Schrödinger, Heisenberg, Interaction Pictures
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•
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Experiment: measurable quantities (variables)
Quantum mechanics: operators (variables) and state functions
Classical mechanics: variables carry time dependence
State at time, t, determined by initial conditions
‘Schrödinger mechanics’: operators time-independent

Ĥ

(t)

i
S (t)
State function carries time-dependence
S
t
*
Â
(t)


(t)
Ô

(t)
Expectation value
S
S
S
‘Heisenberg mechanics’: operators carry time dependence

State function is time-independent i H (0)  0
t
Time evolution operator
S (t)  Û(t,0) H (0) Û time evolution operator
ĤÛ  i

Û
t
Exercise : Prove that Û  e iĤt is a solution to

ĤÛ  i Û provided H is independen t of time
t
Schrödinger, Heisenberg, Interaction Pictures
• Operators in Heisenberg picture
Ô(t)  S (t) ÔS S (t)



S (t)  Û(t,0) H (0) S (t)  Û(t,0) H (0)  H (0)Û  (t)
Ô(t)  H (0)Û  (t,0) ÔS Û(t) H (0)  H (0) Û  (t) ÔS Û(t) H (0)
 H (0) Ô H (t) H (0)
Definition of operator in Heisenberg picture
Ô H (t)  Û  (t) ÔS Û(t)  eiĤt ÔSe- iĤt iff Ĥ is time - independen t
Schrödinger, Heisenberg, Interaction Pictures
• Heisenberg equation of motion



 


Ô H (t) 
Û (t) ÔS Û(t)  Û  (t) ÔS Û(t)  Û  (t) ÔS Û(t)
t
t
t
t


 
Û  -iĤÛ
Û  i ĤÛ  iÛ  Ĥ
t
t

Ô H (t)  i Û  ĤÔS Û - Û  ÔS ĤÛ  i ĤÛ  ÔS Û - Û  ÔS ÛĤ  i ĤÔ H - Ô H Ĥ
t
 

 iĤ, Ô 
(t)  Ô , Ĥ 
 
 
Heisenberg equation of motion
 
OK iff H is time - independen t
H
i

Ô H
t
H
We used Ĥ, Û  0

iX̂ 
i
X̂
e  1  iX̂ 
2
Exercise : Prove this using Û  e- iĤt
2!
 ...

Schrödinger, Heisenberg, Interaction Pictures
• Operators in interaction picture
• Split Hamiltonian H = Ho + H1 Ho is time-independent
 iĤ o t
I (t)  e
S (t) definition
S (t)  e

  (0)
 i Ĥ1  Ĥ o t
S
Equation of motion
i
 

 iĤ o t 
I (t)  i iĤ o I (t)  e
i S (t)
t
t
 iĤ o t
 -Ĥ o I (t)  e
Ĥ o  Ĥ1 S (t)


 iĤ t
Ĥ  Ĥ e iĤ t  (t)
 -Ĥ  (t)  e
 iĤ t
- Ĥ  Ĥ  Ĥ e iĤ t  (t)
e
o
o
I
o
o
o
 iĤ o t
I
o
o
e
1
Ĥ1e
o
 iĤ o t
1
I (t)
I
Schrödinger, Heisenberg, Interaction Pictures
• Operators in interaction picture:
i

I (t)  Ĥ1 (t) I (t)
t
Ĥ1 (t)  e
 iĤ o t

Ĥ1e

i
Exercise: Prove that t Ô I (t)  Ô I , Ĥ o
• Schrödinger picture

i S (t)  Ĥ o  Ĥ1 S (t)
t


• Interaction picture

i I (t)  Ĥ1 t I (t)
t
• Heisenberg picture

i H (t)  0
t
 iĤ o t

ÔS  ÔS
Ô I (t)  e
 iĤ o t
i
ÔSe
 iĤ o t
Ô H (t)  e  iĤt ÔSe  iĤt
i

ÔS  0
t


Ô I (t)  Ô I , Ĥ o
t



i Ô H (t)  Ô H , Ĥ H
t

Schrödinger, Heisenberg, Interaction Pictures
• Time evolution operator
I (t)  Û I t,0 I (0)

 iĤ o t
 iĤ o t
I (t)  Ĥ I (t) I (t) Ĥ I (t)  e
Ĥ1e
t


i Û I t,0 I (0)  Ĥ I (t) Û I t,0 I (0)
i Û I t,0   Ĥ I (t) Û I t,0 
t
t
i
• Integrate to obtain implicit form for U

i  dt' Û I t' ,0    dt' Ĥ I (t' )Û I t' ,0 
t'
0
0
t
t


t
i Û I t,0   Û I 0,0    dt' Ĥ I (t' )Û I t' ,0 
0
t
Û I t,0   1  i  dt' Ĥ I (t' )Û I t' ,0 
0
Û I 0,0   1
Schrödinger, Heisenberg, Interaction Pictures
• Solve by iteration
t
Û I t,0   1  i  dt' Ĥ I (t' )Û I t' ,0
0
t'


