2)V̂I(t2)V̂I(t1)
equation (3.4.11) on the previous page is:∫ tt0dt1∫
t1t0dt2·θ(t1−t2)V̂I(t2)V̂I(t1)
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where θ(t1 − t2) is restricting the to region t2 > t1. Also,
I put the latesttime in the front.
3.4.1 Time Ordering Operator T
We introduce a time ordering operator T to signify the process
we describedabove. When T acts on a set of operators, it
automatically sorts them indescending time order.
For example, if t3 > t2 > t1, then
T (A(t1)B(t3)C(t2)) = B(t3)C(t2)A(t1)
Then
∫ tt0
dt1
∫ tt0
dt2 =T (V̂ (t1)V̂ (t2)) (3.4.12)
= (integration of V̂I(t1)V̂I(t2) when t1 > t2)
+(integration of V̂ (t2)V̂ (t1) when t2 > t1)
= (
∫ tt0
dt1
∫ t1t0
dt2V̂I(t1)V̂I(t2)
+(
∫ tt0
dt2
∫ t2t0
dt1V̂I(t2)V̂I(t1)) (3.4.13)
But equation (3.4.11) on page 6 is equal to equation (3.4.10) on
page 5,
because∫ tt0dt2 ·
∫ t2t0dt1V̂I(t2)V̂I(t1) =
12 ·∫ tt0dt1∫ tt0dt2 · T [V̂2(t1)V̂2(t2)]
Similarly, for arbitrary n, we have∫ tt0
dtn
∫ tnt0
dt(n−1) · · ·∫ tt0
dt1·V̂I(tn) · · · V̂I(t1) =1
n!·∫ tt0
dtn · · ·∫ tt0
dt1·T (V̂I(tn) · · · V̂I(t1))
The factor n! is that we have in total n! different ways to
rearrange t1, · · · , tn.And each arrangement ti1 , · · · , tin
corresponds to a region in [t0, t0]
nwhere
t0 6 ti1 < · · · < tin 6 t.Thus we the most useful
expression for operator Û :
Û(t, t0) =
∞∑n=0
(−i)n
n!
∫ tt0
dt1
∫ tt0
dt2 · · ·∫ tt0
dtnT [V̂ (t1) · · · V̂ (tn)] (3.4.14)
3.5 Adiabatic Switch on
The next idea for interacting many-body system is to use the
notion of adiabaticswitch on. In the Interaction Picture, we assume
our Hamiltonian is:
H = H0 + V
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We wish to extract the eigenstate of H from that of H0. Hence we
imaginethat we have
H ′ = H0 + e−�|t|V
So we have limt→∞H′ = limt→∞H
′ = H0.In this scenario, we have
|Ψ(t = ±∞)〉 = |Φ0〉where |Φ0〉 can be exactly solved.Then using
the evolution operator U , we can bring up the states from −∞
to now t = 0.
|Ψ(t = 0)〉 =U�(0,−∞) |Ψ(t = −∞〉 (3.5.1)=U�(0,−∞) |Ψ(∞)〉
(3.5.2)
This is the solution to H = H0 + V , since e−�·0 = 0.
However, there is a problem with this. Because the real physical
system isin the Hamiltonian H = H0 + V all the time. And our system
dives in theHamiltonian H only at one instant of time. If the
result of over calculation isto be meaningful, it must be
independent on �.
In addition, we have a Gell-Mann and Low theorem, which requires
a lengthproof provided in appendix A. It states that:
If the following quantity exists to all orders in perturbation
theory, (denoting|Ψ0〉 :≡ U�(0,−∞) |Φ0〉. )
Û�(0,−∞) |Φ0〉〈Φ0| Û�(0,−∞) |Φ0〉
≡ |Ψ0〉〈Φ0|Ψ0〉
then |Ψ0〉 is an eigenstate of Ĥ in the following sense:
Ĥ|Φ0〉〈Φ0|Ψ0〉
= E|Ψ0〉〈Φ0|Ψ0〉
The equation (6.44) could be recast into a new form.
