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HW1 - ψ satisfies the anticommutation relations ψ ~x,ψ...

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PHY5667 Problem Set #1 (due Tue Aug 31) (If you are taking a qualifying exam this week, just email me and you will be allowed to hand in this HW by Thursday, Sep 2, without any penalty.) (1) Prove the following identities for the commutator [ , ] and the anticommutator { , } : (a) [ A, BC ] = [ A, B ] C + B [ A, C ] . (b) [ A, BC ] = { A, B } C - B { A, C } . (2) In order to fill in the missing step in the calculation in the lecture, show that ψ ( ~x ) ψ ( ~ y ) ψ ( ~x 1 ) ψ ( ~x 2 ) | 0 i = δ ( ~x - ~x 1 ) δ ( ~ y - ~x 2 ) + δ ( ~x - ~x 2 ) δ ( ~ y - ~x 1 ) | 0 i , assuming that the field operator ψ ( ~x ) satisfies the commutation relations [ ψ ( ~x ) , ψ ( ~ y )] = δ ( ~x - ~ y ) , [ ψ ( ~x ) , ψ ( ~ y )] = [ ψ ( ~x ) , ψ ( ~ y )] = 0 . [Hint: recall what you always have to do in QFT — move annihilation operators to the right!] (3) As we did in class, consider a QFT for a field ψ ( ~x ) (in the Schr¨ odinger picture) with the Hamiltonian density given by H ( ~x ) = - 1 2 m ψ ( ~x ) 2 ψ ( ~x ) + 1 2 Z d 3 ~ y ψ ( ~x ) ψ ( ~ y ) V ( ~x - ~ y ) ψ ( ~ y ) ψ ( ~x ) , but this time assume that
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Unformatted text preview: ψ satisfies the anticommutation relations { ψ ( ~x ) ,ψ † ( ~ y ) } = δ ( ~x-~ y ) , { ψ ( ~x ) ,ψ ( ~ y ) } = { ψ † ( ~x ) ,ψ † ( ~ y ) } = 0 . Let us specialize a two-particle state | φ i ≡ Z d 3 ~x 1 d 3 ~x 2 φ ( t,~x 1 ,~x 2 ) ψ † ( ~x 1 ) ψ † ( ~x 2 ) | i , where φ is just a function (not an operator). (a) Show that φ obeys the Fermi-Dirac statistics, i.e., φ ( t,~x 2 ,~x 1 ) =-φ ( t,~x 1 ,~x 2 ). (b) Show that φ obeys a two-particle Schr¨odinger equation i ∂ t φ ( t,~x 1 ,~x 2 ) = ³-1 2 m ∇ 2 x 1-1 2 m ∇ 2 x 2 + V ( ~x 1-~x 2 ) ´ φ ( t,~x 1 ,~x 2 ) . To receive full credit, be sure to keep track of minus signs associated with the Fermi-Dirac statistics in your calculations. 1...
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