Schrödinger, Heisenberg, Interaction Pictures

• 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• State function carries time-dependence• Expectation value• ‘Heisenberg mechanics’: operators carry time dependence• State function is time-independent• Time evolution operator

(t)t

(t)H SS

i(t)O(t)(t)A SS

*S

Ut

UH

U (0)(t,0)U(t) HS

i

operator evolution time

0(0)t H

i

time of tindependen is H provided

to solution a is that Prove :Exercise

Ut

UH

eU tHi

i

Schrödinger, Heisenberg, Interaction Pictures

• Operators in Heisenberg picture

tindependen-time isiff

picture Heisenberg in operator of Definition

H tHeOtHe(t)UO(t)U(t)O

(0)(t)O(0)

(0)(t)UO(t)U(0)(0)(t)UO(t,0)U(0)(t)O

(t)U(0)(0)(t,0)U(t) (0)(t,0)U(t)

(t)O(t)(t)O

SSH

HHH

HSHHSH

HHSHS

SSS

i-i

Schrödinger, Heisenberg, Interaction Pictures

• Heisenberg equation of motion

tHi-e U

0U,H

H,O(t)Ot

O,H

HO-OHHUOU-UOUHUHOU-UOHU(t)Ot

HUUHUt

UH-Ut

(t)Ut

O(t)U(t)UO(t)Ut

(t)UO(t)Ut

(t)Ot

HH

H

HHSSSSH

SSSH

using this Prove :Exercise

tindependen-time is H iffOK used We

motion of equation Heisenbergi

i

iii

iii

...!2

XXXe2

iii 1

Schrödinger, Heisenberg, Interaction Pictures

• Operators in interaction picture• Split Hamiltonian H = Ho + H1 Ho is time-independent

(t)tHeHtHe

(t)tHeHHHtHe

(t)tHeHHtHe(t)H

(t)HHtHe(t)H

(t)t

tHe(t)H(t)t

(0)tHHe(t)

(t)tHe(t)

Io

1o

Io

1ooo

Io

1oo

Io

S1oo

Io

So

IoI

So1

S

So

I

ii

i-i

ii-

i-

iiiii

i

i

motion of Equation

definition

Schrödinger, Heisenberg, Interaction Pictures

• Operators in interaction picture:

Exercise: Prove that

tHeHtHe(t)H (t)(t)H (t)t

o1

o1I1I

iii

oII H,O (t)Ot

i

• Schrödinger picture

• Interaction picture

• Heisenberg picture

HHHSHH

oIIo

So

II1I

SSSS1oS

H,O(t)Ot

tHeOtHe(t)O 0 (t)t

H,O(t)Ot

tHeOtHe(t)O (t)tH (t)t

0Ot

OO (t)HH (t)t

iiii

iiii

ii

Schrödinger, Heisenberg, Interaction Pictures

• Time evolution operator

• Integrate to obtain implicit form for U

11

0,0U ,0t'U)(t'Hdt' t,0U

,0t'U)(t'Hdt' 0,0Ut,0U

,0t'U)(t'Hdt' ,0t'Ut'

dt'

t,0U(t)H t,0Ut

(0)t,0U(t)H (0)t,0Ut

tHeHtHe(t)H (t)(t)H (t)t

(0)t,0U (t)

I

t

0III

t

0IIII

t

0II

t

0I

IIIIIIII

o1

oIIII

III

i

i

i

ii

iii

Schrödinger, Heisenberg, Interaction Pictures

• Solve by iteration

...)'(t'H)(t'H'dt'dt')(t'Hdt' t,0U

,0''t'U)''(t'H''dt')'(t'H'dt')(t'Hdt' t,0U

,0't'U)'(t'H'dt')(t'Hdt' t,0U

,0t'U)(t'Hdt' t,0U

t

0

t

0

t'

0II

2II

t

0

t'

0

't'

0IIIII

t

0

t'

0IIII

t

0III

ii

iii

ii

i

1

111

11

1

Schrödinger, Heisenberg, Interaction Pictures

• Rearrange the term

operators of product ordered-Time )'(t'H)(t'H'dt'dt'