Û I t,0   1  i  dt' Ĥ I (t' ) 1  i  dt' ' Ĥ I (t' ' )Û I t' ' ,0
0
0


t
t'
t''



Û I t,0   1  i  dt' Ĥ I (t' ) 1  i  dt' ' Ĥ I (t' ' ) 1  i  dt' ' ' Ĥ I (t' ' ' )Û I t' ' ' ,0  

0
0
0

 
t
t
Û I t,0   1   i  dt' Ĥ I (t' )   i 
2
0
t
t'
 dt'  dt' ' Ĥ (t' )Ĥ (t' ' )  ...
I
0
0
I
Schrödinger, Heisenberg, Interaction Pictures
t
t'
0
0
• Rearrange the term  dt'  dt' ' Ĥ I (t' )Ĥ I (t' ' )
t’
t' t
t’=t
t'' t'
 dt'  dt''
t' 0
t’
t’=t’’
dt’’
t’
t'' 0
t'' t
t' t
t'' 0
t' t''
t’=t
dt’
 dt' '  dt'
t’’
t’=t’’
t’’<t’
t’’
t
t'
1
t
t'' t
t' 0
t'' t'
 dt'  dt' '
t’’
t’’=t
t' t
t’’=t
t'
1
t
t
 dt'  dt' ' Ĥ (t' )Ĥ (t' ' )  2  dt'  dt' ' Ĥ (t' )Ĥ (t' ' )  2  dt'  dt' ' Ĥ (t' ' )Ĥ (t' )
I
0

I
0
0
I
I
0
0
I
t'
dt'  dt' ' Ĥ (t' )Ĥ (t' ' ) (t'-t' ' )  Ĥ (t' ' )Ĥ (t' ) (t' '-t')

2
1
t
t
I
0

I
I
I
0
dt'  dt' ' T Ĥ (t' )Ĥ (t' ' )

2
1
t
t
I
0
I
0
I
Time - ordered product of operators
Schrödinger, Heisenberg, Interaction Pictures
• Time evolution operator as a time-ordered product
 i 
 dt  dt ... dt T Ĥ (t )Ĥ (t
n t
Û(t,0) 
n!
t
1
0
t
2
0
n
I
1
I
2

)...Ĥ I (t n )  Te
t
i  dt'Ĥ I (t')
0
0
• Utility of time evolution operator in evaluating expectation value
o Ô H (t) o
o o

o Û(, t)Ô I (t) Û(t,- ) o
o Û(,-) o
Gell  Mann Low Theorem (F 61)
Ĥ oo  Eo defines o
o Exact Heisenberg ground state
F 57
Occupation Number Formalism
• Spin Statistics Theorem
Fermion wave function must be anti-symmetric wrt particle exchange
 (r1s1, r2s2)
2 particles in a 1-D box
 (r1s1r2s2) = - (r2s2 r1s1)
r2s2
r1s1
 (r1s1= r2s2, r2s2) = 0
Occupation Number Formalism
• Slater determinant of (orthonormal) orbitals for N particles
 (r1 , r2 ,..., rN ) 

1
p





P

(
r
)

(
r
)...

(
r
)



1
 p 1 1 2 2 N N p
N! p
1 (r1 ) 1 (r2 )  1 (rN )
1 2 (r1 ) 2 (r2 )  2 (rN )
N!




N (r1 ) N (r2 )  N (rN )
•
•
•
•
•
•
N! terms in wavefunction, N! nonzero terms in norm (orthogonality)
P is permutation operator
Number of particles is fixed
Matrix elements evaluated by Slater Rules
Configuration Interaction methods (esp. in molecular quantum chemistry)
How to accommodate systems with different particle numbers, scattering,
time-dependent phenomena ??
Occupation Number Formalism
• Basis functions e.g. eigenfunctions of mean-field Hamiltonian (M 123, F 12)
No. particles
Label
Symbol
0
o
1
1, 2, 3
1000 , 0100 , 0010
2
12, 13, 23,
1100 , 1010 , 1001
3
123, 124,
…
…
0000
1110 , 1101
…
Occupation Number Formalism
• Fermion Creation and Annihilation Operators
ci n1n 2 ...n i ...   1i n i n1n 2 ...n i-1...
ci n1n 2 ...n i ...   1i 1 - n i  n1n 2 ...n i 1...
i  n1  n 2  ...  n i-1 i.e. number of particles to left of ith
• Boson Creation and Annihilation Operators
a i n1n 2 ...n i ...  n1/2
n1n 2 ...n i ...
i
a i n1n 2 ...n i ...  n i  1
1/ 2
n1n 2 ...n i ...
• Fermion example
c3 1111100   1 1 1101100  1101100
11
c 4 1110110   1
111
1 1111110   1111110
Occupation Number Formalism
•
•
•
•
•
•
Factors outside Fermion kets enforce Pauli Exclusion Principle
No more than one Fermion in a state
Identical particle exchange accompanied by sign change
Sequence of operations below permutes two particles
Accompanied by a change of sign
Also ‘works’ when particles are not in adjacent orbitals
c3 c 2 c3c 2 1111
c 2 1111   1 1011
1
c3 1011   1 1001
1
c 2 1001   1 1101
1
c3 1101   1 1111
11
overall
c3 c 2 c3c 2 1111   1 1111   1111
5
Occupation Number Formalism
• Fermion anti-commutation rules
ci c j  δ ij  c j c i
ci , c j   ci c j  c j ci  δ ij
ci c j  1  c j c i
ci , c j   0
c c  c c
 
