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p137bp10 - Physics 137B Fall 2007 Problem set 10 Fermi...

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Physics 137B, Fall 2007 Problem set 10 : Fermi gases, molecules, and scattering Assigned Friday, 23 November. Due in box Friday, 30 November. 1. Write out for yourself that the probability current j = ¯ h 2 im ( ψ * ψ - ( ψ * ) ψ ) (1) satisfies the “continuity equation” | ψ | 2 ∂t + ∇ · j = 0 . (2) If you get stuck, you can refer to class notes or the derivation in Bransden 3.2. Use this expression for j to show that any wavefunction that is completely real (has no imaginary part) has vanishing probability current. 2. In class we said that the total cross section seen by classical small particles scattering off a large stationary hard sphere of radius r 0 is σ tot = πr 0 2 . Use classical physics to calculate the differential cross section d Ω in this case, as a function of the scattering angle θ . Hint: first show by explaing an appropriate diagram that the impact parameter b is related to θ via b r 0 = cos( θ/ 2) (3) Now obtain d Ω by writing d Ω = 2 πb 2 π sin θ db , (4)
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