p137bp10sol

# p137bp10sol - Physics 137B Fall 2007 Moore Problem Set 10...

This preview shows pages 1–3. 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: Physics 137B, Fall 2007, Moore Problem Set 10 Solutions 1. With the probability current defined by j = ~ 2 im ( ψ * ∇ ψ- ( ∇ ψ * ) ψ ) , we see by straightforward differentiation that ∂ ∂t | ψ | 2 + ∇ · j = ∂ψ * ∂t ψ + ψ * ∂ψ ∂t + ~ 2 im ∇ ψ * · ∇ ψ + ψ * ∇ 2 ψ- ( ∇ 2 ψ * ) ∇ - ∇ ψ * · ∇ ψ = ∂ψ * ∂t ψ + ψ * ∂ψ ∂t + ~ 2 im ψ * ∇ 2 ψ- ( ∇ 2 ψ * ) ψ = ψ * ∂ψ ∂t + ~ 2 im ∇ 2 ψ + ∂ψ * ∂t- ~ 2 im ∇ 2 ψ * ψ . Now the wavefunction ψ = ψ ( x , t ) satisfies the Schr¨ odinger equation i ~ ∂ψ ∂t =- ~ 2 2 m ∇ 2 + V ( x ) ψ , where V ( x ) is a real-valued potential. Thus ∂ψ ∂t + ~ 2 im ∇ 2 ψ = 1 i ~ V ( x ) ψ 1 and ∂ψ * ∂t- ~ 2 im ∇ 2 ψ * =- 1 i ~ V ( x ) ψ * , so ∂ ∂t | ψ | 2 + ∇ · j = ψ * 1 i ~ V ( x ) ψ +- 1 i ~ V ( x ) ψ * ψ = 1 i ~ [ ψ * V ( x ) ψ- V ( x ) ψ * ψ ] = 0 . If the wavefunction ψ is completely real then ψ * = ψ , and the probability current j = ~ 2 im ( ψ ∇ ψ- ( ∇ ψ ) ψ ) = 0 vanishes identically. 2. Let b be the impact parameter and θ the scattering angle of an classical particle scattering off a hard sphere of radius r . Classically, the particle will bounce off...
View Full Document

## This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at Berkeley.

### Page1 / 5

p137bp10sol - Physics 137B Fall 2007 Moore Problem Set 10...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online