Feynman Diagram Methods - NPTELnptel.ac.in/courses/115106085/downloads/Unit_4_Feynman_Diagram_… · Select/Special Topics from ‘Theory of Atomic Collisions and Spectroscopy’

Select/Special Topics from ‘Theory of Atomic Collisions and Spectroscopy’

PCD STiTACS Unit 4

Feynman Diagram Methods 1

P. C. Deshmukh

Department of Physics

Indian Institute of Technology Madras

Chennai 600036

Unit 4 Lecture Number 25

Feynman Diagram Methods

Schrodinger, Heisenberg and Dirac ‘pictures’

Primary references: [1] Fetter and Walecka – Quantum Theory of Many-Particle Systems [2] Raimes – Many Electron Theory

2 PCD STiTACS Unit 4 Feynman Diagram Methods

2

HF-SCF :

2.21 0.916

s s

HF

FEG in jellium potential

Rydr r

E

N

: Bohr unitssr

Average HF energy

per electron

0

4

2

2

2

1 13.60569... 2

1 0.5292... A

meRyd eV

Bohr unitme

Second (and higher) Order Perturbative treatment of the

electron-electron Coulomb interaction however diverges.

electron-electron interaction, reduces the energy BELOW that of

the Sommerfeld gas (of course in the positive jellium potential)

First Order Perturbative treatment of the exchange term SAME RESULT

STiTACS Unit 3 (RPA);

L19, L20; Slides # 96, 97


Bohm & Pines: mid-fiftees

D.Pines (1963) Elementary excitations in solids (Benjamin, NY)

2 42

322

2.21 0.916 3 0.916

2 482

cBP

s s s fs

kE

r r r kr

2

:

2.21 0.916

s s

HF

For free electron gas in jellium potential

Rydr r

E

N

Random Phase Approximation

Need!

Many-body

theory – beyond perturbation

methods

kc : Upper bound to wave number of plasma oscillations

Lower bound to wave length; since oscillations get

damped by the random thermal motion of the electrons.


‘Exact

Solution’ ? “Having no body at all is already too many”

– G. E. Brown

( ) ( ) ( ) ( )

1 1 1( ,.., ) ( ,.., ) ( ,.., )N N N N

N N NH q q q q E q q

2

( )

1 1

1

2

N NiN

i i ji ij

i ZH

m r r

Many

electron

problem

Non – Perturbative Methods / eg. RPA

Alternative techniques:

Configuration interaction methods:

Multi-Configuration Hartree-Fock (MCHF)

Multi-Configuration Dirac-Hartree-Fock (MCDHF)

Feynman Diagram

Methods


The above Hamiltonian

not an explicit function of time

If we can treat this term as if it has a time-

dependence, then the mathematical procedure

that would enables us to do so, would also

provide access to powerful methods using the

INTERACTION (Dirac) “PICTURE”.

2

( )

1 1

1

2

N NiN

i i ji ij

i ZH

m r r


The Nobel Prize in Physics 1933

Erwin Schrödinger, Paul A.M. Dirac

The Nobel Prize in

Physics 1932

Werner Heisenberg

"for the creation of quantum

mechanics, the application of

which has, inter alia, led to the

discovery of the allotropic

forms of hydrogen"

"for the discovery of new productive forms of atomic theory"

Schrodinger picture

Heisenberg picture

Dirac (Interaction) picture

http://www.nobelprize.org/nobel_prizes/physics/laureates/2013/


Schrodinger picture

Heisenberg picture

Dirac (Interaction) picture

‘picture’

‘representation’

, ,H r t i r tt

time-evolution: , ,0 Hi t

r t re

, ,0 Ei t

r t re

2 31 1

1 ......2! 3!

Hi t H H Hi t i t i te

,0 ,0H r E r stationary states

Time dependent wave

function

Schrodinger picture


, ,0 Ei t

r t re

State functions are

TIME DEPENDENT

, , INDEPENDENT

of time

OPERATORS q p H

Schrödinger picture

, ,S SH r t i r tt

Schrödinger equation in the

Schrödinger picture PCD STiTACS Unit 4 Feynman Diagram Methods


UNITARY TRANSFORMATIONS

- leave the ‘physics’ invariant…..

-Rotation of basis in the Hilbert space

-‘Generalized’ rotations from one ‘picture’ to ‘another’

† ; H H

i t i t

O e O e

, ,0H r t r

time independent…

wavefunction at t=0

H H

i t i t

H Se e

Heisenberg Picture

( )

( )

H

H

fn time

fn time



- leave the ‘physics’ invariant…..

-Rotation of basis in the Hilbert space

-‘Generalized’ rotations from one ‘picture’ to ‘another’ † ;

H Hi t i t

O e O e

, , , H

i t

H H S Sr t O r t e r t

, ,0 E

S S

i tr t re

, ,0 ,0 H E

i t

H

i tr t e r re

, ,0H r t r

time independent…

wavefunction at t=0

H H

i t i t

H Se e

but ( )H H H St H H fn t

Heisenberg Picture


H H

i t i t

H Se e

+ H H H H

i t i t i t i t

H S Se e e et t t

+ H H H H

i t i t i t i t

H S S

iH iHe e e e

t

H H H H

i t i t i t i t

H S S

iH ie e e He

t

H H H H

i t i t i t i t

H S S

iH ie e e e H

t

,H H

iH

t

, H Hi Ht


H H

i t i t

H Se e

H H H H

i t i t i t i t

H S S

iH ie e e e H

t

H H H

iH iH

t

note: Hamiltonian same in HP as in SP

H H

i t i t

H S SH e H e H



Hi t

O e

H H

i t i t

H Se e

Heisenberg Picture

( )

( )

H

H

fn time

fn time

Dirac Picture

Interaction

Picture

0 1H H H

, , H

i t

H Sr t e r t

0 0 0

& , , H H H

i t i t i t

I S I Se e r t e r t

I IBOTH and are functions of time

( )

( , ) ( )

S

S

fn time

r t fn time

Schrodinger Picture

oHi t

O e


0 1H H H

0 0

0

( )

, ,

H Hi t i t

I S

Hi t

I S

t e e

r t e r t

oHi t

O e

, ,S SH r t i r tt

Schrodinger equation

0

0 1 , , H

i t

S IH H r t i e r tt

0

, , H

i t

I Se r t r t

Unitary transformation

operator that effects

transformation to the

Dirac/Interaction picture

↑

Soluble part


0

0 1, , , H

i t

S S IH r t H r t i e r tt

0

, , H

i t

I Sr t e r t

0

0 1 0, , , + ,H

i t

S S S IH r t H r t H r t i e r tt

0 0

00 1, , , + ,

H Hi t i t

S S I I

HH r t H r t i i e r t i e r t

t

0

0 1 , , H

i t

S IH H r t i e r tt


0 0

1 , ,H H

i t i t

I IH e r t i e r tt

( ) , ,I I IH t r t i r tt

0

, ,H

i t

S Ir t e r t

0

1 , ,H

i t

S IH r t i e r tt

, ,H r t i r tt

Schrodinger equation

00

1 , ,H

i ti

I

H

I

t

H e r t i re tt

Just “like”



, ,H r t i r tt

Schrodinger equation Just “like”

0 0

1( ) H H

i t i t

IH t e H e

0 0

0

( )

, ,

H Hi t i t

I S

Hi t

I S

t e e

r t e r t

in accordance with

Transformation of only the ‘correlation’ part of

the Hamiltonian that remained to be treated in

the Schrodinger picture formalism.

0 1H H H 0: note the role of However H

The subscript I denotes the transformation

to the INTERACTION PICTURE of the

difficult/correlation part of the

Hamiltonian.


0 0

1( ) H H

i t i t

IH t e H e

0 1H H H

Notation

1H


0 0

1( ) H H

i t i t

IH t e H e

Transformation of only the ‘correlation’ part of

the Hamiltonian that remained to be treated in

the Schrodinger picture formalism.

0 1H H H


0 0

0

( )

, ,

H Hi t i t

I S

Hi t

I S

t e e

r t e r t

0 1

, ( )

is determined

I r t fn time

Time dependence by both H and H


0 0 0

& , , H H H

i t i t i t


oHi t

O e

Unitary transformation

operator that effects

transformation to the

Dirac/Interaction picture

0 1H H H


0 0

1( ) H H

i t i t

IH t e H e

Recall

1: 0

: ( ) 0I

IF H

THEN H t

, 0

time-independent

I

I

i r tt

Interaction/Dirac picture ↔Heisenberg picture

1 1 0 0, ,I It U t t U t t t


0

,

Time Development Operator

U t t

0 0,I It U t t t

,I It U t t t

1 3 1 2 2 3

, 1

, , ,

U t t

U t t U t t U t t

Reference: Fetter and Walecka –

Quantum Theory of Many-Particle Systems,

Chapter 3

FW/Eq.6.11/page 55

0 0

1

0 0

1 , , ,

, ,

U t t U t t U t t

U t t U t t

Existence of UNIT operator

Closure property

Existence of INVERSE

‘GROUP’ properties


0 0,I It U t t t

†

0 0

† 1

0 0

, , 1

, ,

U t t U t t

U t t U t t unitary

†

0 0 0 0, ,I I I It t t U t t U t t t

norm 0 0I I I It t t t



Recall:

0 0,I It U t t t

0 0 0 0( ) , ,I I IH t U t t t i U t t tt

0 For arbitrary I t

0 0( ) , ,IH t U t t i U t tt

Equation of

motion for the

time-development

operator.

PCD STiTACS Unit 4 Feynman Diagram Methods 24

0 0 0

& , , H H H

i t i t i t


time-evolution of Schrodinger state:

, ,0 Hi t

r t re

0

0 0 : , , H

t tiIf initial time is t r t r te

0 0

0

0, , ,H H H

i t i t i t t

I S Sr t e r t e e r t

0 0

0

0, , ,H H H

i t i t i t t

I S Sr t e r t e e r t

0

0

0 0, ,H

i t

S Ir t e r t

0 0

0 0

0, ,H HH

i t i t t i t

I Ir t e e e r t

0 0

0 0

0, H HH

i t i t t i t

U t t e e e

Time Evolution: t0 to t



0 0

0 0

0, H HH

i t i t t i t

U t t e e e

† † †AB B A

0 0

0 0†

0, H HH

i t i t t i t

U t t e e e

† 1

0 0, ,U t t U t t

27 PCD STiTACS Unit 4 Feynman Diagram Methods Questions: [email protected]

0 0( ) , ,IH t U t t i U t tt

Equation of

motion for the

time-development

operator.

