Hilbert Space Methods Used in a First Course in

Quantum Mechanics:

Green’s Functions in Quantum Theory

Victor Polinger

Physics/Mathematics

Bellevue College

02/12/15 – 03/12/15

Outline • Perturbation Theory in Quantum Mechanics.

• Permutation Symmetry of a System of Identical Particles. Bosons and Fermions.

• Second Quantization for Bosons.

• Green’s Functions for an Ideal Gas of Free Particles.

• The Adiabatic Hypothesis. Wick’s Chronological T-Operator.

• Perturbation Series for Green’s Functions.

• Wick’s Theorem.

• Feynman Diagrams.

• Dyson’s Equation and Renormalization.

Perturbation Theory in Quantum Mechanics

Part 1. The Stationary Case. No Time Dependence

Finding approximate solution of the Stationary Schrödinger’s Equation: HΨ = EΨ.

Let is a small parameter. Assume the eigenvalue problem with

H0 is solved, . We are trying to find approximate solutions of

perturbed eigenvalue problem: . Its matrix version is:

It is true if

0 , H H V

0 0 0

0 n n nH E

0H V E

0

0 0 0 0 0 0

0

0 1 2 0 1 22 2

0

, ,

..., ...

m m

m

m m m m m km k km k m

m m m

k k km m

m

m m m m

c

H V c E c V V V V

E E E

E E V

c

c c

E c c c

0 0 0 0

1 0 0 1 1 0

0 0 0 0, ,

k n k n

n nn n n k n k

k nn k n k

V VE V V c k n

E E E E

1 1

kn n kV E E

Perturbation Theory in Quantum Mechanics

Part 2. The Time-Dependent Case.

Finding approximate solution of the Time-Dependent Schrödinger’s Equation:

with being a small parameter. Assume the eigenvalue problem with

H0 is solved, . We are trying to find approximate solutions of perturbed

eigenvalue problem: . Its matrix version is:

It is true if

0 , H H V

0i H Vt

h

0

0 0

0

0 00 0

0 1 22

11

Fi

,

,

.

rst-order corre

..

ction:

km

kn

m m

m

m m m m

m m

i t k mkm k

kkm m

m

i tkkm k

m km km

m m m

n

m

k kn

t c t

H V c E c

E EV t V t V e

c t

ci V t c

t

c i ii V t c t V t dt V e

c t

dt

c t c t

h

hh h

h

t 1 1

kn n kV E E

i Ht

h

0 0 0

0 n n nH E

Permutation Symmetry of a System of Identical Particles. Bosons and Fermions.

Permutation Symmetry of Identical Twins.

Particle 1 Particle 2

1 2 2 1

2

1 2 1 2 1 2

22

1 2 2 1 1 2 2 1

1,2 , ,

1, 2 1,2 , , ,

1 1 1

1 meaning 0 or

1, 2 , , or 1, 2 , ,

i

i

i i i

i

P r r e r r

P P r r e r r r r

e e e

e

P r r r r P r r r r

The Bose-Einstein case The Fermi-Dirac case

Permutation Symmetry of a System of Identical Particles. Non-Interacting Particles.

Bosons (phonons, plasmons, magnons, etc.):

Fermions (electrons, nucleons, mesons, etc.):

1 2

1 2 1 2,mn m n

H H r H r

r r r r

1 2

1 2 1 2 2 1

1 2

1 2

1 2

1,

2

1,

2

mn m n m n

m m

mn

n n

H H r H r

r r r r r r

r rr r

r r

Second Quantization for Bosons

• The Harmonic Approximation: Modeling the true potential energy, V(x), with the parabola, U(x) = ½Kx2.

• Introducing operators a and a† .

As , we have

22 2

222 2 2

1,

1 1

,

2

2 2

2 2

pH m x p m x

p KH Kx K m

m

m m

m

†1 1,

2 2

d da m x a m x

dx dxm m

h h

h h

dp i

dx h

†1 1,

2 2a ip m x a ip m x

m m

h h

Second Quantization for Bosons

Since , we have [a, a†] = aa† - a†a = 1.