〈Φ0| Ĥ|Ψ0〉〈Φ0|Ψ0〉
= E · 〈Φ0|Ψ0〉〈Φ0|Ψ0〉
Since H = H0 + V and 〈Φ0|H0|Ψ0〉 = E0 · 〈Φ0|Ψ0〉, we have
〈Φ0|V |Ψ0〉〈Φ0|Ψ0〉
= E − E0 (3.5.3)
Thus, in the limit �→ 0 we have a way to calculate the energy of
our physicalsystem’s E by 3.5.3. This is also intuitively
satisfying, since when � is small, Ĥ ′
approximates Ĥ in a long enough time. If we imagine that we
turn on H ′ fromt = −∞ very slowly, then we are confident that our
calculation is giving us thereal wave function and its energy.
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4 Todo
Here lists I want to add to this article.First I will define
Green functions 4, which are related to a variety of physical
phenomenons. Then the above idea of adiabatic switch on will
allow me to givea formula for Green functions. I will show that
this formula, combined withperturbative solution to Û and the
Wick’s thoerem 5, gives us a systematicalway of calculating Green
functions. I will then explain Feynman Diagrams 6 ,which will give
a graphical representation of each Green functions. I will try
tounderstand the new path integral way of calculating Feynman
Diagrams7.
A Proof of Gell-Mann and Low Theorem
The theorem states:
Theorem 1. If the following quantity exists to all orders in
perturbation theory:
Û�(0,−∞) |Φ0〉〈Φ0| Û�(0,−∞) |Φ0〉
≡ |Ψ0〉〈Φ0|Ψ0〉
then |Ψ0〉 is an eigenstate of Ĥ in the following sense:
Ĥ|Φ0〉〈Φ0|Ψ0〉
= E|Ψ0〉〈Φ0|Ψ0〉
Proof:Here and afterwards, we will set |Ψ0(�)〉 ≡ Û�(0,−∞)
|Φ0〉.Consider the following:
(Ĥ0 − E0) |Ψ0(�)〉 =(Ĥ0 − E0)Û�(0,−∞) |Ψ0〉 (A.0.4)
=(Ĥ0Û�(0,−∞)− Û�(0,−∞)E0
)|Ψ0〉 (A.0.5)
=(Ĥ0Û�(0,−∞)− Û�(0,−∞)Ĥ0
)|Ψ0〉 (A.0.6)
=[Ĥ0, Û�(0,−∞)] |Ψ0〉 (A.0.7)
The expression for Û�(0,−∞) is explicitly:
Û�(0,−∞) =∞∑n=0
(−i)n
n!
∫ 0−∞
dt1 · · ·∫ 0−∞
dtne�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)] (A.0.8)
4could be found on [2] or [3]5Proof of Wick’s theorem could be
found on [2]6Both [2] and [3] explains Feynman Diagrams, though [2]
explains in more detail.7Could be found on [5].
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So we need [Ĥ0, V̂ (t1) · · · ˆV (tn)] to get [Ĥ0,
Û�(0,−∞)].
Lemma 1. [H,ABC · · · ] = ([H,A]BC · · · ) + (A[H,B]C · · · ) +
· · ·
Proof:Obviously we have:
[H,AB] = [H,A]B +A[H,B] (A.0.9)
Then for more terms we can, for example:
[H,ABC] =[H,AB]C +AB[H,C]
=[H,A]BC +A[H,B]C +AB[H,C]
Thus one may obviously obtain the lemma by using equation
(A.0.9) recur-sively.