0 0

0 0

0, H HH

i t i t t i t

U t t e e e

Formal solution

0 , 0 hence the order of the

operators is important.

In general H H

Next class: Dyson’s Chronological Operator


28

P. C. Deshmukh



Chennai 600036

PCD STiTACS Unit 4 Feynman Diagram Methods



Dyson’s Chronological Operator

Primary references: Fetter and Walecka – Quantum Theory of Many-Particle Systems Raimes – Many Electron Theory


0 0( ) , ,IH t U t t i U t tt

0 0

0 0

0, H HH

i t i t t i t

U t t e e e

Equation of

motion for the

time-development

operator. Formal solution

Today:

we develop an integral equation for the time

development operator

0 0, , ,I Ir t U t t r t

I

I I

I Time evolution

of interaction

picture states

Differential equation for the

Time Evolution Operator


, ' '( ) , ' ( ) , 'I Ii U t t H t U t t H t U t tt

Raimes

– Many Electron Theory

Eq.5.32 page 97

Fetter and Walecka

– Quantum Theory of Many Particle Systems

Eq.6.17, page 56

0 0, ( ) , Ii U t t H t U t tt

31

0 0, ( ) , Ii U t t H t U t tt

0 0i.e. , ( ) , I

iU t t H t U t t

t

0 0

0 0' ', ' ( ') ', '

t t

It t

idt U t t dt H t U t t

t

0 integrating from to : t t

0

0 0 0 0 , , ' ( ') ', t

It

iU t t U t t dt H t U t t

0

0 0 , =1 ' ( ') ', t

It

iU t t dt H t U t t

0 0 , 1 U t t


We are dealing with operators….

….so we must preserve the ‘ordering’ 32 PCD STiTACS Unit 4 Feynman Diagram Methods

0 , time development operators

not ordinary functions of time t

U t t

0

0 0 , =1 ' ( ') ', t

It

iU t t dt H t U t t

Independent variable t appears as UPPER LIMIT

If U were ordinary functions, then such integrals

would be “VOLTERRA INTEGRALS”

Iterative solutions available; which are guaranteed to converge in the case of ordinary functions …… …… we attempt similar procedures in the present case.


0

0 0 , =1 ' ( ') ', t

It

iU t t dt H t U t t

0

'

0 0 ', =1 '' ( '') '', t

It

iU t t dt H t U t t

0 0

'

0 0 , =1 ' ( ') 1 '' ( '') '', t t

I It t

i iU t t dt H t dt H t U t t

0 0 0

2'

0 0, =1 ' ( ') ' ( ') '' ( '') '', t t t

I I It t t

i iU t t dt H t dt H t dt H t U t t

0 0 0

2'

0 0, =1 ' ( ') ' '' ( ') ( '') '', t t t

I I It t t

i iU t t dt H t dt dt H t H t U t t

Order of the operators is important: left ↓right↓


0 0 0

2'

0 0, =1 ' ( ') ' '' ( ') ( '') '', t t t

I I It t t

i iU t t dt H t dt dt H t H t U t t

0 0 0

2'

0, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I It t t

i iU t t dt H t dt dt H t H t

0

''

0 0 '', =1 ''' ( ''') ''', t

It

iU t t dt H t U t t

0 0 0 0

0 0

' '

'

1' '' ( ') ( '') ' '' ( ') ( '')

2

1 ' '' ( ') ( '')

2

t t t t

I I I It t t t

t t

I It t

dt dt H t H t dt dt H t H t

dt dt H t H t

Consider the 3rd term

35

''t

't

' ''t t

0t


0

0

'

''

t t t

t t t

0t

'' 't t

' ''t t

0 0 0 0

0 0

' '

'

1' '' ( ') ( '') ' '' ( ') ( '')

2

1 ' '' ( ') ( '')

2

t t t t

I I I It t t t

t t

I It t


dt dt H t H t

''t t

't t

Fetter & Walecka / Fig.6.1 / page 57

Raimes / Fig.5.1/page 100

Integration is over the variables t’ and t’’

36

''t

't

' ''t t

0t


0

0

'

''

t t t

t t t

0t

'' 't t

' ''t t

0 0 0 0

0 0

' '

'

1' '' ( ') ( '') ' '' ( ') ( '')

2

1 ' '' ( ') ( '')

2

t t t t

I I I It t t t

t t

I It t


dt dt H t H t

''t t

't tFetter & Walecka / Fig.6.1 / page 57

Raimes / Fig.5.1/page 100

Integration is over the variables t’ and t’’


x

x

cx x

( ) 1cx x

Heaviside

step function

; 0

; 1

c

c

x x x

x x x

0 0 0 0

0 0

' '

'

1' '' ( ') ( '') ' '' ( ') ( '')

2

1 ' '' ( ') ( '')

2

t t t t

I I I It t t t

t t

I It t


dt dt H t H t


0 0

0 0

0 0

'

'

' '' ( ') ( '')

1 ' '' ( ') ( '') ' ''

2

1 ' " ( ') ( ")

2

t t

I It t

t t

I It t

t t

I It t

dt dt H t H t

dt dt H t H t t t

dt dt H t H t

0 '' 't t t

Fetter & Walecka / Eq.6.21 / page 57

0 0 0 0

0 0

'

'

1' '' ( ') ( '') ' '' ( ') ( '') ' ''

2

1 ' " ( ') ( ")

2

t t t t

I I I It t t t

t t

I It t

dt dt H t H t dt dt H t H t t t

dt dt H t H t

0 0

0 0

0 0

'

' '' ( ') ( '')

1 ' '' ( ') ( '') ' ''

2

1 '' ' ( '') ( ') '' '

2

t t

I It t

t t

I It t

t t

I It t

dt dt H t H t

dt dt H t H t t t

dt dt H t H t t t


0 0

0 0

0 0

'

' '' ( ') ( '')

1 ' '' ( ') ( '') ' ''

2

1 ' " ( ') ( ") ' "

2

t t

I It t

t t

I It t

t t

I It t

dt dt H t H t

dt dt H t H t t t

dt dt H t H t t t

0 '' 't t t

Fetter & Walecka

Eq.6.21 / page 57

nd

'' '

in 2 term

t t


0 0

0 0

0 0

'

' '' ( ') ( '')

1 ' '' ( ') ( '') ' ''

2

1 '' ' ( '') ( ') '' '

2

t t

I It t

t t

I It t

t t

I It t

dt dt H t H t

dt dt H t H t t t

dt dt H t H t t t

Operators containing the latest time stand farthest to the left.

0 0

0 0

'

' '' ( ') ( '')

1' '' ( ') ( '') ' '' ( '') (

2') '' 'I I

t t

I It t

t t

t tI IH t H t t

dt dt H t H

t H t H t t t

t

dt dt


Combining the two terms:

0 0

0 0

'

' '' ( ') ( '')

1' '' ( ') ( '') ' '' ( '') (

2') '' 'I I

t t

I It t

t t

t tI IH t H t t

dt dt H t H

t H t H t t t

t

dt dt

41


Operators containing the latest time stand farthest to the left.

0 0 0 0

' 1' '' ( ') ( '') ' ''

2( ') ( '')

t t t t

It

I IIt t t

dt dt H t H t dt dt H tT H t

T: Time-ordered product of operators. Operators containing the latest time stand farthest to the left.



0 0 0

2'

0, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I It t t


0 0 0

0 1 2

0

1 2(1

, = . ). !

( ).. ( )

nt t

I It

It

n

n

t

nt

H t H t Hi

U t t dt dt dtn

tT


Generalizing:


1 2 ! ' - ' , ,....., nThere are n time orderings of the time labels t t t


0 0 0

2'

0, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I It t t


Fetter & Walecka / page 57,58


0 0 0

0 1 2

0

1 2(1

, = . ). !

( ).. ( )

nt t

I It

It

n

n

t

nt

H t H t Hi

U t t dt dt dtn

tT

2

'

, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I I

i iU t dt H t dt dt H t H t

1 21 2

0

( ) ( ).. ( )1

, = .. !

nt t t

I n

n

I InTi

U t dt dt d H t H t tt Hn

0Now, let t

1

, =1 with n

n

U t U

1 2 1 2

1 .. ( ) ( ).. ( )

!

nt t t

n n I I I n

iU dt dt dt T H t H t H t

n


1 2 1 2 1 2

2 1 2 1

if

if

T A t B t A t B t t t

B t A t t t

1 2 1 2 1 2

2 1

if

I I I I

I I

T H t H t H t H t t t

H t H t

1 2 1 2 1 2

2 1 2 1

if

if

I I I I

I I

T H t H t H t H t t t

H t H t t t

1 2 : chronologically same! t t


1 2 2 1

I I I IT H t H t T H t H t


0 0 0

0 1 2 1 2

0

1, = .. ( ) ( ).. ( )

!

nt t t

n I I I nt t t

n

iU t t dt dt dt T H t H t H t

n

Consider nth term in this series:

0 0 0

1 2 1 2

1 .. ( ) ( ).. ( )

!

nt t t

n I I I nt t t

idt dt dt T H t H t H t

n

1 2

! ' - '

, ,....., n

There are n time orderings of the time labels

t t t

1 2 one of these n! ways is: ....... nt t t

1 2

1 1

, ,..., are dummy labels that get integrated out

and

( ) .. ( ).. ( ).. ( ) ( ) .. ( ).. ( ).. ( )

n

I I i I j I n I I j I i I n

t t t

T H t H t H t H t T H t H t H t H t


i↓ j↓ i↓ j↓

0 1 0; + full Hamiltonianas t H H H


0 1H H H Perturbation: ‘unfriendly’ part

‘solvable’ part

ADIABATIC “SWITCHING” of the PERTURBATION

0 1

tH H e H

: small (positive) ; in the end 0

0 ; 0 & soluble parttas t e H H

Perturbation is turned on and off very slowly

quasi-static (adiabatic) ;

end-results to be obtained independent of α.

