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Transcript

PHYS6520 Quantum Mechanics II Spring 2013 HW #5 Due to the grader at 5pm on Friday 5 Apr 2013 (1) Consider scattering in one dimension x from a potential V (x) localized near √ x = 0. The initial state is a plane wave coming from the left, that is φ(x) ≡ �x|i� = eikx / 2π (a) Find the scattering Green’s function G(x, x� ), defined in one dimension analogously with (6.2.3). You only need concern yourself with scattering from the past to the future. (b) Take the special case of an attractive δ-function potential, namely h̄2 V (x) = −γ δ(x) 2m with γ>0 Use the Lippman-Schwinger Equation to find the outgoing wave function ψ(x) ≡ �x|ψ (+) �. (c) Determine the transmission and reflection coeﬃcients T (k) and R(k), defined as ψ(x) = T (k)φ(x) for x > 0 and e−ikx ψ(x) = φ(x) + R(k) √ for x < 0 2π Show that |T |2 + |R|2 = 1, as must be the case. (d) Confirm that you get the same result by using grade-school quantum mechanics and matching right and left going waves on the left with a right going wave on the right at x = 0. You’ll need to integrate the Schrödinger equation across x = 0 to match the derivatives. (e) We showed last semester that this potential has one, and only one, bound state. Show that your results for T (k) and R(k) have bound state poles at the expected positions when k is treated as a complex variable. (2) Modern Quantum Mechanics, Problem 6.3 (3) Modern Quantum Mechanics, Problem 6.4. I recommend Chapter 10 of the NIST Digital Library of Mathematical Functions at http://dlmf.nist.gov/ as a good online reference for the properties of spherical Bessel functions. (4) Modern Quantum Mechanics, Problem 6.12