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ps10 - Homework(10 for Physics 316 Modern Physics...

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Homework (10) for Physics 316, Modern Physics I (Hoffstaetter/Drasco/Thibault) Due Date: Friday, 04/15/05 - 9:55 in 132 Rockefeller Hall Exercise 1: A particle in an infinite square well extending between x = 0 and x = L has the wave function Ψ( x, t ) = A (2 sin πx L e - i E 1 ¯ h t + sin 2 πx L e - i E 2 ¯ h t ) , E n = n 2 ¯ h 2 π 2 2 mL 2 . (1) a) Choose A so that the wave function is normalized to 1. b) If a measurement of the energy is made, what are the possible results of the measurement, and what is the probability associated with each? c) When the energy is measured for many particles, each having been prepared in this state, what would the average energy be. This average is called the expectation value. (from [1, 8-2]) Exercise 2: (a) In classical mechanics, we have put dx/dt = p x /m . In quantum mechanics, this is replaced by a corresponding relation between expectation values: d dt h x i = h p x i m . Verify this relation by using the definition of h x i , the Schroedinger equation, and the fact that you
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