Raimes – Many Electron Theory Eq.6.6 page 105

adiabatic switching

control parameter


0 1

tH H e H This mathematical device enables us use the

provisions of the INTERACTION PICTURE

very fruitfully.

adiabatic switching

control parameter

0 1 0; + full Hamiltonianas t H H H

: small (positive) ; in the end 0



0 1

tH H e H

INTERACTION PICTURE

0 0 0

, , H H H

i t i t i t


0 0

1 H H

i t i tt

IH t e e H e

0 0

1( ) H H

i t i t

IH t e H e

Perturbation: ‘unfriendly’ part,

controlled by the mathematical switch

adiabatic switching

control parameter


0 0, ( ) , Ii U t t H t U t tt

0 0 0

0 1 2 1 2

0

1, = .. ( ) ( ).. ( )

!

nt t t

n I I I nt t t

n


n

0 0

1 H H

i t i tt

IH e e H e

(1)

(2)

The following two results remain valid

under adiabatic switching when 1 1

tH e H

The time development operator must explicitly depend on α


0 0,I It U t t t

0 0( ) , ,IH t U t t i U t tt

Interaction

picture

0 0,I It U t t t

0 0

0

, ,

H Hi t i t

I S

Hi t

I S

e e

r t e r t

0 0 0

0 1 2 1 2

0

1, = .. ( ) ( ).. ( )

!

nt t t

n I I I nt t t

n


n

0 0 0

0

1 2 1 2

0

, =

1= .. ( ) ( ).. ( )

!

nt t t

n I I I nt t t

n

U t t


n

0 1

tH H e H

0 1H H H

0 0

1 H H

i t i tt

IH e e H e

0

00

, ,0 HH

S H H

i t i tr t r re e


0 0 0 0

0 0

:

H E

time independent stationary eigenstate of H

Time evolution of the Schrodinger state if there

were no ‘correlations’


Two limits that concern us

and 0t

0 0

1 H H

i t i tt

IH e e H e


0

00

time-evolution: , ,0 HH

S H H

i t i tr t r re e

0 0

0 0

If there were NO correlations, , would evolve as:

, independent of time

I

H Hi t

I

i t

r t

r t e r re

0 1 0 t

tH H e H H

, 0I r tt

0 0 0 0 H E

0 time independent stationary eigenstate of H

How would the corresponding , evolve with time?I r t

0

, , H

i t

I Sr t e r t

,

I r t independent

of time

,0 ,0I Sr r


The eigenstates of H0 remain independent of

time in the interaction picture;

since in this case the perturbation H1=0.

0 1

,

t

As t increases from the perturbation is turned on

H H e H

0 0,I It U t t t

0 0

0 0

If there were NO correlations, , would evolve as:

, independent of time

I

H Hi t

I

i t

r t

r t e r re

When correlation is present


Next class:

What happens in the limit α→0?

Gell-Mann and Low theorem

Questions: [email protected]

Two limits that concerned us: & 0t

0 1

0

t

t

H H e H

H H

0,I r t

0

0

, 0

0, ,

0,

I

I

r t

U t t r t

U t t


54

P. C. Deshmukh



Chennai 600036




Gell-Mann and Low Theorem M. Gell-Mann and F. Low Phys.Rev. 84:350 (1951)

0 0 , 0 0, I r t U t t

0 0 0

, , H H H

i t i t i t


, , H H H

i t i t i t

H S H Se e r t e r t

0 1

tH H e H

, 0

, 0

, 0

, 0

, 0 , 0

H

H

S

S

I

I

r tr t

r t

r

t r

t

r

, ,0 E

S S

i tr t re

Schrodinger picture

0 0 0 0H r E r

55

0 0 ; , St

H H r r

Dirac / Interaction picture

Heisenberg picture


56

1: 0

: ( ) 0I

IF H

THEN H t

, 0

time-independent

I

I

i r tt

Interaction/Dirac picture ↔Heisenberg picture

0 0 0

, , H H H

i t i t i t


, , H H H

i t i t i t

H S H Se e r t e r t

0 1

tH H e H

0

0

1 0 =0, H=H & , ,0

, , = ,0

Ei t

S S

Hi t

I S S

If H r t e r

r t e r t r


57

1 00 1 0

,

i.e. t

t

t

As t increases from the perturbation is tur

H t H

ned

e

on

H e H HH H

0 , I r t

0, 0 0, ,I Ir t U t t r t

0 0

0 0

i.e.

0,

0,

U t t

U t t


Today,

we examine the following question:

What happens in the limit α→0?


Two limits concern us: (1) & (2) 0 t

0 1 0

0

;

: full Hamiltonian (with correlations)

t

t

tH H e H H H

H

Fetter & Walecka,

Quantum Theory of Many-Particle Systems

pages 60, 61

0

0 0

, 0 0, ,

0,

I Ir t U t t r t

U t t

0 0

00 0 0 0

0 1

0 0

0 00 0

(0, ) lim ,

(0, )

;

. .

UIf exists

U

then it is an eigenstate of H H H

i e H E



For ‘PROOF’, see:

Fetter & Walecka

Quantum Theory of

Many-Particle Systems,

page 61

0 1 0 t

tH H e H H

0 0 0 0 H E

Question: 0 1 ?How do we get eigenstate of H H H

0 0 , 0 0, I r t U t t

0 t

from which is an eigen-state of the

unperturbed Hamiltonian:

The eigenstate



0 1 0 t

tH H e H H

0 0 0 0 H E

0 0

00 0 00

(0, )lim

(0, )

U

U

0 1 of H H H develops ADIABETICALLY

0

0 0 0 0 H E

0 0 , 0 0, I r t U t t


0

, , 0 = , 0

0,

H S Ir t r t r t

U t t

The question we had asked: what happens in the limit α→0?

0 1 0 t

tH H e H H

0 0 0 0 H E

0 0

00 0 00

(0, ) lim

(0, )

U

U

00

lim 0, need not be well defined. U t t

Limit of

the RATIO

0 0 , 0 0, I r t U t t

0 1

0

0 t

tH H e H H


0 0

00 0 00

(0, )lim

(0, )

U

U

00

lim 0, need not be well defined. U t t

Limit of

the RATIO

The phase of the numerator diverges in

the limit α→0, but it is nicely cancelled

in the ratio by the denominator. Gell-Mann and

Low theorem

0 0 , 0 0, I r t U t t

The question we had asked: what happens in the limit α→0?


0 0

00 0 0 0

0 1

0 0

0 00 0

(0, ) lim ,

(0, )

;

. .

UIf exists

U


i e H E

Gell-Mann and

Low theorem

0 0

0 0

0 00 0

H E

0 0 0 0

0 00 0

HE

0

0 0 , 0 0, I r t U t t


0 0 0 0

0 00 0

HE

0 0 0 0 1 0 0 0

0 0 00 0 0

+

H HE

0 0 1 0 0 0

0 00 0

H HE

0 0 0 1 0 0 0

0

0 0 00 0 0

+

HE E

0 1 0

0

0 0

HE E

0 0 , 0 0, I r t U t t

0 0 0 0 = H E 0 1 0 t

tH H e H H

From Gellman & Low Theorem:


0 1 0

0

0 0

HE E

0 0 , 0 0, I r t U t t

0 0 0 0 = H E

0 1 0 t

tH H e H H

0 1 0

0

0 0

0,

0,

H U t tE E

U t t

Fetter & Walecka

Quantum Theory of Many-Particle Systems, Eq.6.45, page 61

Raimes

Many Electron Theory, Eq.6.15, page 105

0lim

Can the energy

correction depend on α?


0 1 0

00

0 0

0, lim

0,

H U t tE E

U t t

0 0 00 0

log ,limt

E E i U tt

We now show that:

0 0

0 0

0 00 00

0

,

log , ,

lim limt

t

i U tt

i U tt U t

0 0

0 0

0 0

0 00

0

1

, log ,

,

lim limt

t

U ti U t

ti U t

t

Using:


0 0

0 0

0 00 00

0

,

log , ,

lim limt

t

i U tt

i U tt U t

0 0

0 0

0 00 00

0

, log ,

,lim lim I

tt

H U ti U t

t U t

0 0now: , ( ) , Ii U t t H t U t tt

0 1 0

0 00

0 00 0

0, log , lim

0, lim

t

H U t ti U t

t U t t

0 0 00 0

log ,limt

E E i U tt

Thus:

Raimes / Many Electron Theory / Eq.6.22, page 106

0 1 0

00

0 0

0, but: lim

0,

H U t tE E

U t t


0 0

0 00 0

H E

0 0 0 0 = H E

0 1 0 t

tH H e H H

0 ?E E E

0 00 0

log ,limAdiabaticHypothesis t

E i U tt

?

RayleighSchrodinger

PerturbationTheory

E


0

1 2 1 2

0

, =

1= .. ( ) ( ).. ( )

!

nt t t

n I I I n

n

U t t


n

From STiTACS /U4L25/S45:

0

1 2 1 2

1

, 1

1 .. ( ) ( ).. ( )

!

nt t t

n I I I n

n

U t t


n

0

1

1 2 1 2

, 1

1 .. ( ) ( ).. ( )

!

n

n

nt t t

n n I I I n

U t t U


n

n=0 term


0 00 0

log ,limt

E i U tt

0 0 0 0

1

, 1 n

n

U t U

0

1

1 2 1 2

, 1

1 .. ( ) ( ).. ( )

!

n

n

nt t t

n n I I I n

U t t U


n

0 0

where

n nA U

1 2

1 1 , ,... . ( ), ( ),... . t t

I Ie e etc appear in H t H t etc

0 1

tH H e H remember that:

0 0

1

1 n

n

U

1

1 n

n

A


0 0

1

log , log 1 n

n

U t A

0 0

1

, 1 n

n

U t A

0 0 n nA U

2 3 4 51 1 1 1 log 1 ....

2 3 4 5

1 1

e x x x x x x

for x

2 3

0 0

1 1 1

4 5

1 1

1 1log ,

2 3

1 1 ....

4 5

n n n

n n n

n n

n n

U t A A A

A A


0 00 0

log ,limt

E i U tt

2 3

0 0

1 1 1

4 5

1 1

1 1log ,

2 3

1 1 ....

4 5

n n n

n n n

n n

n n

U t A A A

A A

2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n


2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

0 0 0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

nt t t

n n n I I I n

iA U dt dt dt T H t H t H t

n

2 2 2

1 2 3

1 2 3 1 2 1 3 1 4

2 1 2 3 2 4

3 3 3

1 2 3 1 2 3

2 2 2

1 2 1 3 1 4 2 1 3

2 2 2

2 1 2 3 2 4

0

...1

...2

..

