hw solutions 6

# hw solutions 6 - Physics 519 Homework Set#6 Due in class...

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Physics 519 Homework Set #6 Spring 2013 Due in class 5/22/13 300 pts 1. (100 pts) Sakurai, 7.1: The Lippmann-Schwinger formalism can also be applied to a one-dimensional transmission-reflection problem with finite-range potential, V ( x ) = 0 for 0 < | x | < a only. a. Suppose we have an incident wave coming from the left: x | φ = e ikx / 2 π . How must we handle the singular 1 / ( E - H 0 ) operator if we are to have a transmitted wave only for x > a and a reflected wave and the original wave for x < - a ? Is the E E + i prescription still correct? Obtain an expression for the appropriate Green’s function and write an integral equation for x | ψ + We will assume that the integral equation for the scattering state | ψ (+) has the same form as in 3-dim, and show that this is correct. The form is: | ψ (+) = | φ + GV | ψ (+) , (1) where | φ is the free particle state with wavenumber k , normalized so that x | φ = e ikx / 2 π , V = V ( x ) is the potential (with x the usual QM operator), and G is a Green function for the free Schr¨ odinger operator. Crudely speaking G = 1 E - H 0 , H 0 = p 2 2 m , but the singularities in G need to be regularized by an “ -prescription”. To make things more concrete, let’s rewrite (1) in terms of wavefunc- tions, x | ψ (+) = x | φ + d 3 x G ( x, x ) V ( x ) x | ψ (+) , (2) where G ( x, x ) = x | G | x . Acting on this with ¯ h 2 2 m k 2 + d 2 dx 2 we see that x | ψ (+) satisfies the Schr¨ odinger equation as long as ¯ h 2 2 m k 2 + d 2 dx 2 G ( x, x ) = δ ( x - x ) . (3) Note that this normalization of G is different from that used in the 3-dim case in class. We next need to choose appropriate BC for G , which translates into a pole prescription. First, we want G to be translation invariant, 1

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so that the scattering from a translated potential is described by a translated wavefunction. Thus G ( x, x ) = G ( x - x ) . Second, we want the second term on the r.h.s. of (2) to contain only outgoing waves.
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• Spring '13
• Lin
• mechanics, Work, SCATTERING, Scattering theory, dr r, L-S equation

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