N-particle states (fermions)

Transcription

1 Product states ormalization -particle states (fermions) ) ( ) , 1 2, 2..., Completeness )( Identical particles: symmetric or antisymmetric states Fermions: use antisymmetrizer A 1! Permutation operator: product of two-particle permutations # of two-particle permutations odd/even sign p ( 1) p P

2 Check odd/even permutation ote normalization (6 states) Also note antisymmetry Example for 3 particles ) 2 1 3) ) 3 2 1) ) 1 3 2) o two fermions can occupy the same state!! Example for three bosons (symmetric state) [Check!] ) ) ) 3!2! ) ) ) ) ) ). 3

3 fermions Completeness with ordered single-particle (sp) quantum numbers ot ordered ordered ! ormalization with ordered single-particle (sp) quantum numbers ot ordered determinant 1, 1 2, 2...,

4 ormalized -particle wave function Called a Slater determinant (x 1x 2...x ) 1! x x 1 x x x 1... x. Hard to work with Slater determinants Use occupation number representation or second quantization

5 Motivation: Second quantization Slater determinants tedious to work with Relevant operators change only the quantum numbers of one or two particles (and in exceptional cases three) Consider states that are labeled by the # of particles occupying sp states occupation number representation Allow states in CVS with different # of particles Fock space Includes new state: the vacuum all sp states all antisymmetric two-particle (tp) states.. all antisymmetric -particle states 0 { } up to infinite number of particles.. { 1 2 } { }

6 Alternative writing Vacuum state Sp state Tp state i i j i i j etc. Use ordered states 0 ordered Introduce new operator in Fock space a

7 Particle addition (creation) operator Definition a Takes an antisymmetric -particle state and turns it into an antisymmetric +1-particle state with occupied!!!! ote: i not a state i i1,, new state (may require ordering) Acts on any state Including a 0 and etc. a a

8 Particle removal (destruction) operator Action of adjoint operator? a Consider once M0 M0 M0 ordered M ordered M ordered M M M a M placed in the correct location M a M ( 1) i M i... M 1, 1 2, 2... i, i+1, i..., 1 So a ( 1) i i 1 i+1... if i or a if i,i1,..., Example: a 0 0 ote: again antisymmetric state!

9 Fermion anticommutation relations {a,a } a a + a a, {a,a } {a,a } 0 Easy to demonstrate Rewrite antisymmetric state a a a a a a 0 a 0 i Ensures Pauli principle i a a...a a a...a Occupation numbers n 1 1,n 2 0,n 3 1, 0,..., 0,

10 Examples? One-body operators in Fock space 1 particle in sp space F F Operator completely determined by all F matrix elements -particle space F F (1) + F (2) F () Action of F (i) on a product state i1 F (i) F (i) ) i 1 i i F i i+1... i i i F i 1... i 1 i i+1... )

11 One-body operators (continued) Matrix element i F i same for any particle (dummy variables) Then F )F(1) F () 1 F ) F ) 1 i1 i i F i i 1 i i+1... ) Since F Thus F is symmetric it commutes with the antisymmetrizer i F i i 1 i i+1... A i1 i

12 Fock-space one-body operator Consider Fock-space operator ote the ^ notation This operator accomplishes the same as for any! Use ˆF [ ˆF,a ] F [a a i,a ] i F F (a a a i a i a a ) F a (a a + a a i i ) F a, i F i a i F i a i F a a and apply i ˆF ˆFa a...a [ˆF,a ]a...a 0 + a ˆFa...a [ˆF,a ]a...a 0 + a [ ˆF,a ]...a a a...[ ˆF,a ] i1 i i F i a 1...a i 1 a i a i+1...a 0 i1 i i F i 1... i 1 i i+1...