... ...1

... ...3

....

lim

A A A

A A A A A A A A A

A A A A A AE i

t A A A A A A

A A A A A A A A A

A A A A A A

0

....

...t

1 2 3 ..E E E E 1

indexed

thn order corrections

by the power of H

Observe where the terms for various orders come from!

Messy? Let us look at just the I

order term 74 PCD STiTACS Unit 4 Feynman Diagram Methods

1 2 3 ..E E E E

2 2 2

1 2 3

1 2 3 1 2 1 3 1 4

2 1 2 3 2 4

3 3 3

1 2 3 1 2 3

2 2 2

1 2 1 3 1 4 2 1 3

2 2 2

2 1 2 3 2 4

0

...1

...2

..

... ...1

... ...3

....

lim

A A A

A A A A A A A A A

A A A A A AE i

t A A A A A A

A A A A A A A A A

A A A A A A

0

....

...t

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

1

10 0

limt

E i At


0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

1

10 0

limt

E i At

1 0 0

0 1 1 0

( )

n

t

I

A U

idt H t

0 0

1 ( ) H H

i t i t

IH Ht e e

† † †1 v

2i i jj k l

i j i j k l

H c i f j c c c ij lk c c

0 0

H H

i t i t

I Se e

1 1 tH e H



76

P. C. Deshmukh



Chennai 600036




Correspondence between Adiabatic Switching

technique and Rayleigh-Schrodinger perturbation

theory.

0 0

00 0 0 0

0 1

0 0

0 00 0

(0, ) lim ,

(0, )

;

. .

UIf exists

U


i e H E



For ‘PROOF’, see:

Fetter & Walecka

Quantum Theory of

Many-Particle Systems,

page 61

0 1 0 t

tH H e H H

0 0 0 0 H E

Question: 0 1 ?How do we get eigenstate of H H H

0 0 , 0 0, I r t U t t

0 t


0 0 0 0 = H E 0 1 0 t

tH H e H H

0 ?E E E

From Gellman & Low Theorem:

0 1 0

00

0 0

0, lim

0,

H U t tE E

U t t

0 0 00 0

log ,limAdiabaticHypothesis t

E E E i U tt

We showed that:

?

RayleighSchrodinger

PerturbationTheory

E

79

0 00 0

log ,limt

E i U tt

0 0 0 0

1 1

, 1 1n n

n n

U t U A

0

1

1 2 1 2

, 1

1 .. ( ) ( ).. ( )

!

n

n

nt t t

n n I I I n

U t t U


n

0 0

where

n nA U

1 2

1 1 , ,... . ( ), ( ),... . t t

I Ie e etc appear in H t H t etc

0 1

tH H e H remember that:

2 3

0 0

1 1 1

4 5

1 1

1 1log ,

2 3

1 1 ....

4 5

n n n

n n n

n n

n n

U t A A A

A A



2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

0 0 0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

nt t t

n n n I I I n


n

2 2 2

1 2 3

1 2 3 1 2 1 3 1 4

2 1 2 3 2 4

3 3 3

1 2 3 1 2 3

2 2 2

1 2 1 3 1 4 2 1 3

2 2 2

2 1 2 3 2 4

0

...1

...2

..

... ...1

... ...3

....

lim

A A A

A A A A A A A A A

A A A A A AE i

t A A A A A A

A A A A A A A A A

A A A A A A

0

....

...t

1 2 3 ..E E E E 1

indexed

thn order corrections

by the power of H

Observe where the terms for various orders come from!

Messy! Look at just the I

order term 81 PCD STiTACS Unit 4 Feynman Diagram Methods

1 2 3 ..E E E E

2 2 2

1 2 3

1 2 3 1 2 1 3 1 4

2 1 2 3 2 4

3 3 3

1 2 3 1 2 3

2 2 2

1 2 1 3 1 4 2 1 3

2 2 2

2 1 2 3 2 4

0

...1

...2

..

... ...1

... ...3

....

lim

A A A

A A A A A A A A A

A A A A A AE i

t A A A A A A

A A A A A A A A A

A A A A A A

0

....

...t

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

1

10 0

limt

E i At


0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

1

10 0

limt

E i At

1 0 0

0 1 1 0

( )

n

t

I

A U

idt H t

0 0

1 ( ) H H

i t i t

IH Ht e e

† † †1 v

2i i jj k l

i j i j k l


0 0

H H

i t i t

I Se e

1 1 tH e H


0 0

1 ( ) H H

i t i t

IH Ht e e

† † †1 v

2i i jj k l

i j i j k l


0 0

† i j

H Hi t i t

ece c

Transformation to interaction picture of some

second quantized creation and destruction

operators in some order…..

1 1 tH e H

0 0

† † ( ) ( ) H H

i t i t

i j i je c c e c t c t


0 0

† i j

H Hi t i t

ece c

0 0 0 0 0 0

† † H H H H H H

i t i t i t i t

j

t

i

t

i

i i

jce e e e e ec c c

0 0

† †

, , ( ) ) (H H

i

i j I i

i t

I j

t

c c c t ce te

suppress subscript I

for ‘interaction picture’

for brevity.

0 0

† † ( ) ( ) H H

i t i t

i j i je c c e c t c t


0 0

1 ( ) H H

i t i t

IH Ht e e

† †

1

1 v

2 i j k l

i j k l

H c c ij lk c c

0 0 0 0 0 0

† † H H H H H H

i t i t i t i t

j

t

i

t

i

i i

jce e e e e ec c c

† †1 ( ) ( ) ( ) v ( ) ( )

2 i j

t

I k l

i j k l

H t c t c t ij lk c t c t e

0 1 tH H e H

This recipe would work for

any combination of creation

and destruction operators.

α: adiabatic switching:

1 1 tH e H


† †1 ( ) ( ) ( ) v ( ) ( )

2 i j

t

I k l

i j k l


,( ) ( ) ?k I kc t c t

0 0

0

, ,

H Hi t i t

I S

Hi t

I S

e e

r t e r t

0 0

,

,

( ) to the equation of motion

( )

I k

H Hi t i t

I k k

c t solution

i c t i e c et t


0 0

,

,

( ) to the differential equation:

( ) with ( ) ( ) ;

I k

H Hi t i t

I S I I k S k

c t solution

i t i e e t c t ct t

0 0 0 0

0 0 H H H H

i t i t i t i t

S S

H Hi i e e e i e

0 0 0 0

0 0 H H H H

i t i t i t i t

S SH e e e H e

0 0

0,H H

i t i t

Se H e

0,I t H

, , 0( ) ( ),I k I ki c t c t Ht

0 0

0

( ) to the differential equation:

( ) ,

I

H Hi t i t

I S I

t solution

i t i e e t Ht t

0 0

, ,( ) ( )H H

i t i t

I k j jk j k I k

j

i c t e c e c tt


, , 0( ) ( ),I k I ki c t c t Ht

0 0

0,H H

i t i t

ke c H e

0 0

†,H H

i t i t

k j j j

j

e c c c e

†

0since j j j

j

H c c

0 0

†

, ( ) ,H H

i t i t

I k j k j j

j

i c t e c c c et

† † †

:

, , 0 , 0r s rs r s r sa a a a a a

for fermion operators

† † † † †

† †

,k j j k j j j j k jk j k j j j k

jk j j k j j j k

c c c c c c c c c c c c c c c

c c c c c c c


, ,( ) ( )I k k I ki c t c tt

, ( ) ki t

I k kc t c e

† †

, ( ) ki t

I k kc t c e

† †1 ( ) ( ) ( ) v ( ) ( )

2 i j

t

I k l

i j k l


† †1( ) v

2

ji k l

i j

i ti t i t i t t

I k l

i j k l

H t c c ij lk c c e e e e e

† †1( ) v

2

i j k l

i j

i t t

I k l

i j k l

H t c c ij lk c c e e

Raimes


* *

1 2 1 2 1 2 1 2v v ,i j l kij lk dq dq q q q q q q


1

10 0

limt

E i At

1 0 1 0

0 1 1 0

( ) t

I

A U

idt H t

† †1( ) v

2

i j l k

i j

i t t

I k l

i j k l


α switch

1

1

1 0 1 0

† †

, , ,0 1 0

1v

2 i j

i j l k

t k lt

i j k l

i t

A U

c c ij lk c cidt e

e

↓ Note: ↓ integration variable is t1


1

1 0 1 0

† †

, , ,0 1 0

1v

2

i j

i j l k

t k lt

i j k l

i t

A U

c c ij lk c cidt e

e

1 1 11 1 1

1 1 1

1

i j l kt t ti t i ti t tt

i j l k

I dt e e dt e e dt e

where

1 1 1 1

1 1

1

1 1

ti t t i t t

t i t e e e eI dt e

i i

1

† †

1 0 1 0 0 0

, , , 1

v 2 i j

i t

k l

i j k l

i eA U c c ij lk c c

i


2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

1

10 0

limt

E

i At

1

† †

1 0 1 0 0 0

, , , 1

v 2 i j

i t

k l

i j k l

i eA U c c ij lk c c

i

1† †1

0 0 1

, , , 1

1 v

2 i j

i t

k l

i j k l

A ic c ij lk c c i e

t i

1† †10 0 0 0

, , ,

v ( )2 i j

i t

k l I

i j k l

A i ic c ij lk c c e H t

t

† †1( ) v

2

i j l k

i j

i t t

I k l

i j k l



1

1 † †

0 0

, , , 10

0

v 2

limi j

i t

k l

i j k lt

i eE i c c ij lk c c

t i

1

0 0

† †

0 0

, , ,

'

1= v

2 i j k l

i j k l

E H

c c c c ij lk

α0 not relevant for

first order correction;

but not so for higher

order terms…..