Using above lemma, one has:
[Ĥ0, V̂ (t1) · · · V̂ (tn)] = [Ĥ0, V̂ (t1)]V̂ (t2) · · ·+ · ·
·+ V̂ (t1) · · · V̂ (tn−1)[Ĥ0, V̂ (tn)] (A.0.10)
Also, in Interaction Picture, one has:
i∂V̂ (t)
∂t= [V̂ (t), Ĥ0]
or:1
i
∂V̂ (t)
∂t= [Ĥ0, V̂ (t)]
Hence the above commutators are turned into partial
derivatives:
[Ĥ0, V̂ (t1) · · · V̂ (tn)] =1
i
(∂
∂t1+ · · ·+ ∂
∂tn
)V̂ (t1) · · · V̂ (tn) (A.0.11)
We need an additional fact:
T [
(∂
∂t1+ · · ·+ ∂
∂tn
)V̂ (t1) · · · V̂ (tn)] =
(∂
∂t1+ · · ·+ ∂
∂tn
)T [V̂ (t1) · · · V̂ (tn)]
(A.0.12)
This can be seen on an example. Notice that T [V̂ (t1)V̂ (t2)] =
θ(t1−t2)V̂ (t1)V̂ (t2)+θ(t2 − t1)V̂ (t2)V̂ (t1), where θ(t) is
Heaviside step function. We also have:∂∂tθ(t) = δ(t). So when t1 6=
t2, T
∂∂t1
= ∂∂t1T , i.e. it commutes with T .When t1 = t2, this is
obviously also correct as well. Hence:
T [∂
∂t1V̂ (t1)V̂ (t2)] =
∂
∂t1T [V̂ (t1)V̂ (t2)]
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Similar arguments will give us equation (A.0.12), since T is
essentially a seriesof θ(t) functions.
Using the above fact and A.0.11, one turns A.0.7 into (Denote∫
0−∞ dt1 · · ·
∫ 0−∞ dtn
as (Integrate)):
(Ĥ0 − E0) |Ψ0(�)〉 =[Ĥ0, 1 +
∞∑n=0
(−i)n
n!(Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)]
]=
∞∑n=1
(−i)n
i · n!(Integrate) e�(t1+···+tn)
(∂
∂t1+ · · ·+ ∂
∂tn
)T [V̂ (t1) · · · V̂ (tn)]
=
∞∑n=1
− (−i)n−1
n!(Integrate) e�(t1+···+tn)
(∂
∂t1+ · · ·+ ∂
∂tn
)T [V̂ (t1) · · · V̂ (tn)]
(A.0.13)
Integration by parts gives us, for example:
∫ 0−∞
dt1e�t1
∂
∂t1T [V̂ (t1)] = e
�t1T [V̂ (t1)]
∣∣∣∣0∞− �∫ 0∞dt1e
�t1T [V̂ (t1)]
Similarly:
(Integrate) e�(t1+···+tn)∂
∂t1T [V̂ (t1) · · · V̂ (tn)]
= V̂ (0) (Integrate without t1) e�(t2+···+tn)T [V̂ (t2) · · · V̂
(tn)]
− � (Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)]
(A.0.14)
For other ” ∂∂ti ”, the procedure is essentially the same, and
they all contributeto the same value, as can be seen by relabeling
the dummy indices ”t1 · · · tn”.Hence A.0.13 is:
(Ĥ0 − E0) |Ψ0〉 =∞∑n=1
− (−i)n−1
n!· n×{
V̂ (0) (Integrate without t1) e�(t2+···+tn)T [V̂ (t2) · · · V̂
(tn)]
− � (Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)]}|Φ0〉
(n for n same contributions from
∫ 0−∞
dt1 · · ·∫ 0−∞
dtn.)
