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Fall 09 Quiz 1 - Quantum Mechanics I Physics 3143 Quiz 1...

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Unformatted text preview: Quantum Mechanics I, Physics 3143: Quiz 1, Fall 2009 Please answer all questions. Start each question on a new page. Clear explanations are required for partial credit. 1. Calculate the commutator [(12, of], given that the harmonic oscillator raising and lowering operators satisfy the commutation relation [a, all] : 1. NOTE: In Grifliths’s text a —> a- and ar I—> a... 2. Consider the wave function @(x, 0) = Ae“%°‘$2+mx+i7, where o}, [i and ’y are real constants. (a) Show that the normalization constant satisfies, |A| = g. (b) Find the expectation values, (at), (3:2) and (p). (c) What is the average momentum if the system state is instead given by the wave function M520) = A’e‘iaxQSiMfim + 7) :2 NOTE: In wave mechanics p = ~i7ia/ 6:13. The normalized Gaussian probability distri— bution in one dimension may be written 3. A particle of mass m, confined in the region 0 < a: < a, is prepared in the initial state with normalized wave function me) = gene) + sees». (a) Find the time-dependent wave function @(rr, t). (13) Find the expectation value of energy (H). (c) What are the possible results of a measurement of energy '3 NOTE: The stationary states, 1042:) = fisinmwx/a), which form a complete, or- thonormal set, satisfy the energy eigenvalue equation Hibn = Enibn, n = 1,2,3,..., with En = h2u2n2/(2ma2). ...
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