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Unformatted text preview: 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 (x, t) = A(2 sin x -i E1 t 2x -i E2 t h2 2 . e h + sin e h ) , E n = n2 L L 2mL2 (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 = px /m. In quantum mechanics, d this is replaced by a corresponding relation between expectation values: dt x = px . Verify m this relation by using the definition of x , the Schroedinger equation, and the fact that you can write -i x for px . h d (b) Can you obtain the quantum-mechanical counterpart of Newton's second law: dt px = - V . x (From [1,8-19]) Exercise 3: One can represent a real function f (x) as integral over sin and cos functions, f (x) =
0 C(k) cos(kx)dk +
0 S(k) sin(kx)dk . (2) How are the functions C(k) and S(k) related to the Fourier transform F (k) of f (x)? Exercise 4: Given a potential wall with V1 < V0 and 0 V (x) = V0 V1 unequal potential plateaus on both sides, i.e. for for for x<0 0x<L Lx (3) Determine the transmission coefficients for waves coming from x < 0 and for waves coming from x > 0. Exercise 5: Read [1, Sec. 10]  An Introduction to Quantum Physics, French and Taylor, Norton, W. W. & Company, Inc. (1990) ...
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