)'-t'(t')(t'H)'(t'H)'(t'-t')'(t'H)(t'H'dt'dt'

)(t'H)'(t'H'dt'dt')'(t'H)(t'H'dt'dt')'(t'H)(t'H'dt'dt'

'dt'dt' dt''dt' 'dt'dt'

)'(t'H)(t'H'dt'dt'

t

0

t

0II

t

0

t

0IIII

t

0

t

t'II

t

0

t'

0II

t

0

t'

0II

tt'

0t'

t't'

t''t'

t't'

0't'

tt'

't't'

tt'

0t'

t''t'

0't'

t

0

t'

0II

2

1

2

1

2

1

2

1

T

t’

t’’

dt’’t’

t’’=t

t’=t’’t’=t

t’

t’’t’’

dt’

t’’=t

t’=t’’t’=t

t’’<t’

Schrödinger, Heisenberg, Interaction Pictures

• Time evolution operator as a time-ordered product

• Utility of time evolution operator in evaluating expectation value

57 F )(t'Hdt'

t

0It

0

t

0nI2I1I

t

0n21

n

e)(tH)...(tH)(tHdt...dtdtn!

(t,0)U

i

TTi

state ground Heisenberg Exact defines

61) (F TheoremLow MannGell

EH

)-,(U

)(t,-U(t)Ot),(U(t)O

o

oooo

oo

oIo

oo

oHo

Occupation Number Formalism

• Spin Statistics Theorem Fermion wave function must be anti-symmetric wrt particle exchange

r22

r11

(r11= r22, r22) = 0

(r11, r22)

2 particles in a 1-D box (r11r22) = - (r22 r11)

Occupation Number Formalism

• Slater determinant of (orthonormal) orbitals for N particles

• 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 ??

)()()(

)()()()()()(

N!1

1 )()...()(P!N

1),...,,(

NN2N1N

N22212

N12111

ppNN2211

ppN21

rrr

rrrrrr

rrrrrr

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

2 12, 13, 23,

3 123, 124,

… … …

0000

0010,0100,1000

1001,1010,1100

1101,1110

Occupation Number Formalism

• Fermion Creation and Annihilation Operators

• Boson Creation and Annihilation Operators

• Fermion example

......nnn1n......nnna

......nnnn......nnna

n...nni

......nnnn-11......nnnc

......nnnn1......nnnc

i212/1

ii21i

i211/2ii21i

1-i21

1i21ii

i21i

1-i21ii

i21i

ith of left to particles of number i.e.

11111101111110111110110c

11011001101100111111100c111

4

113

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

1111111111111cccc

111111101c

110111001c

100111011c

101111111c

1111cccc

52323

113

12

13

12

2323

overall

Occupation Number Formalism

• Fermion anti-commutation rules

sign changes operators Fermion ngneighbouri of nPermutatio

2233232233232323

ji

ji

ijijjiji

cccc0.c.cccc-δccccc

0c ,c

0c ,c

δ ccccc ,c

ji cccc

ji c cc c

ji cccc ji cc1cc

ccδcc

ijji

ijji

ijjiijji

ijijji

operator number n cc ii

ii

Number operator counts particles in a particular ket (as an eigenvalue)

Exercise:(1) Prove Fermion anti-commutation rules using defining relations(2) Apply to |10> and |11> for i=j=1; i=1,j=2 and commentijji cccc

Occupation Number Formalism

• Fermion particle, , and hole, , operators

• Virtual (empty, unoccupied) states– particle annihilation operator destroys Fermion above F

– particle creation operator creates Fermion above F

• Filled (occupied) states– hole annihilation operator creates Fermion below F

– hole creation operator destroys Fermion below F

• Fermi vacuum state– all states filled below F

• Commutation relations from Fermion relations

ia ib

01111100000occ.no.