Then

Therefore,

As aa† - a†a = 1, we have aa† = 1 + a†a and, therefore,

, 1x x xx x x

2

†

2

†

22

22

1

2

1

2

1

2

1

2

aa ip m x ip m xm

a

p im xp px m xm

p im xp px

a ip m x ip

m

m x

m x

m

h

h

h

h

2 22 2

2 22 2 †

†

†

† 1

2

1

p im xp px m x p im xp px m xm

p m x

aa

p m x m aa a am

a a

h

hh

22 † † 1

22 1 2p m x m a a m a a h h

Second Quantization for Bosons

Plugging into , we

come to

• Theorem 1: Let ψ(x) be eigenfunction of H = ħω(a†a + 1) with the eigenvalue E.

Then Ψ(x) = a†ψ(x) is another eigenfunction of H with the eigenvalue E + ħω

• Proof: H Ψ = H a†ψ = ħω(a†a + 1)a†ψ = ħω(a†aa† + 1a†)ψ

Plugging aa† = 1 + a†a, we have

H Ψ = ħω[a†(1 + a†a) + 1a†]ψ = a†ħω[a†a + 1 + 1]ψ

= a†[ħω(a†a + 1) + ħω]ψ = a†[H + ħω]ψ

= a†[E + ħω]ψ = [E + ħω]a†ψ = [E + ħω]Ψ

221

2H p m x

m

† 12

H a a h

22 † 1

22p m x m a a h

Second Quantization for Bosons

• Theorem 2: Let ψ(x) be eigenfunction of H = ħω(a†a + 1) with the eigenvalue E.

Then Ψ(x) = aψ(x) is another eigenfunction of H with the eigenvalue E - ħω

• Proof: H Ψ = Haψ = ħω(a†a + 1)aψ = ħω(a†aa+ 1a)ψ

Plugging a†a = aa† - 1, we have

H Ψ = ħω[(aa† - 1)a + 1a]ψ = aħω[a†a + 1 - 1]ψ

= a [ħω(a†a + 1) - ħω]ψ = a [H - ħω]ψ

= a [E - ħω]ψ = [E - ħω]aψ = [E + ħω]Ψ

• How deep can we go down in energy?

Total energy:

So, there is a minimal energy, E0, the system can have.

The corresponding state, ψ0, is called ground state: aψ0 = 0.

The “ladder” of equidistant energy

levels of a harmonic oscillator

2 2

min min2 2

p pE V x V V

m m

0

1 1 0

2 2

d da m x m x

dx dxm m

h h

h h

Second Quantization for Bosons

• Ground state: We come to separable differential equation,

Its solution is

• Excited states:

• n the number of “elementary excitations”

00 0

dm x

dx

h

2

1 4

0 0 0 with and 2

x m mx N e N

h h

2 2

1 4

†

1 0 1 1

1 1with

2 2

x xd ma m x Ne N xe N

dxm

h

hh

The first four stationary states of the harmonic oscillator

1 1 ††

22 n H aE n a a an h h

The Heisenberg representation

• Time dependence:

• The Schrödinger representation The Heisenberg representation

• Wave function of a system of several non-interacting oscillators:

• Ground state: |0, 0, 0, … , 0 = |0

1 2 31 2 3 1 2 3 1 2 3, , , ,n n nx x x x x x n n n

Occupation-number representation

Vacuum state

*

, , iEt i t

i t i t

i x t x t x e e xt

x e A x e x dxA

H

H H

Hh h

h h

h

, iEt i tx t x e e x

A A x

Hh h

†

, ,

,

, , , , ,

i t

i t i t

x x U t e

A x t e A x e U t A x U t

A x t A x t H A x t H HA x tt

H

H H

h

h h

Interaction representation

So far, . What if ? Assume we know solutions of the unperturbed eigenvalue problem,

.

We want to solve the time-dependent Schrödinger equation for the perturbed case,

Expanding the perturbed wave function in terms of the orthonormal basis set of unperturbed states,

Consider the function

This change is achieved using the operator of unitary transformation

Respectively, for operators,

0H x H x V x

0 00 0 0 0

0 , , , , ni E t

n n n n nH x t E x t x t e x

h

0 01,i H t i H t

A x t A x A xU U e e

h h%

0, ,H x t H x V x t

,, ,

x t iH x t x t

t h

,x t

0 0

0 00 0

0, , ni E t

n n n n

n n

i H t i H t

n n n n

n n

x t c t x t c t e x

c t e x e c t x

h

h h

0 00 0

0 , ,i H t i i H t

n n n n

n n

H tx t x t e c t c te x xe

h h h%

0i H tU e

h

This is the interaction reperesentation

Interaction representation. Part 2.

