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

This preview shows pages 1–2. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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

{[ snackBarMessage ]}

### Page1 / 3

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online