ii cbii cb

ii ca ii ca

0scommutator b a, all

0b ,b 0a ,a

0b ,b 0a ,a

δb ,b δa ,a

jiji

jiji

ijjiijji

Occupation Number Formalism

• Number of holes or particles is not specified• Relevant expectation value is wrt Fermi vacuum • Specify states by creation/annihilation of particles or holes wrt

ijijijijijji

iiii

iiii

i

i

i

i

0bb0000bb-00bb0

b00b b00b

a00a a00a

00b

00b

00a

00a

conjugates Hermitian

hole create

vacuum Fermi in holedestroy cannot

particle create

vacuum Fermi in particledestroy cannot

00

Occupation Number Formalism

• Coordinate notation

• Matrix Mechanics

• Occupation Number (Second Quantized) form

• One body (KE + EN) and two body (EE) terms

lkjiijkljik

j2kiij

lkjiijkljiij

jik

j2kijiij

k

2k

φφ'

1φφ V φ1φφφ21 O

cccc Vcc OH

φ)(Vφφφ21φ)(HφH

)(V2

)(H

2

1

rrμr

rr

rr

μ

• Potential scattering of electron (particle) or hole

• Electron-electron scattering Electron-hole scattering Hole-hole scattering

• Scattering + electron-hole pair creation

Occupation Number Formalism

j

iijij aaO

jiij bbO

aaaaV kjiijkl

babaV kjiijklbbbbV kjiijkl

bbaaV kjiijkl

i

j

Left out – Right out – Left in – Right in

ji

k ℓ

ℓi

k j

ℓk

i j

ℓi

k

j

Field Operators

• Field operators defined by

)'δ()'()φ(φ)'()φ(φ

cccc)'()φ(φ

c)(φc)'(φc)'(φc)(φ

)(ψ)'(ψ)'(ψ)(ψ)'(ψ),(ψ

)(φ

(t)c

(t)c)(φt),(ψ

(t)c)(φt),(ψ

i*iijj

*i

ijjij*i

i*ijjjji

*i

j

j

jj

*j

jj

j

rrrrrr

rr

rrrr

rrrrrr

r

rr

rr

picture dingeroSchr in operators for relations nCommutatio

nHamiltonia field-mean of ionseigenfunct

pictures Heisenberg or ninteractio dependent-time

Field Operators

• Commutation relations

included is dependence-time when

relations useful

0)t',(ψ),t',(ψ )t',(ψ),t',(ψ

)t'-(t)'-()t',(ψ),t',(ψ

)(ψ)'(ψ)'(ψ)(ψ

)(ψ)'(ψ )'(ψ)(ψ)(ψ)'(ψ)'-( )'(ψ)(ψ

0)'(ψ),(ψ)'(ψ),(ψ

)'-()'(ψ),(ψ

rrrr

rrrr

rrrrrrrr

rrrrrr

rrrr

rrrr

Field Operators

• Hamiltonian in field operator form

• Heisenberg equation of motion for field operators

78 M 19, Fbetween difference NB )()'('-

1)'()('ddV

)()(O)(dO

ccccVccO

)(ψ)'(ψ'-

1)'(ψ)(ψ'dd)(ψ)(O)(ψdH

lk*j

*iijkl

j*iij

lkjiijkljiij 2

1

2

1

rrrr

rr rr

rrrr

rrrr

rr rrrrrr

t),(ψt),(H-t),(Ht),(ψ

t),(Ht),,(ψt),(ψt

HHHH

HHH

