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# 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 x < 0 ψ > = Se ikx x > 0 We must satisfy the continuity and jump conditions at x = 0 : ψ < (0) = ψ > (0) , ψ < (0) = ψ > (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 ) = 1 img k ¯ h 2 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 = T e . The result is simply T = 1 1 + ( mg k ¯ h 2 ) 2 , δ = 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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define the phase to be 0 when g 0 . This corresponds to taking tan 1 to lie between π/ 2 and π/ 2 . The limiting phase shifts are then given by E 0 E → ∞ g > 0 π/ 2 0 g < 0 π/ 2 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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hw206a - Physics 512 Homework Set#6 Due Monday March 3...

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