3.12 Let E, denote the bound-state energy eigenvalues of a one-dimensional system and let yn(x) denote the corresponding energy eigenfunctions. Let ¥(x, 1) be the wave function of the system, normalised to unity, and suppose that at t = 0 it is given by 1 eiaivi(x) + 1 :el":/2(x) + 1 iaz e"V3(x) ¥(x,1 = 0) = where the a; are real constants. (a) Write down the wave function ¥(x, 1) at time t. (b) Find the probability that at time t a measurement of the energy of the system gives the value E2. (c) Does (x) vary with time? Does (px) vary with time? Does E = (H) vary with time?

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