 
c , c 

i

j 
i
j
j
i
i j
c i c j  c j ci
i j
i j
0
ci c j  c j ci
i j
Permutatio n of neighbouri ng Fermion operators changes sign


c3 c 2 c3c 2  c3 δ 32 - c3c 2 c 2  c3 .0.c 2  c3 c3c 2 c 2

c
 i ci  n̂ i number operator
i
Number operator counts particles in a particular ket (as an eigenvalue)
Exercise:
(1) Prove Fermion anti-commutation rules using defining relations
(2) Apply ci cj  cj ci to |10> and |11> for i=j=1; i=1,j=2 and comment
Occupation Number Formalism
• Fermion particle, a i , and hole, b i , operators
• Virtual (empty, unoccupied) states
– a i  c i particle annihilation operator destroys Fermion above eF
– a i  ci particle creation operator creates Fermion above eF
• Filled (occupied) states
– bi  ci hole annihilation operator creates Fermion below eF
– bi  ci hole creation operator destroys Fermion below eF
• Fermi vacuum state 1111100000
– all states filled below eF
occ.no.
 0
• Commutation relations from Fermion relations
a , a   δ b , b   δ
a , a   0 b , b   0
a , a   0 b , b   0
i

j 
i

j 
i
j 
i
j 

i

j 

i
ij

j 
ij
all a, b commutator s  0
Occupation Number Formalism
• Number of holes or particles is not specified
• Relevant expectation value is wrt Fermi vacuum 0
• Specify states by creation/annihilation of particles or holes wrt 0
a i 0  0 cannot destroy particle in Fermi vacuum
a i 0  0 create particle
b i 0  0 cannot destroy hole in Fermi vacuum
b i 0  0 create hole
Hermitian conjugates
a 0 
b 0 

 0 a i


i
i
i
 0b
a 0 
b 0 

i

 0 ai

i

 0 bi
0 b i b j 0  0  ij - b j b i 0   ij 0 0  0 b j b i 0   ij
Occupation Number Formalism
• Coordinate notation
 2k
Ĥ(r )   
 V̂(r )
2
k
• Matrix Mechanics
1
H ij  φ i Ĥ(r ) φ j    φ i  2k φ j  φ i V̂(r ) φ j
2 k
• Occupation Number (Second Quantized) form
1
Ĥ  O ij ĉ i ĉ j  Vijkl ĉi ĉ j ĉ k ĉ l
2
1
1
2
O ij    φ i  k φ j   φ i
φj
2 k
r

μ
μ
1
Vijkl  φ i φ j
φk φl
r  r'
• One body (KE + EN) and two body (EE) terms
Occupation Number Formalism
• Potential scattering of electron (particle) or hole
j

ij j i
Oa a

ij i
O b bj
i
i
j
• Electron-electron scattering Electron-hole scattering Hole-hole scattering
i
i
j
Vijkla i a j a k a 
ℓ
Vijkla i b j a k b
ℓ
k
ℓ
k
Vijklbi b j bk b
k
j
i
j
• Scattering + electron-hole pair creation
i
Vijkla i a j b k b
j
ℓ
Left out – Right out – Left in – Right in
k
Field Operators
• Field operators defined by
ψ̂(r, t)   φ j (r )ĉ j (t)
j
ψ̂  (r, t)   φ*j (r )ĉ j (t)
j
ĉ j (t) time - dependent interactio n or Heisenberg pictures
φ (r) eigenfunct ions of mean - field Hamiltonia n
j
dinger picture
Commutatio n relations for operators in Schr o
ψ̂


(r ), ψ̂(r ' )   ψ̂  (r )ψ̂(r ' )  ψ̂(r ' )ψ̂  (r )
 φ*i (r )ĉ i φ j (r ' )ĉ j  φ j (r ' )ĉ jφ*i (r )ĉ i

 φ*i (r )φ j (r ' ) ĉ i ĉ j  ĉ jĉ i

 φ*i (r )φ j (r ' ) ij  φ*i (r )φ i (r ' )  δ(r  r ' )
Field Operators
• Commutation relations
ψ̂