Consider time dependence of . Its time derivative is

Thus, . Converting it into an integral

equation, we have:

,, ,

x t iV x t x t

t h

%% %

0 0

0 0

0

0

0 0 0

0

0

,, ,

, ,

,

, ,

, ,

i H t i H t

i H t i H t

i H t

i H t

i H t i H t i H t

x tx t x t

t t t

ix t H x t

h

iH H x t

h

iV x t x t

h

i

e e

ie H e

e

V x t

e

e e e x th

h h

h h

h

h

h h h

h

%

0 , ,i H t

tex t x h%

0

1 1 1,( )

t

n V xi

t t t dt

% % %%h

The Adiabatic Hypothesis. Wick’s Chronological T-Operator. Part 1.

The Adiabatic Hypothesis: Assume λV (x) turns on very slowly from V(x, -∞) = 0 to V(x,

t) at a finite time t. This means V (x, t) = eεtV (x) with ε =+0. Then H (x, t) =H0(x) + λV (x, t).

Picard’s Method of Successive Approximations:

0

1 1 1,( )

t

n V xi

t t t dt

% % %%h

1

0 0

1 1 1

1 2 2 2 1

1 2 3

1

0 0

0

2

0

1

,

, ( )

(

,

, ,

,

) ( )

...

( )

t

n n n

tt

n n n

nn n

k

n

nn n

k

it t t t dt

i it t t t dt dt

t t t t

it

V x

V x V x

dt V x dt

S

V

t

t

%

% %

%

% % % % %h

% %h h

% % % %

%h

% %

%

%

%

11 2

2 3

0

3( ) ( )... ( ), , , ktt tt

k k nt dt t dtV x x tx V

%% %

The Adiabatic Hypothesis. Wick’s Chronological T-Operator. Part 2.

The evolution operator, , is:

Here t > t1 > t2 > t3 >… . The connected integrals could be simplified if would be just a c-number, not an operator. At different times, the commutator is NOT zero:

1 1 2

1 2

2 3

1 1 1 2 1 21 3 32, ,1 ( ) ( ) ( , ) ( ) ( ) ( ) .., ,

,

.,

t t tt t t

V x dt Vi i i

x dt V x dt V x dt V x dtt dt t t t

S t

Vt tx

% % % % % %h h h

0, n nt S t %%

0 0( ), i iH t H t

t V xV x e e

h h%

0 1 0 1 0 2 0 2 0 2 0 2 0 1 0 1

0 1 0 1 2 0 2 0 2 0 1 2 0 1

1 2 1 2 2 1( ), ( ) ( ) ( ) ( ) ( ), , , , ,

0

,

i i i i i i i iH t H t H t H t H t H t H t H t

i i i i i iH t H t t H t H t H t t H t

t t t t t t

e V x e e V x e e V x e e V x e

e V x e V

V x V x V x V x V x V

x e e V x e V x e

x

h h h h h h h h

h h h h h h

% % % % % %

The Adiabatic Hypothesis. Wick’s Chronological T-Operator. Part 3.

Consider the 2nd-order term.

Thus,

1

2 2

1 2 1 2 2 11 2 2 1 1 2( ) ( ) ( ), , , , , ( ) , ( ) ( )

tt t t t t

t t

dt dt V x V x dt dt V x V x dt dtt t t t V x V xt t

% % % % % %

Changing the order of integration Renaming t1 t2

1 1

2

1 2 1 2 2 1

1 2

1 2 1 2 1 2, , , , , ,1 1

( ) ( ) ( ) ( ) ( ) ( )2 2

Here

t tt t t t

t

dt dt V x V x dt dt V x V x dt dt V x V xt t t t t t

t t

% % % % % %

2 1 Here t t

Operators are chronologically ordered

The Adiabatic Hypothesis. Wick’s Chronological T-Operator. Part 4.