rrrr

rrr

i

Field Operators

• One-body part

tHe )(ψ)(O tHe t),(Ht),,(ψt),(ψt

)(ψ)(O

)(ψ)(O)-(d)(ψ)(O)(ψ),(ψdψH H ψ

)(ψ)(O)(ψ)(ψdH ψ

)(ψ)(O)(ψ)(ψd )(ψ)(ψ)(O)(ψd ψH

tHe H ψ tHe tHe H tHetHe ψ tHe Hψ

tHe H tHe H

)(ψ)(O)(ψdH

HHH

11111111SSSS

1111SS

11111111SS

SSSSHH

SH

1111S

iii

iiiiii

ii

rrrrr

rr

rrrrrrrrrr

rrrrr

rrrrrrrrrr

rrrr

Field Operators

• Two-body part

)(ψ)(ψ1)(ψ)(ψ),(ψddψH H ψ

)(ψ)(ψ1)(ψ)(ψ)(ψddH ψ

)(ψ)(ψ1)(ψ)(ψ)(ψdd-)(ψ)(ψ1)(ψd ψH

)(ψ)(ψ)-()(ψ)(ψ

)(ψ)(ψ1)(ψ)(ψ)(ψdd

)(ψ)(ψ)(ψ1)(ψ)(ψdd ψH

)(ψ)(ψ1)(ψ)(ψddH

1221

2121SSSS

1221

2121SS

1221

212111

11SS

222

1221

2121

1221

2121SS

1221

2121S

2

1

2

1

2

1

2

1

2

1

2

1

rrrr

rrrrr

rrrr

rrrrr

rrrr

rrrrrrrrr

rr

rrrrrr

rrrr

rrrrr

rrrrr

rrrr

rrrr

rrrr

2(-1) nspermutatio 2

Field Operators

• Two-body part continued

• ½ factors included

tHe )(ψ)(ψ1)(ψd tHe t),(Ht),,(ψt),(ψt

)(ψ)(ψ1)(ψd

)(ψ)(ψ1)(ψ)-(dd

)(ψ)(ψ1)(ψ)(ψ),(ψddψH H ψ

11

11HHH

22

22

1221

2121

1221

2121SSSS

iii

rrrr

rrrrr

rrrr

rr

rrrr

rrrrr

rrrr

rrrrr

Wick’s Theorem

• Time-dependence of Fermion operators in interaction picture

tec(t)c

tec...ct2!

tcc(t)c

ccδc,H

0cccc cccδ cccccc

ccccccc,H

... tc,H,H2!

tc,Hc

tHi-ectHie(t)c

ccH

iii

iii

2i

2

iiii

iik

kkikio

kiikkkikikikikk

kkkiikkkio

2ioo

2

ioi

oi

oi

kkkko

i

iii

ii

since

Picture nInteractio

• Time-ordered products of 1-body part of Hamiltonian

Wick’s Theorem

jijijijiji

jjjiii

jiji,

ijjiji,

ij1

j*iijji

ji,ijji

ji,j

*i

ii

i1

bb||ab||ba||aa cc

b||ac b||ac

e cc V(t)(t)cc V)t(H

)()(h)(dV cc Vcc )()(h)(d

c)()(ψ )(ψ)(h)(ψdH

)tj-i(

i

rrrr rrrr

rrrrrr

ji ba

i

jF

ji abj

iF

ji bb

ji

F

ji aa

ji

F

Wick’s Theorem

• Time-dependence of particle and hole operators in interaction picture

• Time-ordered product of two operators (M 155)

)t(tG0)t()a(ta0

00)(t)a(ta0

t-te te tεete tεe0aa00)t()a(ta0

)t(t )t()a(ta

)t(t )t()a(ta)t()a(ta

12o1l2k

2k1l

12kkl

1l2k1l2klk1l2k

212k1l

121l2k1l2k

iT

i-iiii

T

state vacuum Fermi over value nExpectatio(-))/fermionsbosons( to applies sign

teb(t)b teb(t)b

tea(t)a tea(t)a

kkk

kkk

kkk

kkk

ii

ii

Wick’s Theorem

• Time-ordered products of more than two operators (M 155)