(r ), ψ̂(r ' )    (r - r ' )
ψ̂(r ), ψ̂(r ' )  ψ̂  (r), ψ̂  (r ' )  0
useful relations
ψ̂  (r )ψ̂(r ' )   (r - r ' )  ψ̂(r ' )ψ̂  (r )
ψ̂(r )ψ̂(r ' )   ψ̂(r ' )ψ̂(r )
ψ̂  (r )ψ̂  (r ' )  ψ̂  (r ' )ψ̂  (r )
ψ̂(r, t' ), ψ̂


(r, t' )    (r - r ' ) (t - t' )
ψ̂(r, t' ), ψ̂(r, t' )


 ψ̂  (r, t' ), ψ̂  (r, t' )   0
when time - dependence is included
Field Operators
• Hamiltonian in field operator form
1
Ĥ   dr ψ̂ (r )Ô(r )ψ̂(r )   drdr ' ψ̂ (r )ψ̂ (r ' )
ψ̂(r ' )ψ̂(r )
2
r - r'
1



1
 Oijĉi ĉ j  Vijklĉi ĉ j ĉ k ĉ l
2
Oij   dr i* (r )Ô(r ) j (r )
Vijkl   drdr ' i* (r ) *j (r ' )
1
 k (r ' )l (r ) NB difference between F 19, M 78
r - r'
• Heisenberg equation of motion for field operators



i ψ̂ H (r, t)  ψ̂ H (r, t), Ĥ H (r, t)
t
 ψ̂ H (r, t)Ĥ H (r, t) - Ĥ H (r, t)ψ̂ H (r, t)
Field Operators
• One-body part
Ĥ S   dr1ψ̂  (r1 )Ô(r1 )ψ̂(r1 )
Ĥ H  e  iĤt Ĥ S e  iĤt
ψ̂ H Ĥ H  e  iĤt ψ̂S e  iĤt e  iĤt Ĥ S e  iĤt  e  iĤt ψ̂S Ĥ S e  iĤt
Ĥ S ψ̂S   dr1ψ̂  (r1 )Ô(r1 )ψ̂(r1 )ψ̂(r )    dr1ψ̂  (r1 )ψ̂(r )Ô(r1 )ψ̂(r1 )
ψ̂S Ĥ S   dr1ψ̂(r )ψ̂  (r1 )Ô(r1 )ψ̂(r1 )


ψ̂S Ĥ S  Ĥ S ψ̂S   dr1 ψ̂(r ), ψ̂  (r1 )  Ô(r1 )ψ̂(r1 )   dr1 (r - r1 )Ô(r1 )ψ̂(r1 )
 Ô(r )ψ̂(r )
i



ψ̂ H (r, t)  ψ̂ H (r, t), Ĥ H (r, t)  e  iĤt Ô(r )ψ̂(r ) e  iĤt
t
Field Operators
• Two-body part
Ĥ S 
1
2


d
r
d
r
ψ̂
(
r
)
ψ̂
 1 2 1 (r2 )
Ĥ S ψ̂S 
1
2

1
ψ̂(r2 )ψ̂(r1 )
r1  r2


d
r
d
r
ψ̂
(
r
)
ψ̂
(r2 )
1
2
1

1
2
1
ψ̂(r2 )ψ̂(r1 )ψ̂(r )
r1  r2


d
r
d
r
ψ̂
(
r
)
ψ̂
 1 2 1 (r2 )ψ̂(r)
1
ψ̂(r2 )ψ̂(r1 ) 2 permutatio ns (-1) 2
r1  r2
ψ̂  (r2 )ψ̂(r )   (r - r2 )  ψ̂(r )ψ̂  (r2 )
Ĥ S ψ̂S 
1
2

d
r
ψ̂
(r1 )
1

1
1
1
ψ̂(r )ψ̂(r1 ) -  dr1dr2 ψ̂  (r1 )ψ̂(r )ψ̂  (r2 )
ψ̂(r2 )ψ̂(r1 )
r  r1
2
r1  r2
1
ψ̂S Ĥ S   dr1dr2 ψ̂(r )ψ̂ (r1 )ψ̂ (r2 )
ψ̂(r2 )ψ̂(r1 )
2
r1  r2
1




ψ̂S Ĥ S  Ĥ S ψ̂S   dr1dr2 ψ̂(r ), ψ̂  (r1 )  ψ̂  (r2 )
1
ψ̂(r2 )ψ̂(r1 )
r1  r2
Field Operators
• Two-body part continued


ψ̂S Ĥ S  Ĥ S ψ̂S   dr1dr2 ψ̂(r ), ψ̂  (r1 )  ψ̂  (r2 )
  dr1dr2 (r - r1 )ψ̂  (r2 )
  dr2 ψ̂  (r2 )
i

