hw206a - Physics 512 Homework Set #6 Due Monday, March 3...

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Physics 512 Winter 2003 Homework Set #6 – Due Monday, March 3 1. One dimensional scattering. Consider scattering from a potential V ( x )= ( x ). If we send in a particle from the left ψ inc ( x e ikx x< 0 there will be an amplitude for both reflection and transmission. The transmitted part may be written ψ trans = S ( E ) e ikx = Te i ( kx + δ ) x> 0 where T is the (real) transmission coefficient and δ the phase shift ( E is the energy). a ) Find the transmission coefficient and phase shift and show that T is insensitive to the sign of g . What are the limiting values of δ for E 0 and E →∞ (consider both positive and negative g )? Although the incident wavefunction is e ikx , there must be a reflected one as well. Thus the complete wavefunction may be given as ψ ( x ± ψ < = e ikx + Be ikx 0 ψ > = Se ikx 0 We must satisfy the continuity and jump conditions at x =0 : ψ < (0) = ψ > (0) 0 < (0) = ψ 0 > (0) + 2 mg ¯ h 2 ψ (0) This gives a set of equations 1+ B = S, ik (1 B ikS + 2 mg ¯ h 2 S which may be solved to give S ( E ² 1 img k ¯ h 2 ³ 1 ,E = ¯ h 2 k 2 2 m (1) The transmission coefficient and phase shift is obtained by rewriting S ( E ) in terms of a magnitude and phase, S = . The result is simply T = 1 ( mg k ¯ h 2 ) 2 = tan 1 mg k ¯ h 2 Since g enters squared in T , the transmission coefficient is insensitive to the sign of g . However this is not the case for the phase shift. As for the limiting behavior of δ , we note that there is always a 2 π phase ambiguity present. However we can 1
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defne the phase to be 0 when g 0 . This corresponds to taking tan 1 to lie between π/ 2 and 2 . The limiting phase shiFts are then given by E 0 E →∞ g> 0 20 g< 0 In general, we may understand the sign oF the phase shiFt as being related to the attractive versus repulsive nature oF the potential.
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This note was uploaded on 02/15/2011 for the course PHYS 512 taught by Professor Unknow during the Winter '03 term at Cornell College.

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hw206a - Physics 512 Homework Set #6 Due Monday, March 3...

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