= −V̂ (0)Û�(0,−∞) |Φ0〉 (the first term)
+ �
∞∑n=1
(−i)n−
(n− 1)!(Integrate) e�(t1+···+tn)T [V̂ (t1) · · · V̂ (tn)]
|Φ0〉
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The second term is dealt with as follows:
(−i)n−1
(n− 1)!(Integrate) T [V̂ (t1) · · · V̂ (tn)](assuming V̂ = g ·
V̂ ′)
=(−i)n−1
(n− 1)!(Integrate) gnT [V̂ ′(t1) · · · V̂ ′(tn)]
= ig∂
∂g
(−i)n
n!gn (Integrate) T [V̂ ′(t1) · · · V̂ ′(tn)]
= ig∂
∂g
(−i)n
n!gn (Integrate) T [V̂ (t1) · · · V̂ (tn)]
Hence:
second term = �
∞∑n=1
ig∂
∂g
(−i)n
n!(Integrate) T [V̂ (t1) · · · V̂ (tn)] |Φ0〉
= i�g∂
∂g
∞∑n=0
(−i)n
n!(Integrate) T [V̂ (t1) · · · V̂ (tn)] |Φ0〉
(since∂
∂g(zeroth term) = 0)
= i�g∂
∂gÛ�(0,−∞) |Φ0〉
= i�g∂
∂g|Ψ0(�)〉
Together we have (noting that V̂ (0) is actually the interacting
part of realHamiltonian V̂ ):
(Ĥ0 − E0) |Ψ0(�)〉 = −V̂ |Ψ0(�)〉+ i�∂
∂g|Ψ0(�)〉 (A.0.15)
Write it in another way:
(Ĥ0 + V̂ − E0) |Ψ0(�)〉 = (Ĥ − E0) |Ψ0(�)〉
= i�g∂
∂g|Ψ0(�)〉 (A.0.16)
Multiplying left with 〈Φ0|〈Φ0|Ψ0(�)〉 , we have:
RHS = i�g〈Φ0| ∂∂g |Ψ0(�)〉〈Φ0|Ψ0(�)〉
(since∂
∂g〈Φ0| = 0)
= i�g
∂∂g 〈Φ0|Ψ0(�)〉〈Φ0|Ψ0(�)〉
= i�g∂
∂gln 〈Φ0|Ψ0(�)〉 (A.0.17)
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-
while:
LHS =〈Φ0|Ĥ|Ψ0(�)〉〈Φ0|Ψ0(�)〉
= E − E0 (A.0.18)
So we have:
i�g∂
∂gln 〈Φ0|Ψ0(�)〉 = E − E0 (A.0.19)
From above expression, we notice that lim�→0+ � ln 〈Φ0|Ψ0(�)〉 is
finite, otherwisewe would have E = E0 when �→ 0+, which is absurd.
Hence, ln 〈Φ0|Ψ0(�)〉 ∝ 1� ,or 〈Φ0|Ψ0(�)〉 ∝ e1/�.
Now we calculate
(H − E0 − i�g∂
∂g)|Ψ0(�)〉〈Φ0|Ψ0(t)〉
=(Ĥ − E0 − i�g ∂∂g ) |Ψ0(�)〉
〈Φ0|Ψ0(�)〉+ i�g
|Ψ0(�)〉 ∂∂g 〈Φ0|Ψ0(�)〉
〈Φ0|Ψ0(�)〉2
= (by A.0.16 the first term vanishes) 0 +|Ψ0(�)〉〈Φ0|Ψ0(�)〉
· i�g ∂∂g
ln 〈Φ0|Ψ0(�)〉
= (by A.0.19)|Ψ0(�)〉〈Φ0|Ψ0(�)〉
· (E − E0) (A.0.20)
That is: (Ĥ − E − i�g ∂
∂g
)|Ψ0(�)〉〈Φ0|Ψ0(�)〉
= 0 (A.0.21)
Now by assumption |Ψ0(�)〉〈Φ0|Ψ0(�)〉 is finite when �→ 0+, we
have:
Ĥ|Φ0〉〈Φ0|Ψ0〉
= E|Ψ0〉〈Φ0|Ψ0〉
(A.0.22)
Q.E.D.
References
[1] http://we-taper.github.io/
[2] Alexander L. Fetter, John Dirk Walecka, Quantum Theory of
Many-ParticleSystems
[3] Gerald D. Mahan, Many-Particle Physics (3rd)
[4] https://en.wikipedia.org/wiki/Planck_units (accessed June 1,
2016).
[5] Negele & Orland, Quantum Many-particle Systems.
14
http://we-taper.github.io/https://en.wikipedia.org/wiki/Planck_units
NoteNatural UnitsInteraction PictureEquation of Motion for
ObservablesEquation of Motion for wave functionThe evolution
operator in Interaction picturePerturbative Solution for Time
Ordering Operator T
Adiabatic Switch on
TodoProof of Gell-Mann and Low Theorem