1
ψ̂(r2 )ψ̂(r1 )
r1  r2
1
ψ̂(r2 )ψ̂(r1 )
r1  r2
1
ψ̂(r2 )ψ̂(r )
r  r2


1
ψ̂ H (r, t)  ψ̂ H (r, t), Ĥ H (r, t)   e  iĤt  dr1ψ̂  (r1 )
ψ̂(r1 )ψ̂(r ) e  iĤt
t
r  r1
• ½ factors included
Wick’s Theorem
• Time-dependence of Fermion operators in interaction picture
Ĥ o   e k ĉ k ĉ k
k
ĉ i (t)  e
iĤ o t
ĉ i e
- iĤ o t

 ĉ i  i Ĥ o , ĉi
Interactio n Picture
2

i
t  Ĥ
2!
Ĥ o , ĉ i   e k ĉ k ĉ k ĉ i  ĉ i ĉ k ĉ k



k

ĉ k ĉ k ĉ i  ĉ k ĉ i ĉ k   δ ki  ĉ i ĉ k
Ĥ , ĉ     e δ
o
i
k


2
,
Ĥ
,
ĉ
t
 ...
o
o
i

ĉ
k
since
ĉ  e i ĉi
ki k
k
ĉ i (t)  ĉ i
2

i
e t 2 ĉ
 ie ĉ t 
ĉ i (t)  ĉ i e
i i
 ie i t
2!
i
i  ...  ĉ i e
 ie i t
ĉ k ĉ i  ĉi ĉ k  0
Wick’s Theorem
• Time-ordered products of 1-body part of Hamiltonian
Ĥ1   dr ψ̂  (r )ĥ(r )ψ̂(r ) ψ̂(r )   i (r )c i
i
   dr i* (r )ĥ(r ) j (r ) c i c j   Vij ci c j
i, j
i, j
Ĥ1 ( t )   Vij c (t)c j (t)   Vij c c j e

i

i
i, j
c i  a i || b i
Vij   dr i* (r )ĥ(r ) j (r )
i (e i -e j )t
i, j
c j  a j || b j
c i c j  a i a j || a i b j || b i a j || b i b j
a i b j
a i a j
eF
j
i
i
eF
j
eF
j
bi bj
bi a j
eF
i
j
i
Wick’s Theorem
• Time-dependence of particle and hole operators in interaction picture
â k (t)  â k e
 ie k t
b̂ k (t)  b̂ k e
 ie k t
â k (t)  â k e
 ie k t
b̂ k (t)  b̂ k e
 ie k t
• Time-ordered product of two operators (M 155)
T a k (t 2 )a l ( t1 )  a k (t 2 )a l ( t1 ) (t 2  t1 )


 a l (t 1 )a k ( t 2 ) (t 1  t 2 )
 sign applies to bosons(  )/fermions (-)
Expectatio n value over Fermi vacuum state
- ie k  t 2 - t1 
 iε k t 2  ie l t 1
 iε k t 2  ie l t 1
0 a k (t 2 )a ( t1 ) 0  0 a k a 0 e
e
e
e
  kl e

l
0 a l (t 1 )a k (t 2 ) 0  0

l
0 T a k (t 2 )a l ( t1 ) 0  iG o (t 2  t1 )
Wick’s Theorem
• Time-ordered products of more than two operators (M 155)


T â k (t) â l (t' )b̂ m (t' ' )b̂ n (t' ' )   1 x operators arranged so that
P
time decreases from left to right
  1 x operators arranged so that all ĉ  (or â  or b̂)
P
to left of ĉ (or â or b̂  ) for equal times
P is number of permutatio ns of neighbouri ng operators required to reach
a particular time order
Example


T â k (t 2 )â p (t) â q (t) â r (t' )â s (t' )â l (t 1 )
 (1) 0 â k (t 2 )â p (t) â q (t) â r (t' )â s (t' )â l (t 1 )
for t 2  t  t'  t1
 (1) 4 â k (t 2 )â r (t' )â s (t' )â p (t) â q (t) â l (t 1 )
for t 2  t'  t  t1
 (1) 5 4 â l (t 1 )â p (t) â q (t) â r (t' )â s (t' )â k (t 2 ) for t1  t'  t  t 2
 (1) 5 4 4 â l (t 1 )â r (t' )â s (t' )â p (t) â q (t) â k (t 2 ) for t1  t'  t  t 2
Wick’s Theorem
• Time-ordered products of more than two operators (M 155)
Wick' s theorem asserts that a time ordered product minus
a ' normal ordered' product is a scalar called a ' contractio n'


N â k (t 2 )â l (t 1 )  (1)1 â l (t 1 )â k (t 2 )
All creation operators are to the left of all annihilati on operators in an N product
Its importance is that its expectatio n value with the Fermi vacuum is always zero
â k (t 2 )â l (t 1 )  â k e
ie k t 2
â l e
ie k t1
 â k â l e
ie k t 2
e
ie k t1
â â  â â eie t eie t   eie t eie t From commutatio n rules
â â  (1) â â eie t eie t   eie t eie t for t  t
T â (t )â (t ) N â (t )â (t )  â (t )â (t ) (M 364, F 83)
k

l
k

l
k

l
k 2
2
k 2
k 1
kl
1

l
k 1
k
1

l
k 2
k 1
k
k 2
k 1
kl
k
2

l
1

k
2
2

l
1
1
Wick’s Theorem
• Time-ordered products of more than two operators (M 155)