1† †10 0 0 0

, , ,

v ( )2 i j

i t

k l I

i j k l

A i ic c ij lk c c e H t

t

† †1( ) v

2

i j l k

i j

i t t

I k l

i j k l


with

10 0 0 0

0

0 'I

t

A i iH t H

t

Eq.6.35/Raimes

Page 108


1 † †

0 0

, , ,

1 v

2 i j k l

i j k l

E c c c c ij lk

Now, before we consider higher order terms,

recapitulate that:

1 0 1 0

0 1 1 0

( ) t

I

A U

idt H t

10 1 1 0

0 0

( )

( )

t

I

I

A idt H t

t t

iH t

Resulted in:


1 † †

0 0

, , ,

1 v

2 i j k l

i j k l

E c c c c ij lk

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

1 2 3 ..E E E E

indexed th

In order corrections by the power of H

2 3

1 1 1

4 5

1 1 0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n nt

A A A

E it

A A

Now, we shall consider higher order terms:


0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

22

2 10 0

1

2lim

tE i A A

t

22

2 10 00 0

2

lim limt t

iE i A A

t t



2

'

, =1 ' ( ') ' '' ( ') ( '') .. t t t

I I I


1 21 2

0

( ) ( ).. ( )1

, = .. !

nt t t

I n

n

I InTi


Equivalent form:

1

1 2 1 22

1 ( ) ( )

t t

I Idt dt H t H t

2nd order term

: we shall use

, : switcht

Note

U t e

1 1 tH e H

98 PCD STiTACS Unit 4

Feynman Diagram

Methods

22

2 10 00 0

2

lim limt t

iE i A A

t t

2 0 2 0 2 0 2 0; hence A U A Ut t

1

2 0 1 1 2 2 02

1 ( ) ( )

t t

I IA dt H t dt H tt t

1

2 0 1 1 2 2 02

1 ( ) ( )

t t


0 1 1 0 0 0from Slide 94: ( ) ( )t

I I

i idt H t H t

t

2 0 2 2 02

1 ( ) ( )

t

I IA H t dt H tt

99

0 1 0 t

tH H e H H

0 0

1 ( )

( ) 0

H Hi t i t

t

I

I

H t e e H e

H t

2 0 2 2 02

1 ( ) ( )

t

I IA H t dt H tt

0

2 0 2 2 020

1 ( 0) ( ) I I

t

A H t dt H tt

0

2 0 020

1 ( 0) ( ) I I

t

A H t dt H tt

Using t instead of t2 ↑



0

2 0 020

1 ( 0) ( ) I I

t

A H t dt H tt

0 0

1 1 ( ) i.e. ( 0)H H

i t i tt

I IH t e e H e H t H

0

2 0 1 020

1 ( ) I

t

A H dt H tt

0 00

2 0 1 1 020

1

H Hi t i t

t

t

A dt H e e H et

0 00

2 0 1 1 020

1

H Ei t i t

t

t

A dt H e H e et

Raimes



0 00

2 0 1 1 020

1

H Ei t i t

t

t

A dt H e H e et

0 0

*

0 1 1 0 0 1 1 0 ' H H

i t i t

H e H dV H e H

space integral

0 0

*

0 1 1 0 1 0 1 0 ' H H

i t i t

H e H dV H e H

1 0

0

1 0 1 0

0 0

m m

m

m m m m

m m

H c

H H


0 0

*

0 1 1 0 1 0 1 0 ' H H

i t i t

H e H dV H e H

1 0 1 0

0

m m

m

H H

0

0

0 1 1 0

*

0 1 1 0

0 0

'

Hi t

Hi t

n n m m

n m

H e H

dV H e H

0

*

0 1 1 0

0 0

' H

i t

n m n m

n m

H H dV e


*

0 1 1 0

0 0

' mE

i t

n m n m

n m

H H e dV

0

*

0 1 1 0

0 0

' H

i t

n m n m

n m

H H dV e

0

0 1 1 0 H

i t

H e H

nm

0

0 1 1 0 0 1 1 0

0 0

mH E

i t i t

n m nm

n m

H e H H H e


0

0 1 1 0 0 1 1 0

0

nH E

i t i t

n n

n

H e H H H e

0 2

0 1 1 0 1 0

0

nH E

i t i t

n

n

H e H H e

0 00

2 0 1 1 020

1

H Ei t i t

t

t

A dt H e H e et

020

2 1 020 0

1

nE Ei t i t

t

n

t n

A dt H e e et


020

2 1 020 0

1

nE Ei t i t

t

n

t n

A dt H e e et

02 0

2 1 020 0

1

nE E ii t

n

t n

A H dt et

02 0

2 1 020 0

1

nE E ii t

n

t n

A H dt et

0

0

0

0

0 0

n

n

E Ei tE E i t

i t

n n

e edt e

E E i i E E ii


02 0

2 1 020 0

1

nE E ii t

n

t n

A H dt et

0

0

0

0

0 0

n

n

E Ei tE E i t

i t

n n

e edt e

E E i i E E ii

2

1 0

2

0 0 0

1

n

t n n

HA

t i E E i

22

2 10 00 0

2

lim limt t

iE i A A

t t

?



107

P. C. Deshmukh



Chennai 600036

Feynman Diagrams





1 2 3 ..E E E E

2 2 2

1 2 3

1 2 3 1 2 1 3 1 4

2 1 2 3 2 4

3 3 3

1 2 3 1 2 3

2 2 2

1 2 1 3 1 4 2 1 3

2 2 2

2 1 2 3 2 4

0

...1

...2

..

... ...1

... ...3

....

lim

A A A

A A A A A A A A A

A A A A A AE i

t A A A A A A

A A A A A A A A A

A A A A A A

0

....

...t

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

Chronological operator


1

1 † †

0 0

, , , 10

0

v 2

limi j

i t

k l

i j k lt

i eE i c c ij lk c c

t i

1

0 0

† †

0 0

, , ,

'

1= v

2 i j k l

i j k l

E H

c c c c ij lk

α0 not relevant for

first order correction;

but not so for higher

order terms…..

1 0 1 0 0 1 1 0 ( ) t

I

iA U dt H t

1 1 1 1

1 1

1

1 1

ti t t i t t

t i t e e e eI dt e

i i

10 1 1 0 0 0 ( ) ( )

t

I I

A i idt H t H t

t t

1

10 0

limt

E i At


0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

22

2 10 0

1

2lim

tE i A A

t

22

2 10 00 0

2

lim limt t

iE i A A

t t

111 PCD STiTACS Unit 4

Feynman Diagram

Methods

22

2 10 00 0

2

lim limt t

iE i A A

t t

2 0 2 0 2 0 2 0; hence A U A Ut t

1

2 0 1 1 2 2 02

1 ( ) ( )

t t


2 0 2 2 02

1 ( ) ( )

t

I IA H t dt H tt

at t=0


0 00

2 0 1 1 020

1

H Ei t i t

t

t

A dt H e H e et

0 0

*

0 1 1 0 1 0 1 0 ' H H

i t i t

H e H dV H e H

1 0 1 0 1 0

0 0 0

m m m m m m

m m m

H c H H

020

2 1 020 0

1

nE Ei t i t

t

n

t n

A dt H e e et


02 0

2 1 020 0

1

nE E ii t

n

t n

A H dt et

0

0

0

0

0 0

n

n

E Ei tE E i t

i t

n n

e edt e

E E i i E E ii

2

1 0

2

0 0 0

1

n

t n n

HA

t i E E i

22

2 10 00 0

2

lim limt t

iE i A A

t t

?


1 0 1 1 0 ( ) t

I

iA dt H t

10 0 ( ) I

A iH t

t

2

1 0 1 1 0 0 02

2 ( ) ( )

t

I IA dt H t H tt

02

1 0 1 1 0 0 020

2 ( ) ( 0)I I

t

A dt H t H tt

2 1

1 12A

A At t

2

10 0

?2

limt

iA

t

?

( 0)t


0 0

1 1 ( ) ( ) 0 ( 0)H H

i t i tt

I I IH t e e H e H t H t H

02

1 0 1 1 0 0 020

2 ( ) ( 0)I I

t

A dt H t H tt

02

1 0 0 0 1 020

2 ( ) I

t

A dt H t Ht

0 00 0

0 0 010( ) H H

i t i tt

Idt dt tH e e H e

0 00 0

0 0 0 1 0 ( ) E E

i t i tt

Idt H t dt e e H e

116

PCD STiTACS Unit 4 Feynman Diagram

Methods

02

1 0 0 0 1 020

2 ( ) I

t

A dt H t Ht

0 00 0

0 0 0 1 0 ( ) E E

i t i tt

Idt H t dt e e H e

0

0 1 0 tdt e H

0

0 1 0 te

H

0 1 0 H

2 0 1 0

1 0 1 020

2

t

HA H

t


22

2 10 00 0

2

lim limt t

iE i A A

t t

2 0 1 0

1 0 1 020

2

t

HA H

t

2

1 0

2

0 0 0

1

n

t n n

HA

t i E E i

2

1 0 0 1 02

0 1 020 0

0

1 2

2lim

n

n n

H HiE i H

i E E i

2 2

1 0 0 1 02

0 00

+ lim

n

n n

H HiE

E E i


2 2

1 0 0 1 02

0 00

+ lim

n

n n

H HiE

E E i

Individually, the first and the third term on r.h.s. blow

up as α→0, but these two terms cancel each other

happily.

2 2 2

0 1 0 1 0 0 1 02

10 0 00

+ lim

n

n n

H H HiE

E E i E E i

↑ n=0 term

2

1 02

1 00

limn

n n

HE

E E i

n=0 term ↓


2

1 0

1 0

n

n n

H

E E

α→0 was needed to see correspondence with

Rayleigh-Schrodinger perturbation theory.

Raimes Many Electron Theory, Eq.6.48, page 111

2

1 02

1 00

limn

n n

HE

E E i

Same result holds good for higher order terms.

Same as 2nd order

Rayleigh-Schrodinger

perturbation theory.


If it is the same result as Rayleigh-Schrodinger

Perturbation Theory, what is the advantage?

Combined with time-dependent methods and

FEYNMAN DIAGRAMS, the present method

offers tremendous convenience, specially in

addressing higher order corrections.

2

1 02

1 0

n

n n

HE

E E

PCD STiTACS Unit 4 Feynman

Diagram Methods

121

The technique of adiabatic switching,

and

addressing the perturbations

using the methods we discussed

can be applied to many other situations;

not just a many-electron system.


1

2 0 1 1 2 2 02

1 ( ) ( )

t t

I IA dt H t dt H t

We have seen the second order term:

Let us get an advance glance at the nth

order term:

1 2

, , , , , , , , ,

2 2

....

1 2

....2

v

v

....

v

1

.... 1

1....