212kqpsr1l445

212ksrqp1l45

121lqpsr2k4

121lsrqp2k0

1lsrqp2k

P

Pnmlk

ttt' t )(ta(t)a(t)a)(t'a)(t'a)(ta )1(

ttt' t )(ta)(t'a)(t'a(t)a(t)a)(ta )1(

ttt' t )(ta(t)a(t)a)(t'a)(t'a)(ta )1(

tt't t )(ta)(t'a)(t'a(t)a(t)a)(ta )1(

)(ta)(t'a)(t'a(t)a(t)a)(ta

P )b a ( c

)b a ( c x 1

x1)'(t'b)'(t'b)(t'a(t)a

for

for

for

for

Exampleorder time particular a

reach to required operators ngneighbouri of nspermutatio of number is times equal forororof left to

ororall that so arranged operators

right to left from decreases time that so arranged operators

T

T

Wick’s Theorem

• Time-ordered products of more than two operators (M 155)

83) F 364, (M

for

rules ncommutatio From

zero always is vacuum Fermi the withvalue nexpectatio its that is importance Itsproduct an in operators onannihilati all of left the to are operators creation All

n'contractio' a called scalar a is product ordered' normal' a minus product ordered time a that asserts theorem sWick'

)(ta)(ta)(ta)(ta)(ta)(ta

tte ee eaa)1(aa

e ee eaaaa

e eaaea ea)(ta)(ta

)(ta)(ta)1()(ta)(ta

1l2k1l2k1l2k

121k2k

kl1k2k

kl1

lk

1k2kkl

1k2kkllk

1k2klk

1kl

2kk1l2k

2k1l1

1l2k

tttt

tttt

tttt

NT

NN

iiii

iiii

iiii

Wick’s Theorem

• Time-ordered products of more than two operators (M 155)

)t-(tG)t-(tG)t-(tG0)(tˆ)(tˆ0

00)(ta)(tb00)(ta)(tb0

tt)t-(tG e0)(tb)(tb0

tt 00)(tb)(tb0

tt 0)(ta)(ta)(ta)(ta

)(ta)(ta)1()(ta)(ta

tt )(ta)(ta)1()(ta)(ta

12o12-o12o12

1l2k1l2k

2112o12k

kl1l2k

121l2k

211l2k1l2k

2k1l1

1l2k

212k1l1

1l2k

)t(t

iiiT

TT

iT

T

NT

N

T

i

that so

order timeany for

that and

for

for

thatShow :Exercise

for

order time either for

for

qlpkqlpkqlkplkqp

2qklp

3

qlpkqlpkqlkpqlkklp2

lqpkqlpklqkpqlkp2

lqqlpkkplqpk

ijijji

121lqp2k

lqpk1lqp2k

1qp2

aaaaaa1)(aaaa1)(

aaaaaaa-a1)(

aaaaaaaa1)(

aa-aa-aaaa

aaaa

ttteaaaa)(ta(t)a(t)a)(ta

0)(tˆ(t)a(t)a)(tˆ0

tttt

result. desired the are way this in created ns'contractio' function delta Thezero. are products- of elementsmatrix vacuum Fermi emerge. terms

function delta al Additionproducts.- i.e. operators, onannihilati all ofleft to are operators creation all that so operators permute :Strategy

using product the Evaluate

for

NN

T

T

Tiiii

Wick’s Theorem

• Longer products of operators (M 155)

qplkqpkllkpq3

pklq4

qplkklplkklq3

qplkplkq3

qplklpqklqppqklqpk

qpqpqpqp

1lql

2kpk

1lqp2kqlpk

1lqp2kqlpkqlpkqlkplkqp

2qklp

3

121lqp2k

aa1)(bb1)(baab1)(

aa-baa-b1)(

aabaab1)(aaabba1)(abb-aabba

ab,bb,baaat

eee00

e0aaaaaa1)(aaaa1)(0

ttt0)(ta(t)a(t)a)(ta0

)t(tt)(ttttt

tttt

?contributethey do What

. , contains time at acting nHamiltoniabody 1- The

operator onannihilati an be must time (single) latest the and operator creationa be must time (single) earliest the because zero yieldsorder time otherAny

for

iiiiii

iiii

T

Wick’s Theorem

• Longer products of operators (M 155)