Thus, the second-order term can be written in the following form:

Similar T-ordering can be done in all other terms. Then

1 1

1

1 2 1 2 1 2

1

1 2 2

1

1 2

1 2 1 2 1 2

1 2 2

1 2

1 1( ) ( ) ( ) ( ) ( ) ( )

2 2

1( ) ( ) ( )

2

, , , , , ,

, , ,

1( ) ( )

2, ,

t tt t t t

t

tt t

t

t

dt dt V x V x dt dt V x V x dt dt V x V x

dt V x dt V x dt V x

dt dt

t t t t t t

t t t

t tV x V

T

x

T T

T

% % % % % %

% % %

% %

2

1 1

1( ),

2

t

t

t dtTV x

%

2 32 3

1 1 1 1 1 1

1 1

1 11 ( ) ( ) ( ) ...

2 3!

e

, , ,

,

,

xp ( )

t t t

t

V x VTi i i

t dt t dt t dt

it dt

x V x

V xT

S t

%h h

%h

% %h

Gian Carlo Wick 1909 - 1992

Definition of T-ordering: TA(t1)B(t2) = A(t1)B(t2) if t1 > t2

TA(t1)B(t2) = B(t2)A(t1) if t1 < t2

In general,

2

1

2 1 1 1 2 1, exp ( ) , ,

t

t

iS t t t dtV x t tT

%h

Also, S(t2, t1)S(t1, t2) = 1. Then S(t2, t1) = S-1(t1, t2). Besides, S(∞, t)S(t, -∞) = S(∞, -∞). Therefore, S(t, -∞) = S-1(∞, t) S(∞, -∞).

Quantum Transitions Due to a Periodic Perturbation

Let, say, a linearly polarized electromagnetic wave with Ex(t) = E0e

iΩt is incident on a quantum system, say, an atom. If d is the electric dipole moment of the quantum system, the respective coupling is the dot product E ∙ d = E0dxe

iΩt . It generates quantum transitions from the ground state, Ψ0, to excited states Ψn. This is observed as absorption of the electromagnetic wave by the quantum system. Intensity of the absorption is proportional to

Plugging , it can be transformed in the following way:

Here I(t) is The Fourier transform of the absorption intensity I(Ω),

2

00 0 0 2 20

1 with , = lim n

n x n n

n

E EI d

h

0 02 2

0 0

1 1 1

2 2 2

n ni t i ti t i t

x n x n

n n

I d e dt e d e dt e I t dt

0

0

1

2

ni t

n e dt

00

0

2

0 0 0

0 0 0 0

nn

n

i E E ti t

x n x n n x

n n

i E t i E t i Ht i Ht

x n n x x n n x

n n

I t d e d d e

e d e d e d e d

h

h h h h

Correlation Functions and Green’s Functions

0 0 0 0

i Ht i Ht i Ht i Ht

x n n x x n n x

n n

I t e d e d e d e d

h h h h

This is Heisenberg time dependence,

dx(t)

This is just 1 Thus, I(t) = Ψ0|dx(t)dx(0)|Ψ0 Equivalently, I(t) = 0|dx(t)dx(0)|0 In general, for any two operators A and C, we have IAC(t1 - t2) = A(t2)|C(t1) = 0|A(t2)C(t1)|0

This is the so-called “correlation function”

The T-ordered (chronological) Green’s function is: DAC(t1 - t2) = (i/ħ)0|TA(t2)C(t1)|0.

Changing to the interaction representation, Â(t2) = S-1(-∞, t2) A(t2)S(-∞, t2)

And Ĉ(t2) = S-1(-∞, t1)C(t1)S(-∞, t1), we come to DAC(t1 - t2) = (i/ħ)0|T S-1(-∞, t2)Â(t2) S(-∞, t2)S

-1(-∞, t1) Ĉ(t1)S(-∞, t1)|0 = (i/ħ)0|S-1[TÂ (t2)Ĉ(t1)]S|0 Here S = S(-∞, ∞). At the same time, S(-∞, ∞)|0 = eiα|0. Then 0|S-1 = 0|e-iα → 0|S(-∞, ∞)|0 = eiα. Then e-iα = 0|S(-∞, ∞)|0-1 . Therefore, 0|S-1 = 0|S(-∞, ∞)|0-1 0|.

Thus,

2 1

2 1

ˆ ˆ0 0( )

0 0AC

A t C t SD

Tt t

S