....

n

n

i j k l p q r s u w x y

n

n

i n t

n

i

ij lk

pq sr

uw yx

i n

i

e

i n

† † † † † †0 0

....i j pk l q r s u w x yc c c c c c c c c c c c


complex? 0 0 n nA U

http://www.google.ca/url?sa=i&source=images&cd=&cad=rja&docid=fFRD-OGwK3r0ZM&tbnid=FoPbxFkNK9G-mM:&ved=0CAgQjRw&url=http%3A%2F%2Fwww.zazzle.com%2Ffunny_feynman_diagram_skin_for_laptop-134378033666831985&ei=bbWmUvvnG8a42wWCiYDICg&psig=AFQjCNH9lQu66oSES9-x4enD4wHHEz1lmA&ust=1386743533521082

Mediator

(Boson)

PHOTON


“I learned very early the difference between

knowing the name of something, and

knowing something.” ― Richard P. Feynman (1918-1988)

http://www.goodreads.com/author/quotes/1429989.Richard_P_Feynman

Downloaded on December 09, 2013 particle creation

particle destruction

Vertex is where an interaction between

interacting particles is indicated

hole creation

hole destruction

http://www.goodreads.com/author/quotes/1429989.Richard_P_Feynman


Our interest in the present course:

Select/Special Topics

in the

Theory of Atomic Collisions and Spectroscopy

Study of electron-correlation effects in atomic

collision / photoabsorption processes -

-Random Phase Approximation

-Many Body Perturbation Theory

-Configuration Interactions …… etc.

-RRPA: Relativistic Random Phase Approximation

-MCTD: MultiConfiguration Tamm Dancoff method

-MQDT: Multichannel Quantum Defect Theory etc…



† †

1

1 v

2 i j k l

i j k l

H c c ij lk c c

0 1

tH H e H

We shall use the INTERACTION PICTURE

formalism.

0 0

0

, ,

H Hi t i t

I S

Hi t

I S

e e

r t e r t

0 0

1 ( ) H H

i t i tt

IH t e e H e

0 0

1( ) H H

i t i t

IH t e H e

Adiabatic switching on

of the interaction.


0 0

1 ( )

( ) 0

H Hi t i t

t

I

I

H t e e H e

H t

† †1 ( ) ( ) ( ) v ( ) ( )

2 i j

t

I k l

i j k l


0 1 tH H e H

α: adiabatic switch

control parameter

0 00 0

log ,limt

E i U tt

From U4L26/Slide 66:

Correction to the energy due to the interaction between

the many-particle electron system.


0 00 0

log ,limt

E i U tt

2 3

0 0

1 1 1

4 5

1 1

1 1log ,

2 3

1 1 ....

4 5

n n n

n n n

n n

n n

U t A A A

A A

2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

From U4L26

STiTACS


1

† †

1 0 1 0 0 0

, , , 1

v 2 i j

i t

k l

i j k l

i eA U ij lk c c c c

i

Slide 80 / U4L27 ; Raimes / Many Electron

Theory / Eq. 7.18, page 114

2

'

, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I I


1 21 2

0

( ) ( ).. ( )1

, = .. !

nt t t

I n

n

I InTi


Two equivalent forms of the

Time Evolution Operator:

0 0

1 , since ( ) H H

i t i tt

IWe must use U t H t e e H e

1 i j l k

We shall now consider the 2nd order term; use (A)

(A)

(B)


2

'

, =1 ' ( ') ' '' ( ') ( '') .. t t t

I I I


1

2

2, 1 2 1 2, = ( ) ( )t t

I I

iU t dt dt H t H t

1

2 0 2 0

2

0 1 2 1 2 0

( ) ( ) t t

I I

A U

idt dt H t H t

† †1 ( ) ( ) ( ) v ( ) ( )

2 i j

t

I k l

i j k l


1 t

e 2

te

* *



1

2

2 0 1 2 1 2 0 ( ) ( ) t t

I I

iA dt dt H t H t

† †1 ( ) ( ) ( ) v ( ) ( )

2 i j

t

I k l

i j k l


* *


1

1

2

1 2

† †

1 1 1 1

0 02, , , , , , † †

2 2 2 2

( ) ( ) ( ) ( )v1

2 v

( ) ( ) ( ) ( )

i j

p

t t

k l

t

i j k l p q r s

q r s

t

dt dt

c t c t c t c tij lk

epq sr

c t c t c t c t

e

2 A

* *

1 2 1 2 1 2 1 2v v ,p q s rpq sr dq dq q q q q q q


1

1

2

1 2

† †

1 1 1 1

0 02, , , , , , † †

2 2 2 2

( ) ( ) ( ) ( )v1

2 v

( ) ( ) ( ) ( )

i j

p

t t

k l

t

i j k l p q r s

q r s

t

dt dt

c t c t c t c tij lk

epq sr

c t c t c t c t

e

2 A

, ( ) ki t

I k kc t c e

† †

, ( ) ki t

I k kc t c e

11 1 1 2 2 2

1 2

2, , , , , ,

† † † †0 0

v1

2 v

i j p

t ti t t i t t

i j k l p q r s

k l q r s

dt e e dt e eij lk

pq src c c c c c c c

2 A

1 i j l k 2 p q r s


11 1 1 2 2 2

1 2

2, , , , , ,

† † † †0 0

v1

2 v

i j p

t ti t t i t t

i j k l p q r s

k l q r s

dt e e dt e eij lk


2 A

1 i j l k 2 p q r s

11 1 1 2 2 2

1 2 ?t t

i t t i t tdt e e dt e e

12 2 2

2 ?t

i t tdt e e

12 2 2 1

1 12 22 2 2

2 2

2 2

ti t i t

t t i ti t t e edt e e dt e

i i


11 1 1 2 2 2

1 2 ?t t

i t t i t tdt e e dt e e

2 11

2 2 2

2

2

i t

ti t t e

dt e ei

2 1

2 11 1 1 1 1 1

1 1

2 2

1i t

t t i ti t t i t tedt e e dt e e e

i i

1 21

1 1 1 2 2 2

2

1 2

2 1 2

1

2

i tt t

i t t i t t edt e e dt e e

i i

1 2 12

1

2

1 t i tdt e

i

1 2 12 1

1 1 1

2

1

2 2 1 2

1

2

ti ti t

ti t t e e

dt e ei i i


11 1 1 2 2 2

1 2

2, , , , , ,

† † † †0 0

v1

2 v

i j p

t ti t t i t t

i j k l p q r s

k l q r s

dt e e dt e eij lk


2 A

1 i j l k 2 p q r s

1 21

1 1 1 2 2 2

2

1 2

2 1 2

1

2

i tt t

i t t i t t edt e e dt e e

i i

1 2

2

† † † †0 02 2, , , , , ,

1 2

1

v1

2 v

2

i j p

i ti j k l p q r s k l q r s

iij lk

c c c c c c c cepq sr

i

2 A

Raimes / Many Electron Theory / Eq. 7.20, page 115 Next: generalize the

pattern for nth term…..


1 2

2

† † † †0 02 2, , , , , ,

1 2

1

v1

2 v

2

i j p


iij lk


i

2 A

nth order term….

One may generalize the pattern we have

seen above….

2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

0 0

0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n n

nt t t

n I I I n

A U


n

1 2

, , , , , , , , ,

2 2

....

1 2

....2

v

v

....

v

1

.... 1

1....

....

n

n


n

n

i n t

n

i

ij lk

pq sr

uw yx

i n

i

e

i n

† † † † † †0 0



complex? 0 0 n nA U

Questions:

[email protected]

http://www.google.ca/url?sa=i&source=images&cd=&cad=rja&docid=fFRD-OGwK3r0ZM&tbnid=FoPbxFkNK9G-mM:&ved=0CAgQjRw&url=http%3A%2F%2Fwww.zazzle.com%2Ffunny_feynman_diagram_skin_for_laptop-134378033666831985&ei=bbWmUvvnG8a42wWCiYDICg&psig=AFQjCNH9lQu66oSES9-x4enD4wHHEz1lmA&ust=1386743533521082


139

P. C. Deshmukh



Chennai 600036


I Order Feynman Diagrams




2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

0 0 0 1 2 1 2 0

1 .. ( ) ( ).. ( )

!

n

t t t

n n n I I I n


n

1 2

, , , , , , , , ,

2 2

....

1 2

....2

v

v

....

v

1

.... 1

1....

....

n

n

n


n

n

i n t

n

iA

ij lk

pq sr

uw yx

i n

i

e

i n

† † † † † †0 0

....

i j pk l q r s u w x yc c c c c c c c c c c c


Transformation of particles (electrons)

to particles

and holes 1

0 : single Slater determinant

of elements ( )k

q

The present technique can be easily extended for more

complex systems, such as electron gas in a periodic lattice

potential.

For FREE ELECTRONS: Fermi surface is a sphere


Transformation of particles (electrons) to excited particle

states above Fermi surface, and vacant hole states below it.

For FREE ELECTRONS: Fermi surface is a sphere.

occupied; vacantF Fk k k k

↑ Occupied and unoccupied states are described

simply, but it can still be easily done in other cases.

vectors: lie inside Fermi sphere of radius Fk k

10 : single Slater determinant of elements ( )

kq

143 PCD STfTACS Unit 2 Many-body theory, electron correlations, Feynman-Goldstone diagrams

12 Z

2 2 2 4 2

1 1 1 1 3 1

2 2 2 2 2

1 2 2 2 3SD s s p p s

2 2 2 4 2

2 1 1 1 3 1

2 2 2 2 2

1 2 2 2 3SD s s p p p

Slater

determinant

….. Many different Slater determinants

can be used!

Multi-configuration Hartree-Fock:

CI: Configuration Interaction


Bubbles

in boiling

water

Creation

and

Annihilation

of particles

and holes

…

above/below

EF

1 2

, , , , , , , , ,

2 2

....

1 2

....2

v

v

....

v

1

.... 1

1....

....

n

n


n

n

i n t

n

i

ij lk

pq sr

uw yx

i n

i

e

i n

† † † † † †0 0



2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

2

'

, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I I


nA


occupied and vacantF Fk k k k

states within the Fermi sphere: UNEXCITED

If an unexcited (i.e. below the Fermi level) state is unoccupied by an electron,

then it is called a “hole” state.

Destruction of an electron in an unexcited (i.e. below

the Fermi level) ↔ creation of a hole.

