hw2 - 4(Griffiths Problem 1.16 Show that d dt −∞ ∞...

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Assignment #2 PHYS-446 due in class on 26 September 2008 1. A particle in free space is initially in a wave packet described by  x = 1 / 4 e − x 2 / 2 What is the expectation value of the momentum? 2. (Shankar, Problem 4.2.3) Show that if a state  x has mean momentum P , then the combination e ip 0 x /  x has mean momentum P p 0 . 3. (Griffiths, Problem 1.8) Suppose you add a constant V 0 to the potential energy of a system. In classical mechanics this doesn't change anything, but what about quantum mechanics? Show that the wave function picks up a time-dependent phase factor: e iV 0 t / . What effect does this have on the expectation value of a dynamical variable?
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Unformatted text preview: 4. (Griffiths, Problem 1.16) Show that d dt ∫ −∞ ∞ 1 ∗ 2 dx = for any two (normalizable) solutions to the Schrödinger equation, 1 x ,t and 2 x,t . 5. (Griffiths, Problem 1.9) A particle of mass m is in the state x ,t = Ae − a [ mx 2 / ℏ it ] where A and a are positive real constants. (a) Find A . (b) For what potential energy function V x does satisfy the Schrödinger equation? (c) Calculate the expectation values of X ,X 2 ,P and P 2 . (d) Find x and p . Is their product consistent with the uncertainty principle?...
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