21

qp12k

lk1l

ql2k

pkpq

121l12k

1lqp2k

1lqp2k

qp12k

lk

qpqp

12klk121lqp2k

qplkqpkllkpq3

pklq4

lqpk

ttt

eeeeV

ttt0)(ta(t)H)(ta0

00)(ta(t)a(t)b)(ta0

0)(ta(t)b(t)a)(ta0

ee

ee00ttt0)(ta(t)b(t)b)(ta0

aa1)(bb1)(baab1)(abba

0)t(t)t(t)t(t

0)t(t

)t()t(t

order? time other the about What

for

Similarly d.annihilate is oneonly

and created are holesparticles/ 3 as zero expect

for

iiii

ii

ii

T

T

T

T

Wick’s Theorem

• Longer products of operators (M 155)

0)t(t)t(t)t(t

0)t(t)t(t)t(t

0)t(t)t(t)t(ttttt

tttt

eeeeV0)(ta(t)H)(ta0

eeeeV0)(tb(t)H)(tb0

eeeee0bbbb0

bb1)(bbbb1)(bbb-b1)(bbbb1)(

bb1)(1)(bb1)(bb1)(bbbb1)(

bb-bb-bb-1)(

bbbbbb1)(bbb-bbbbb

e0bbbb0V

ttt )(tb(t)H)(tb1)(0)(tb(t)H)(tb0

qp12l

kl1l

ql2k

kppq1l12k

qp12l

kl1l

ql2k

kppq1l12k

qp12l

kl1l

ql2k

kp2kqp1l

kqpl

klpq4

plkq5

pkllkq4

pklq4

qpklqplk2

kpql2

qlpk3

kplq3

pklq4

qpkllkkppkqllq2

qpklkpql2

kqppqlkqpl

2kqp1lkqplpq

212k11l23

1l12k

iiii

iiii

iiiiiiii

iiii

T

T

T

for

Wick’s Theorem

• Longer products of operators (M 155)

Wick’s Theorem

• Diagrammatic interpretation of results

)t-(tG)t-(tG)t-(tG0)(tˆ)(tˆ0

)t-(tG

tt 0 tt 1-

tt k)e-(k)t-(t)(tb)(tb-

)t-(tG

tt 0 tt 0 tt )ek-(k)t-(t)(ta)(ta

12o12-o12o12

12o

21

21

2112k

F212k1k

12o

12

12

1212k

F121k2k

)t(t

)t(t

iiiT

i

i

i

i

that so

for for for

for for for

t1 particle created

t2 particle destroyed

t1 particle created

t2 particle destroyed

t1

t2

. called are and arguments time equal have nscontractio These part. particle the c.f. terms extra creates of part hole The

V)elementmatrix (omitting

? for defineWhy

loops Fermion (t)H

)(tb(t)b(t)b)(tb

)(tb(t)b(t)b)(tb

0)( )(tb(t)a(t)a)(tb

)(ta(t)b(t)b)(ta

)(ta(t)a(t)a)(ta

0)(tˆ(t)H)(tˆ0

tt -1)(tb)(tb- and 0 )(ta)(ta

1

21

21

21

12

12

112

212l1k1l2k

T

Wick’s Theorem

• Diagrammatic interpretation of results

Fermion loops

Scattering by V

loop. Fermion the accomodate to writtenhave we

function sGreen' a as factor each Interpret inserted. been have )previously had we whatchange tdon' (which factors

2-o

-o12o1o2opq1l12k

-o12

-o1

-o2

-opq1l12k

qp12l

kl1l

ql2k

kppq1l12k

qp12l

kl1l

ql2k

kppq1l12k

1l12k1l12k112

(-1)t)-(tGe

t)-(tG)t-(tG)t-(tG)t-(tGV0)(ta(t)H)(ta0

t)-(tG)t-(tG)t-(tG)t-(tGV0)(tb(t)H)(tb0

)1(

eeeeV0)(ta(t)H)(ta0

eeeeV0)(tb(t)H)(tb0

0)(ta(t)H)(ta00)(tb(t)H)(tb00)(tˆ(t)H)(tˆ0

)1(0

)1(

)1(

0)t(t)t(t)t(t

0)t(t)1(

)t(t)1(

)t(t)1(

i

iiiiT

iiiiT

T

T

TTT

i

iiii

iiii

Wick’s Theorem

• Diagrammatic interpretation of results

not! do WaleckaFetter which from to factor a attaches Mattuck N.B.automated! be to needs operators 4 than more containing products of Evaluation