Creation of an electron in an unoccupied unexcited

(i.e. below the Fermi level) ↔ destruction of a hole.

states above (“outside; in the momentum-space”)

the Fermi sphere: EXCITED

How would you create a hole state?

How would you now destroy that hole state?



, ( ) ki t

I k kc t c e

† †

, ( ) ki t

I k kc t c e

Electron

destruction

and

creation

operators

k ka c† †

k ka c†

k kb c

†

k kb c

hole

destruction

and

creation

operators

Electron

operators

act at all ε

particle

operators

act at ε > εF

hole

operators

act at ε ≤ εF

, ( ) ki t

I k ka t a e

† †

, ( ) ki t

I k ka t a e

, ( ) ki t

I k kb t b e

† †

, ( ) ki t

I k kb t b e

147

particle and hole

operators in the

interaction picture

particle

destruction

and

creation

operators



k ka c† †

k ka c

particle

destruction

and

creation

operators

†

k kb c

†

k kb c

hole

destruction

and

creation

operators

particle

operators

act at ε > εF

hole

operators

act at ε ≤ εF

, ( ) ki t

I k kb t b e

† †

, ( ) ki t

I k kb t b e

†

k k

F

c b

if k k

† †

k k

F

c a

if k k

k k

F

c a

if k k

†

k k

F

c b

if k k


Feynman diagrams/graphs

(2005)

QED: electrons exchange virtual photons which mediate

the interaction between the electrons.

The electromagnetic interaction is treated at the level of

quantum theory.

positron: electron propagating backward in time

particle lines

point upwards hole lines point

downwards

AMO Physics: no positrons; but there are ‘hole’ states…

……vacant states normally occupied by electrons


Feynman diagram/graph

Time: increases from bottom to the top….

……. (alternative conventions exist)

Vertex: intersection of photon wavy line and the trunk.

In AMP, evolution of atomic states is represented by

vertical solid lines.

Atomic state lines: sometimes referred to as ‘trunk’ of the

diagram.

Feynman – 1948 Spring at Pocono Manor Inn (Pennsylvania)

Present treatment: Goldstone J, 1957, Proc. Roy. Soc. A239 267

RAIMES: MANY ELECTRON THEORY, Chapter 7


particle lines

point upwards

hole lines point

downwards

Wiggly lines: mediators of the interaction

photons

Vertex: indicates where the interaction occurs

particle and hole lines go in or out

of a vertex……

….depending on particle/hole

creation/destruction……

Principal elements of a Feynman diagram/graph

It does not matter whether these lines lean

toward left or right; or do not lean at all….

Line pointing

down, out of the

vertex:

destruction of a

hole in an

unexcited state

ε<εF

Line pointing up, out

of the vertex:

creation of a particle

in an excited state

ε>εF

Mediator

(Boson)

PHOTON


particle creation

particle destruction Vertex is where an interaction is indicated

hole creation

hole destruction

†; i i Fa

Line pointing up,

Into the vertex:

destruction of a particle

in an excited state ε>εF

; i i Fa

†; j j Fb

; j j Fb

Line pointing

down, into the

vertex:

creation of a hole

in an unexcited

state ε<εF

1 2

, , , , , , , , ,

2 2

....

1 2

....2

v

v

....

v

1

.... 1

1....

....

n

n


n

n

i n t

n

i

ij lk

pq sr

uw yx

i n

i

e

i n

† † † † † †0 0



2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

2

'

, =1 ' ( ') ' '' ( ') ( '') .... t t t

I I I


nA


1

1 0 1 0

† †

0 0

, , , 1

v 2 i j

i t

k l

i j k l

A U

i eij lk c c c c

i

† †

0 0v i j k lij lk c c c c

FIRST ORDER FEYNMAN DIAGRAMS

1 i j l k

, , ,

for some particular

i j k l


1

1 0 1 0

† †

0 0

, , , 1

v 2 i j

i t

k l

i j k l

A U

i eij lk c c c c

i

† †

0 0v i j k lij lk c c c c

j,k: Index on the right i,l : Index on the left

j,l : Inner index

i,k : Outer index

Consider: εi, εj > εF

εl, εk < εF † † † † † † i j i jk l k lc c c c a a b b

Illustration 1


† †

0 0v i j k lij c c c clk


j,l : Inner index

i,k : Outer index



particle

creation

particle

destruction

hole creation

hole destruction

kl

Right vertex Left vertex

i j

Look at inner/outer indices


† †





particle

creation

particle

destruction

hole creation

hole destruction

kli j

l

k

i

j


† †



Consider: εi, εj < εF

εl, εk > εF † † i j k l i j k lc c c c bb a a

particle

creation

particle

destruction

hole creation

hole destruction

kli j

l

k i

j

j


† †



Consider: εi, εj < εF

εl, εk > εF † † i j k l i j k lc c c c bb a a

kli

l

k i

j

li


† †



Consider: εi, εj , εk > εF

εl < εF † † † † † i j i j lk l kc c c c a a a b

particle

creation

particle

destruction

hole creation

hole destruction

kRight vertex Left vertex

j

Look at inner/outer indices

Outer Inner Inner Outer

l

i j

k

l

i


† †



Consider: εi, εj , > εF

εl, εk > εF † † † † i j i jk l k lc c c c a a a a

particle

creation

particle

destruction

hole creation

hole destruction

k

j


l

k

i

j



εl, εk > εF

† † † † i j i jk l k lc c c c a a a a

particle

creation

particle

destruction

hole creation

hole destruction

k l

ij

l

i

k

j

left right

Interchange vertices

V V

† † † † j i il k j l kc c c c a a a a

equivalent Fig.7.8/page123

Raimes/MET

If either i=j or k=l, the

term becomes zero



εl, εk > εF


kl

ij

l

i

k

j

Interchange

two lines

i j

† † † † j i ik l j k lc c c c a a a a

Opposite

sign

Fig.7.9/page124

Raimes/MET


term becomes zero

Questions:

[email protected]


164

P. C. Deshmukh



Chennai 600036


Some more I order Feynman Diagrams

II Order RING Diagrams



l

i


† †




εl, εk > εF † † † † i j i jk l k lc c c c a a a a

particle

creation

particle

destruction

hole creation

hole destruction

k

j


l

k

i

j



εl, εk > εF


particle

creation

particle

destruction

hole creation

hole destruction

k l

ij

l

i

k

j

left right

Interchange vertices

V V

equivalent Fig.7.8/page123

Raimes/MET


term becomes zero



εl, εk > εF


kl

ij

l

i

k

j

Interchange

two lines

i j

Opposite

sign

Fig.7.9/page124

Raimes/MET

i

i


† †

0 0v i j l ij l c c c ci i

j,l: Index on the right SAME! i,i : Index on the left

Consider: εi < εF & εj, εl > εF

† † † † i j jl i i l ic c c c ba a b

particle

creation

particle

destruction

hole creation

hole destruction

l

j


l

i

j

l

ji

ki


† †

0 0v i j i kj k c c c ci i

j,i: Index on the right

SAME index i at outer left and also at outer right

Consider: εi < εF & εj, εk > εF

† † † † i j ji k i i kc c c c ba b a

particle

creation

particle

destruction

hole creation

hole destruction

ji


i

j

k

k i

ji

ki


† †

0 0v i j i kj k c c c ci i

j,i: Index on the right

SAME index i at outer left and also at outer right

Consider: εi < εF & εj, εk > εF

† † † † i j ji k i i kc c c c ba b a

j i


k i

ji

j

ki

i

i

k

j

k

j

RAIMES / Fig.7.7 / p.123


k

ji

i

RAIMES / Many Electron Theory

Fig.7.7 / p.123

ji

i

i


† †

0 0v i j j ij j c c c ci i

i,j: same on the right SAME! i,i : Index on the left

Only two indices: εi , εj < εF † † † † i j j i i j j ic c c c bb b b

particle

creation

particle

destruction

hole creation

hole destruction

j

j


i j

Two unexcited particles below the Fermi surface

interact; but they do not change their respective states.

ii and jj

† †

0 0v i j i jj i c c c ci j

j

i


j,i: on the right SAME! i,j : Index on the left

Only two indices: εi , εj < εF

† † † † i j i j i j i jc c c c bb b b

hole creation

hole destruction

j

i


Two unexcited particles below

the Fermi surface interact; and in

this example they exchange their

respective states: ij and ji

† † † †

0 0 0 0

1

,

i j i jj i i j

i j F

ONLY

c c c c and c c c c

with CAN CONTRIBUTE TO A


1

1 0 1 0

† †

0 0

, , , 1

v 2 i j

i t

k l

i j k l

A U

i eij lk c c c c

i

† †

0 0

† †

0

For - ,

i j

i j

k l

k l

c c c c to be non zero

c c c c must be non zero

0 has neither holes nor excited particles


† †


ji

† †


Only two indices:

εi , εj < εF

Opposite

signs

Direct

or

Coulomb

Exchange


† †

0 0 1i j j ic c c c

† †

0 0 1i j i jc c c c

, .For i j both the elements go to zero

i j


1

1 0 1 0 1

† †

0 0

, , , 1

v 2 i j

i j k l

i t

k l

i j k l

A U

i eij lk c c c c

i

Total contribution to A1 from all Direct/Coulomb terms

1

1 0 0

,v 0 & 1

2direct

tA

i j

i j

i eC ij ij

Total contribution to A1 from all Exchange terms

1

1 0 0

,v 0 & 1

2

t

A

exchange

i j

i j

i eC ij ji

, i j F


1

,

,v

2direct

i j i

tA

i j

i j

i eC ij ij

1

,

,

( 1)v

2exchange

i j i

tA

i j

i j

i eC ij ij

1

10 0

limt

E i At

1

1 0 1 0 1

† †

0 0

, , , 1

v 2 i j

i j k l

i t

k l

i j k l

A U

i eij lk c c c c

i


1

10 0

limt

E i At

1

,

0 , 0

( 1) v + v

2 2lim

i j i

ti j

i j t

i i eE i ij ij ij ji

t

1

,

0 , 0

1 v v

2lim

i j i

t

i j t

eE ij ij ij ji

t

1

,

0 , 0

1 v v

2lim

i j i

t

i j t

eE ij ij ij ji

1


ji

εi , εj < εF

Direct or Coulomb

Exchange

1

,

,

1 v v

2

i j i

i jE ij ij ij ji

I order corrections: Hartree- Fock

Next: II & higher order Feynman diagrams


2 3

1 1 1

4 5

1 1

0

0

1 1

2 3

1 1 ....