N â (t )â (t )  (1) â (t )â (t ) for either time order
T â (t )â (t ) N â (t )â (t )  0 for t  t
T â k (t 2 )â l (t 1 )  (1)1 â l (t 1 )â k (t 2 ) for t1  t 2
k
k

l
2
2

l
1
1
1
k

l
1
k
2

l
1
2
1
2
Exercise : Show that


0 T b̂ (t )b̂ (t ) 0
0 T b̂ k (t 2 )b̂ l (t 1 ) 0  0 for t 2  t1

k
and that

2
l
1

  kle
ie k (t 2  t1 )

 iG o (t 2 - t1 ) for t1  t 2

0 T b̂ k (t 2 )â l (t 1 ) 0  0 T b̂ k (t 2 )â l (t 1 ) 0  0 for any time order
so that


0 T ˆ (t 2 )ˆ  (t 1 ) 0  iG o (t 2 - t1 )  iG -o (t 2 - t1 )  iG o (t 2 - t1 )
Wick’s Theorem
• Longer products of operators (M 155)
0 T ˆ (t 2 )â p (t) â q (t)ˆ  (t 1 ) 0
T â k (t 2 )â p (t) â q (t) â l (t 1 )  â k â p â q â l e
ie k t 2 ie p t ie q t ie l t1
for t 2  t  t1
Evaluate the T  product using â i â j  â j â i   ij
Strategy : permute operators so that all creation operators are to left
of all annihilati on operators, i.e. N - products. Addition al delta function
terms emerge. Fermi vacuum matrix elements of N - products are zero.
The delta function ' contractio ns' created in this way are the desired result.
â k â p â q â l  - â p â k   pk - â l â q   lq 
 (1) 2 â p â k â l â q  â p â k lq   pk â l â q   pk lq
 (1) 2 â p - â l â k   lk â q  â p â k ql   pk â l â q   pk ql
 (1) 3 â p â l â k â q  (1) 2 â p â q lk  â p â k ql   pk â l â q   pk ql
Wick’s Theorem
• Longer products of operators (M 155)


0 T â k (t 2 )â p (t) â q (t) â l (t 1 ) 0 for t 2  t  t1 
 0 (1) â â â k â q  (1) â â q lk  â â k ql   â â q   pk ql 0 e
 
p l
3
 0 0  pk qle

p
2
ie k t 2 ie p t ie q t ie l t1

p
  pk e

pk l
ie k (t 2  t)
 qle
ie k t 2 ie p t ie q t ie l t1
ie l (t  t1 )
Any other time order yields zero because the earliest (single) time must be a
creation operator and the latest (single) time must be an annihilati on operator
The 1- body Hamiltonia n acting at time t contains â p â q , â p b̂ q , b̂ p b̂ q , b̂ p â q .
What do they contribute ?


â k b̂ p b̂ q â l  â k - b̂ q b̂ p   qp â l  (1)â k b̂ q b̂ p â l  â k â l qp  (1) 3 b̂ q â k â l b̂ p  â k â l qp




 (1) 3 b̂ q - â l â k   lk b̂ p  - â l â k   lk  qp
 (1) 4 b̂ q â l â k b̂ p  (1) 3 b̂ q b̂ p lk  (1)â l â k qp   lk qp
Wick’s Theorem
• Longer products of operators (M 155)
â k b̂ p b̂ q â l  (1) 4 b̂ q â l â k b̂ p  (1) 3 b̂ q b̂ p lk  (1)â l â k qp   lk qp


0 T â k (t 2 )b̂ p (t) b̂ (t) â (t 1 ) 0 for t 2  t  t1  0 0  lk e
  lke

q

l
ie k (t 2  t1 )
 qpe
i (e p e q )t
ie k (t 2  t1 )
 qpe i 0


0 T â k (t 2 )â p (t) b̂ q (t) â l (t 1 ) 0 expect zero as 3 particles/ holes are created and
only one is annihilate d. Similarly


0 T â (t )Ĥ (t) â (t ) 0 for t
0 T â k (t 2 )b̂ p (t) â q (t) â l (t 1 ) 0  0
k
2
1

l
ie (t  t )
1
2
 t  t1
ie (t  t )
ie (t  t )
 Vpq   pk e k 2  qle l 1   lk e k 2 1  qpe i 0 


What about the other time order? t1  t  t 2
Wick’s Theorem
• Longer products of operators (M 155)


