A particle in the infinite square well has as its initial wave function an even mixture of...

A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states:

(a) Normalize ψ(x,0). (That is, find A. This is very easy, if you exploit the orthonormality of ψ1 and ψ2 Recall that, having normalized ψ at t = 0, you can rest assured that it stays normalized—if you doubt this, check it explicitly after doing part (b).)

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(b) Find ψ (x, t)and |ψ(x, t)|2, Express the latter as a sinusoidal function of time, as in Example 2.1. To simplify the result, let ω ≡ π2ħ/2ma2.

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(c) Compute ‹x›. Notice that it oscillates in time. What is the angular frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a/2, go directly to jail.)

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(d) Compute ‹p›. (As Peter Lorre would say, “Do it ze kveek vay, Johnny!”)