4 5

limn n n

n n n

n n

n n t

A A A

E it

A A

22

2 10 00 0

2

lim limt t

iE i A A

t t

From: Slide # 95

U4, L28

11 1 1 2 2 2

1 2

2, , , , , ,

† † † †0 0

v1

2 v

i j p

t ti t t i t t

i j k l p q r s

k l q r s

dt e e dt e eij lk


2 A From U4L29/S134


22

2 10 00 0

2

lim limt t

iE i A A

t t

1

1 0 1 0 1

† †

0 0

, , , 1

v 2 i j

i j k l

i t

k l

i j k l

A U

i eij lk c c c c

i

II & higher order Feynman diagrams

1 2

2

† † † †0 02 2, , , , , ,

1 2

1

v1

2 v

2

i j p


iij lk


i

2 A

Raimes / Many Electron Theory / Eq. 7.20, page 115


From U4L29/S134

1 2

2

† † † †0 02 2, , , , , ,

1 2

1

v1

2 v

2

i j p


iij lk


i

2 A

Raimes / Many Electron Theory / Eq. 7.20, page 115

† † † †

0 0

† † † †

0

0

to be non-zero,

must have

non-zero projection on

i j p

i j p

k l q r s

k l q r s

For c c c c c c c c

c c c c c c c c

….. and that makes us pick terms with

appropriate – not all – values of i,j,k,l,p,q,r,s


↑ t

time

Fig.7.14 from page

129 of Raimes,

Many Electron Theory

Second order

graphs which

may

contribute to

A2

ONLY THESE

contribute to

ground state

energy


↑ t

time

ONLY THESE

contribute to

ground state

energy

(b) is obtained from (a)

and, (d) from (c),

by interchanging the

vertices at one of the two

interaction dashed (wiggly) lines.

particle

creation


particle

destruction

hole

destruction

↑ t time

t1

t2

t1

t2

Ring Diagrams

come from ‘direct’

‘coulomb’ terms

m n

n m

p q

q p

hole

creation

q

particle

creation


particle

destruction

hole

creation

hole

destruction

t1

t2

↑ time

t1

t2

p n

m p

m

q n

Exchange Diagram

corresponding to the

2nd order Ring Diagram

particle

creation


particle

destruction

hole

creation

hole

destruction

↑ t

time t1

t2

t1

t2

2nd order ring diagrams with exchange

particle

creation


particle

destruction

hole

creation

hole

destruction

↑ t t1

t2

t1

t2


No. of closed loops, λ=2

No. of hole lines, µ=2




Questions:

[email protected]


191

P. C. Deshmukh



Chennai 600036


II and Higher Order Feynman Diagrams

Relativistic Random Phase Approximation (RRPA) Linearized Time-Dependent Dirac-Hartree-Fock method



particle

creation


particle

destruction

hole

creation

hole

destruction

↑ t t1

t2

t1

t2







ji

i

i


† †


i,j: same on the right SAME! i,i : Index on the left


† † † † i j j i i j j ic c c c bb b b

particle

creation

particle

destruction

hole creation

hole destruction

j

j


i

j Two unexcited particles below the Fermi surface

interact; but they do not change their respective states.

ii and jj


ji

εi , εj < εF

First order Direct or

Coulomb term

1

,

,

1 v v

2

i j i

i jE ij ij ij ji



41 1 1

† †


j

i




hole creation

hole destruction

j

i



the Fermi surface interact; and

they exchange their respective

states: ij and ji

† †


j

i




hole creation

hole destruction

j

i



the Fermi surface interact; and

they exchange their respective

states: ij and ji


No. of hole lines, µ=2 31 1 1


† † † †

0 0 .... i j i jk l k lc c c c c c c c

nth order term has the vacuum-vacuum

matrix element:

Its possible values are 0, +1, -1,

no matter what n

If for a Feynman graph, the no. of closed loops = λ,

and the no. of hole lines = µ,

then the contribution of this graph to < > is given by

1

Proof: based on Wick’s theorem


nA nth order term has

the vacuum-

vacuum matrix

element:

1 2

, , , , , , , , ,

2 2

....

1 2

....2

v

v

....

v

1

.... 1

1....

....

n

n


n

n

i n t

n

i

ij lk

pq sr

uw yx

i n

i

e

i n

† † † † † †0 0



1 2 3

, , , , , , , , ,

3

† † † †01 2 3

2 3

3

2

v

v

v

3

1

2

1

i j p

n


i t

k l q r

i

ij lk

pq sr

uw yx

e

i c c c c c c c c

i

i

† † 0s u w x yc c c c

3 A 3rd order term has

the vacuum-

vacuum matrix

element:


Raimes

Many Electron Theory Page 126 3rd order graphs ↑ t

Linked and Unlinked Graphs


Raimes Linked and Unlinked Graphs



Raimes Linked and Unlinked Graphs





0 .. 1,2,..DHF i i ih V u u i N

1 1 1 1

From STiAP U4L23/S119

( ) ( ) ( ) = i HF i i if r u q V u q u q

1 1

2

( )

(1) .. .. .. ( )

(1) .. .. .. ..1

.. .. .. .. |!

.. .. .. .. ..

(1) .. .. .. ( )

N

N N

u u N

u

N iN

u u N

2

0 ( 1, 1)Ze

h p m cr

Hartree-Fock Eq

Relativistic

Dirac-Hartree-Fock Eq

Relativistic Random Phase Approximation - RRPA W.R.Johnson and C.D.Lin Phys.Rev.A Vol.20,No.3, Sept. 1979

From STiAP

U3L14/S64

PCD STiTACS

Unit 4 Feynman Diagram Methods

207


( )

1 1

2

(1) .. .. .. ( )

(1) .. .. .. ..1

.. .. .. .. |!

.. .. .. .. ..

(1) .. .. .. ( )

N

N N

u u N

u

N iN

u u N

1

n m

n m

n m

ˆ uiG (r) (r)u

ˆ uF (r) (r)r

1 1

2 2

1 1

2 2

1

2

2

2

'

'

j , m m

j m

j , m m

for j

j mˆY (r)

j

j mˆY (r)

j

1 1

2 2

1 1

2 2

1

2

1

2 2

1

2 2

'

'

j , m m

j m

j , m m

for j

j mˆY (r)

j

j mˆY (r)

j

From STiAP: U3L18 / Slides 266, 267

Johnson & Lin Phys Rev A 1979


0

............... 1,2,..

DHF i i ih V u u

i N

Relativistic

Dirac-Hartree-Fock Eq

32

1

( ) [( ) ( ) ]

| |

N

DHF j j j j

j

d rV u r e u u u u u u

r r

( ) ( ) ( ) ( ) ...,i t i ti i i iu r u r w r e w r e

1 1

2

( )

(1) .. .. .. ( )

(1) .. .. .. ..1

.. .. .. .. |!

.. .. .. .. ..

(1) .. .. .. ( )

N

N N

u u N

u

N iN

u u N

v v i t i te e

Application of a

time-dependent

external field

with v , A †

and v v results in

higher harmonics↑

If we rebuild the (Time-Dependent) Dirac-Hartree-Fock

scheme with all the higher harmonics, we get NON-LINEAR

TIME-DEPENDENT DIRAC-HARTREE-FOCK Equations.

(1)

0( ) (v )i i i ij j

j

h V w V u u 1,2,..., .i N


Expanding TD-DHF equations in powers of the

external field, and retaining only first order terms, we get Linearized TD-DHF equations:

(1)

32

1

( )

[( ) ( ) ( ) ( ) ]| |

i

N

j j i j j i j i j j i j

j

V u r

d re u w u w u u w u u u u w

r r

maintain orthogonality of perturbed orbitals

with the occupied orbitals

ij

jw

( )iu r

Lagrange multipliers

↑DHF potential

includes electron-electron correlations

L-TD-DHF


(1)

32

1

( )

[( ) ( ) ( ) ( ) ]| |

i

N

j j i j j i j i j j i j

j

V u r

d re u w u w u u w u u u u w

r r

(1)

0( ) ( v )i i i ij j

j

h V w V u u ↑ driving terms

(1)

0( ) ;i i i ij j i

j

h V w V u u w 1,2,..., .i N

Basic RRPA equations are obtained by dropping the

‘driving’ terms

1,2,..., .i N

i

i

w

w

: Positive frequency components final state correlations

: Negative frequency components initial state correlations

B-RRPA


(1)

0( ) ( v )i i i ij j

j

h V w V u u ↑ driving terms 1,2,..., .i N

i

i

w

w

: Positive frequency components final state correlations

: Negative frequency components initial state correlations

(1)

0( ) ;i i i ij j i

j

h V w V u u w 1,2,..., .i N

B-RRPA

Inhomogeneous equation

: solutions provide excitation spectrum

both discrete and continuum


We solve the L-TD-DHF equations in terms of the

solutions of the B-RRPA equations.

3 † †

1

v vN

i i i i

i

T e d r w u w u

Transition amplitude from the initial state to an

excited state described by the RRPA functions

of frequency induced by

is given by:

iw

v v i t i te e



Relativistic Quantum

Theory of Atoms

and Molecules,

I.P.Grant

Ch.9.10

Fig.9.8

time

Time

forward

ring

diagram

Time

forward

exchange

diagram

Time

backward

exchange

diagram

Time

backward

ring

diagram

Uncorrelated

matrix

element


218 PCD STiTACS Unit 4 Feynman Diagram Methods Questions: [email protected]

https://www.physics.iitm.ac.in/~labs/amp/

Walter R.Johnson | with Vojislav Radojevic | with Steven T. Manson

RRPA ; R-MCTD

Radojevic,Gagan Pradhan, PCD, Karan Govil,Jobin Jose,S.Sunil Kumar, N.M.Murty, N.Shanthi, H.Chakraborty, Alak Banik, Aarthi Ganesan, Ashish Kumar, Tanima Banerjee, Hari Varma, Ankur Mandal, Soumyajit Saha, K.Sindhu

time

Documents

Feynman Diagram Methods - NPTELnptel.ac.in/courses/115106085/downloads/Unit_4_Feynman_Diagram_… · Select/Special Topics from ‘Theory of Atomic Collisions and Spectroscopy’