0 T b̂ k (t 2 )Ĥ1 (t) b̂ l (t 1 ) 0  (1) 3 2 b̂ l (t 1 )Ĥ1 (t) b̂ k (t 2 ) for t1  t  t 2

q

k
  Vpq 0 b̂ l b̂ p b̂ b̂ 0 e

ie l t1 ie p t ie q t ie k t 2

 b̂ l b̂ p b̂ q b̂ k  b̂ l - b̂ q b̂ p   qp b̂ k  (1) 2 b̂ l b̂ q b̂ p b̂ k  b̂ l b̂ k  qp


 

 (1) 2 - b̂ q b̂ l   ql - b̂ k b̂ p   kp  - b̂ k b̂ l   kl  qp
 (1) 4 b̂ q b̂ l b̂ k b̂ p  (1) 3 b̂ q b̂ l  kp  (1) 3 b̂ k b̂ p ql  (1) 2  ql kp  (1) 2 b̂ k b̂ l  qp   kl qp


(1) 4 b̂ q b̂ l b̂ k b̂ p  (1) 4 b̂ q - b̂ k b̂ l   kl b̂ p  (1) 5 b̂ q b̂ k b̂ l b̂ p  (1) 4 b̂ q b̂ p kl

q

k
 0 b̂ l b̂ p b̂ b̂ 0 e
ie l t1 ie p t ie q t ie k t 2


0 T â (t )Ĥ (t) â (t ) 0
  kpe
ie k (t 2  t )
 qle
0 T b̂ k (t 2 )Ĥ1 (t) b̂ l (t 1 ) 0  Vpq   kpe

ie k (t 2  t )
 Vpq   kpe

ie k (t 2  t )
k
2
1

l
1
 qle
 qle
ie l (t  t1 )
  kle
ie l (t 2  t1 )
 qpe i 0
ie l (t  t1 )
  kle
ie l (t 2  t1 )
ie l (t  t1 )
  kle
ie l (t 2  t1 )
 qpe i 0 

 qpe i 0 

Wick’s Theorem
• Diagrammatic interpretation of results


â k (t 2 )â k (t 1 )   (t 2 - t1 ) (k - k F )e
ie k (t 2  t1 )
for t 2  t1
0
for t 2  t1
0
for t 2  t1

o
 iG (t 2 - t1 )


- b̂ k (t 1 )b̂ k (t 2 )   (t 1 - t 2 ) (k F - k)e
ie k (t 2  t1 )
for t1  t 2
 -1
for t1  t 2
0
for t1  t 2
 iG o (t 2 - t1 )
t2 particle destroyed
t1 particle created
t2 particle destroyed
t1 particle created
t2
so that


0 T ˆ (t 2 )ˆ  (t 1 ) 0  iG o (t 2 - t1 )  iG -o (t 2 - t1 )  iG o (t 2 - t1 )
Wick’s Theorem
• Diagrammatic interpretation of results




Why define â k (t 2 )â l (t 1 )  0 and - b̂ k (t 1 )b̂ l (t 2 )  -1 for t1  t 2?


0 T ˆ (t 2 )Ĥ1 (t)ˆ  (t 1 ) 0 (omitting matrix element V)


â  (t 2 )â  (t) â  (t) â  (t 1 )


 â (t 2 )b̂ (t) b̂
 

(t) â (t 1 )
Scattering by V

 b̂  (t 1 )â  (t) â(t) b̂  (t 2 ) ( 0)

Fermion loops

 b̂  (t 1 )b̂  (t) b̂  (t) b̂  (t 2 )



 b̂ (t 1 )b̂ (t) b̂ (t) b̂
 
(t 2 )
The hole part of Ĥ1 (t) creates extra terms c.f. the particle part. These
contractio ns have equal time arguments and are called Fermion loops.
Wick’s Theorem
• Diagrammatic interpretation of results










ie (t  t )
ie (t  t )
0 T ˆ (t 2 )Ĥ1 (t)ˆ  (t 1 ) 0  0 T b̂ k (t 2 )Ĥ1 (t) b̂ l (t 1 ) 0  0 T â k (t 2 )Ĥ1 (t) â l (t 1 ) 0
ie (t  t )
0 T b̂ k (t 2 )Ĥ1 (t) b̂ l (t 1 ) 0  Vpq  ( 1) kpe k 2 ( 1) qle l 1  ( 1) kle l 2 1  qpe i

ie (t  t )
ie (t  t )
ie (t  t )
0 T â k (t 2 )Ĥ1 (t) â l (t 1 ) 0  Vpq   kpe k 2  qle l 1   kle l 2 1  qpe i 0 


(1) factors (which don' t change what we had previously ) have been inserted.
Interpret each factor as a Green' s function


0 T â (t )Ĥ (t) â (t ) 0

iG

- t) 
0 T b̂ k (t 2 )Ĥ1 (t) b̂ l (t 1 ) 0  Vpq iG -o (t 2 - t )iG -o (t - t1 )  iG -o (t 2 - t1 )( 1)iG -o (t - t)
k
2
1

l
1
 Vpq

o
(t 2 - t )iG o (t - t1 )  iG o (t 2 - t1 )( 1)iG -o (t
we have written e i 0  ( 1)iG -o (t - t)  (-1)2 to accomodate the Fermion loop.
Wick’s Theorem
• Products with more than one intermediate time


