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). ...
View Full Document
- Fall '10