13 Examples Density operator for particles (r) i1 (r r i ) Second-quantized form: choose { r,m s } basis In Fock space ˆ(r) m s,m s d 3 r 1 d 3 r 1 r 1 m s (r r op ) r 1 m s a r 1 m s a r1 m s a rm s a rms m s Kinetic energy ˆT T a a p 1 m 1 p 2 m 2 p 1 m 1 p2 op 2m p 2m 2 a p 1 m 1 a p2 m 2 p 1 m 1 p2 1 2m a p 1 m 1 a p1 m 1

14 Consider Determine ˆ ˆ,a i More examples a a a a,a i a i Therefore ˆ Change of basis a 0 Can be done for any state in Fock space a Also a a a 0 a

15 Similar strategy -particles Consider Two-body operators in Fock space V V )( V )( V (1, 2)+ V (1, 3)+ V (1, 4)+...+ V (1,)+ V (2, 3)+ V (2, 4)+...+ V (2,)+ V (3, 4)+...+ V (3,)+.... V ( 1,) V (i, j) 1 V (i, j) 2 i<j1 V (i, j) 1.. i.. j.. ) Matrix elements do not depend on the selected pair i j ij ( i j V i j) 1.. i 1 i i+1.. j 1 j j+1.. ) ( i j V i j) identical for any pair as long as quantum numbers are the same, so V ) i<j1 i j ( i j V i j) 1... i... j... )

16 V More on two-body operators ote: symmetric and therefore commutes with antisymmetrizer As a consequence V Fock-space operator i<j1 i j ( i j V i j) 1... i... j... ˆV 1 2 ( V )a a a a accomplishes the same result for any particle number! ote ordering

17 Two-body operator Use [ ˆV,a i ] 1 2 ( V )a a [a a,a i ]...a a (a a a i a i a a )...a a (a (, i a i a ) a i a a )...a a (a, i, i a ) 1 2 ( V i )a a a 1 2 ( V i )a a a ( V i )a a a i j i ( i j V i i )a i a j a i ote ( V )( V ) since V (i, j) V (j, i)

18 Use to show Two-body operators ˆV ˆVa 1 a 2...a 0 i1 a 1...[ ˆV,a i ]...a 0 i1 i i i ( i i V i i )a 1...a i a i a i a i+1...a 0 i1 j>i i j ( i j V i j)a 1...a i...a j...a 0 Employ j i f( j, i )[a j a i,a j ] j f( j, j)a j Often used ˆV 1 4 V a a a a with Check! V ( V ) ( V ) ˆV

19 Most common operator otation often used Hamiltonian Ĥ ˆT + ˆV ms (r) a rms T a a ( V )a a a a Use and In this basis (r 1 m s1 r 2 m s2 V (r, r ) r 3 m s3 r 4 m s4 ) (r 1 r 3 ) (r 2 r 4 ) Ĥ ms rm s T r m s rm s p2 2m r m s msm s d 3 r 2 ms (r) d 3 r d 3 r i 2m rm s p r m s 2 2m 2 rm s r m s 2 2m 2 (r r ) m s,m s 2m 2 ms (r) ms (r) m s (r )V ( r m s 1,ms 3 ms 2,ms 4 V ( r 3 r 4 ) r ) ms (r ) m s (r) appears as second quantization

20 Symmetric -boson states Symmetrizer Symmetrized states with n Completeness (unrestricted) Ordered sp quantum numbers ormalization S 1! p P Bosons ordered 1/2! S ) n!n! n!n!...! , 1 2, 2..., ot ordered [n!..n!..] 1/2 P 1 p 1 2 p 2.. p

21 Boson addition and removal operator otation /2! S ) n n... n!n!... As for fermions Two particles Otherwise a 0 when a a 0 n 2 1 a a 0 2 General case n n...n 1 [n!n!...n!] 1/2 a n a n... a n 0 Contrast with fermions: for bosons commutation relations [a,a ]a a a a, [a,a ][a,a ]0 from the requirement to generate symmetric states

22 As for oscillators Properties of boson operators and a n n...n n +1 n +1n...n a n n...n n n 1 n...n As for fermions and ˆF ˆV 1 2 F a a ( V )a a a a