852HW11 - HOMEWORK ASSIGNMENT 11 PHYS852 Quantum Mechanics...

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Unformatted text preview: HOMEWORK ASSIGNMENT 11 PHYS852 Quantum Mechanics II, Spring 2009 New topics covered: Bosonic quantum field theory . 1. Using | ( n ) ) = 1 N ! integraldisplay d 3 r 1 ... d 3 r n n ( vector r 1 ,... ,vector r n ) ( vector r 1 ) ... ( vector r n ) | ) , ( vector r ) = ( vector r ) ( vector r ) , and N = integraldisplay d 3 r ( vector r ) ( vector r ) , prove the following properties of Fock-space quantum states: a.) Show that | ( n ) ) is an eigenstate N , with eigenvalue n . b.) Use your result from a.) to prove that ( ( m ) | ( n ) ) = 0 for n negationslash = m . c.) Show that ( (2) | (2) ) = 1 only if 2 ( vector r,vector r ) is properly symmetrized. d.) Compute ( ( vector r ) ) and ( ( ( vector r )) 2 ) for | (2) ) . What is the condition on 2 ( vector r 1 ,vector r 2 ) to have a well-defined density (i.e. ( vector r ) = 0). f.) Show that i planckover2pi1 d dt | (2) ( t ) ) = H | (2) ( t ) ) leads to the usual Schrodinger equation for the two-particle wavefunction 2 ( vector r 1 ,vector r 2 ,t ), where H = integraldisplay d 3 r ( vector r ) bracketleftbigg planckover2pi1 2 2 M 2 + V 1 ( vector r ) bracketrightbigg ( vector r ) + 1 2 integraldisplay d 3 rd 3 r...
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852HW11 - HOMEWORK ASSIGNMENT 11 PHYS852 Quantum Mechanics...

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