. at pair the of onannihilati and at

pair hole-electron an of creation contains

. and times, teintermedia two least at are there if result zero-non a yieldand

operators creation pair hole-electron are in like terms However,

operators. onannihilati an creation of

numbers unequal are there because zero, yield

like terms time, teintermedia one with In

UV

tt'

0)(tb)(t'b)(t'a(t)b(t)a)(tb0

t't

(t)H(t)b(t)a

0)(tb(t)b(t)a)(tb0

,0)(tb(t)H)(tb0

pq

1lsrqp2k

1qp

1lqp2k

1l12k

-i

T

Wick’s Theorem

• Products with more than one intermediate time

order. normal inalready is it because ncontractio a yieldnot does

Example

commute.

since operator ngneighbouri of typeany for true is This ns.permutationeighbour of number the is where(-1) of factor a generate order this into

operators of pair a bring to required nsPermutatio (2) operators. hole particle

are whenleft, the on operator onannihilati ngneighbouri a withpermutedis right the on operator creation a whengenerated is ncontractio A (1)

:Observe

(t)a(t)a

)t-(tGt)-(tG(-1))(ta(t)a(t)a)(ta0)(ta(t)a(t)a)(ta0

etc. 0, ]b,a[ ]b,a[ ]b,b[ ]a,a[

P

qp

1o2o0

1lqp2k1lqp2k

jijijiji

P

ii

orboth

Wick’s Theorem

• Products with more than one intermediate time

t)-(tG))(-1t-(tG(-1)

(t)b(t)b)(ta)(ta)1(0)(ta(t)b(t)b)(ta0

)t-(tG)t)(-1-(tG)(-1(-1)

)(tb(t)b(t)b)(tb)1(0)(tb(t)b(t)b)(tb0

t)-(tG))(-1t-(tG)(-1(-1)

(t)b(t)b)(tb)(tb)1(0)(tb(t)b(t)b)(tb0

t)-(tG))(-1t-(tG(-1)

(t)b(t)b)(ta)(ta)1(0)(ta(t)b(t)b)(ta0

o12o2

qp1l2k2

1lqp2k

1o2o1

1kpq2l1

1kqp2l

o12o2

qp1k2l2

1kqp2l

o12o2

qp1l2k2

1lqp2k

ii

ii

ii

ii

Examples

Wick’s Theorem

• Products with more than one intermediate time

• Normal ordered product of operators (M 364, F 83)

• Contraction (contracted product) of operators

00ABDEGFC010GDEFABC0

p ABDEGFC1GDEFABC

p

p

N

Noperators ngneighbouri of esinterchang No.

Wick’s Theorem

21

212k1l1

2k1l1l2k

1212k

kl

1l2kkllk

122k1l1

1l2k1l2k

t t 0 t t)(ta)(ta1--)(ta)(ta)(ta)(ta

t ttt

eδ

teteaa aa

t t)(ta)(ta1--)(ta)(ta)(ta)(ta

AB - ABBA

i

ii

NT

Wick’s Theorem

• Contraction (contracted product) of operators

• For more operators (F 83) all possible pairwise contractions of operators• Uncontracted, all singly contracted, all doubly contracted, …

• Take matrix element over Fermi vacuum

• All terms zero except fully contracted products

211l2k

1212k

kl1l2k

t t 0)(tb)(tb

t ttt

eδ)(tb)(tb

i

]Z.XY..W.UV[...Z]XYW...UV[

Z]XY[UVW...W...XYZ]UV[][UVW...XYZ][UVW...XYZ

NN

NNNT

0]Z.XY..W.UV[0...0][UVW...XYZ00][UVW...XYZ0 NNT