hw5 - E V(c For a potential such as this which does not go...

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Assignment #5 PHYS-446 due in class on 7 November 2008 1. (Griffiths, problem 2.21) A free particle has the initial wave function  x , 0 = Ae a x where A and a are positive real constants. (a) Normalize  x , 0 . (b) Find  k . (c) Construct  x ,t , in the form of an integral. (d) Discuss the limiting cases ( a very large, and a very small). 2. Consider a wavepacket that describes a neutrino, which is massless to a good approximation, so that E=pc . Show that such a wavepacket does not spread with time. 3. (Griffiths, Problem 2.34) Consider the “step” potential: V x = { 0 x 0 V 0 x 0 (a) Calculate the reflection coefficient, for the case E V 0 , and comment on the answer. (b) Calculate the reflection coefficient for the case
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Unformatted text preview: E V . (c) For a potential such as this, which does not go back to zero to the right of the barrier, the transmission coefficient is not simply ∣ F ∣ 2 / ∣ A ∣ 2 (with A the incident amplitude and F the transmitted amplitude), because the transmitted wave travels at a different speed. Show that T = E − V E ∣ F ∣ 2 ∣ A ∣ 2 for E V . Hint: You can figure it out using Equation 2.98, or from the probability current (Problem 2.19). What is T for E V ? (d) For E V , calculate the transmission coefficient for the step potential, and check that T R = 1 ....
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