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

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

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

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