In 0 T b̂ k (t 2 )Ĥ1 (t) b̂ l (t 1 ) 0 , with one intermedia te time, terms like
0 b̂ k (t 2 )â p (t) b̂ q (t) b̂ l (t 1 ) 0 yield zero, because there are unequal numbers
of creation an annihilati on operators.
However, terms like â p (t) b̂ q (t) in Ĥ1 (t) are electron - hole pair creation operators
and yield a non - zero result if there are at least two intermedia te times, t and t'.
0 b̂ k (t 2 )â p (t) b̂ q (t) â r (t' )b̂ s (t' )b̂ l (t 1 ) 0 contains creation of an electron - hole pair
at t' and annihilati on of the pair at t.
Evaluation of products containing more than 4 operators needs to be automated!
N.B. Mattuck attaches a factor -i to Vpq from U which Fetter  Walecka do not!
Wick’s Theorem
• Products with more than one intermediate time
Observe :
(1) A contractio n is generated when a creation operator on the right is
permuted with a neighbouri ng annihilati on operator on the left, when both are
particle or hole operators.
(2) Permutatio ns required to bring a pair of operators
into this order generate a factor of (-1) P where P is the number of neighbour
permutatio ns. This is true for any type of neighbouri ng operator since
[â i , â j ]  [b̂ i , b̂ j ]  [â i , b̂ j ]  [â i , b̂ j ]  0, etc. commute.
Example




0 â k (t 2 )â p (t) â q (t) â l (t 1 ) 0  â k (t 2 )â p (t) â q (t) â l (t 1 )  (-1)0 iG o (t 2 - t)iG o (t - t1 )
â p (t) â q (t) does not yield a contractio n because it is already in normal order.
Wick’s Theorem
• Products with more than one intermediate time
Examples














q


l
0 â k (t 2 )b̂ p (t) b̂ q (t) â l (t 1 ) 0  (1) 2 â k (t 2 )â l (t 1 )b̂ p (t) b̂ q (t)
 (-1)2 iG o (t 2 - t1 )(-1)iG o (t - t)




0 b̂ l (t 2 )b̂ p (t) b̂ q (t) b̂ k (t 1 ) 0  (1) 2 b̂ l (t 2 )b̂ k (t 1 )b̂ p (t) b̂ q (t)
 (-1)2 (-1)iG o (t 2 - t1 )(-1)iG o (t - t)



q
0 b̂ l (t 2 )b̂ p (t) b̂ (t) b̂
 
k

(t 1 ) 0  (1) b̂ l (t 2 )b̂ (t) b̂ p (t) b̂
1
 
k
(t 1 )
 (-1)1 (-1)iG o (t 2 - t)(-1)iG o (t - t1 )


0 â k (t 2 )b̂ p (t) b̂
 
q

l

(t) â (t 1 ) 0  (1) â k (t 2 )â (t 1 )b̂ p (t) b̂
 (-1)2 iG o (t 2 - t1 )(-1)iG o (t - t)
2
 
q
(t)
Wick’s Theorem
• Normal ordered product of operators (M 364, F 83)


p

N ABC DEF G   1 C  F G  ABDE




p  No. interchang es of neighbouri ng operators


0 N ABC DEF G 0   1



p
0 C  F  G  ABDE 0  0
• Contraction (contracted product) of operators
A  B  T AB - N AB
â k (t 2 )â l (t 1 )  â k (t 2 )â l (t 1 ) - - 1 â l (t 1 )â k (t 2 )
1
 â k â l  â l â k e
 δ kle
 ie k t 2  i e l t 1
e
 ie k  t 2  t1 
t 2  t1
â k (t 2 )â l (t 1 )  â l (t 1 )â k (t 2 ) - - 1 â l (t 1 )â k (t 2 )
1
0
t 2  t1
t1  t 2
t1  t 2
Wick’s Theorem
• Contraction (contracted product) of operators

k

l
b̂ (t 2 )b̂ (t 1 )  δ kle
b̂ k (t 2 )b̂ l (t 1 )  0
 ie k  t 2  t1 
t 2  t1
t1  t 2
• For more operators (F 83) all possible pairwise contractions of operators
• Uncontracted, all singly contracted, all doubly contracted, …
T [UVW...XYZ ]  N [UVW...XYZ ]  N [ UVW...XYZ]  N [UVW... XYZ] 
N [ UVW...XYZ]  ...  N [ UV W...XY.Z]
• Take matrix element over Fermi vacuum
0 T [UVW...XYZ ] 0  0 N[UVW...XYZ ] 0  ...  0 N[UV W...XY.Z] 0
• All terms zero except fully contracted products