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**Problem 2.6 A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: Y (x, 0) = A[V(x) 2(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy if you exploit the orthonormality of and 2. Recall that, having normalized V at t = 0, you can rest assured that it stays normalized-if you doubt this, check it explicitly after doing part b.) (b) Find V(x, t) and | (x, t)2. (Express the latter in terms of sinusoidal functions of time, eliminating the exponentials with the help of Euler's formula: e'e = cos 0 +i sin 0.) Let w = n?h/2ma?. (c) Compute (x). Notice that it oscillates in time. What is the frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a/2, go directly to jail.)

**Problem 2.6 A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: Y (x, 0) = A[V(x) 2(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy if you exploit the orthonormality of and 2. Recall that, having normalized V at t = 0, you can rest assured that it stays normalized-if you doubt this, check it explicitly after doing part b.) (b) Find V(x, t) and | (x, t)2. (Express the latter in terms of sinusoidal functions of time, eliminating the exponentials with the help of Euler's formula: e'e = cos 0 +i sin 0.) Let w = n?h/2ma?. (c) Compute (x). Notice that it oscillates in time. What is the 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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**Problem 2.6 A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: V (x, 0) = A[V1 (x) + 2(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy if you exploit the orthonormality of and 2. Recall that, having normalized at t = 0, you can rest assured that it stays normalized-if you doubt this, check it explicitly after doing part b.) (b) Find V (x, t) and | (x, t)|2. (Express the latter in terms of sinusoidal functions of time, eliminating the exponentials with the help of Euler's formula: ei cos e +i sin 0.) Let w = n'h/2ma?. (c) Compute (x). Notice that it oscillates in time. What is the frequency of the oscillation? What is the amplitude of the oscillation? (If your amplitude is greater than a/2, go directly to jail.)
Transcribed Image Text:**Problem 2.6 A particle in the infinite square well has as its initial wave function an even mixture of the first two stationary states: V (x, 0) = A[V1 (x) + 2(x)]. (a) Normalize (x, 0). (That is, find A. This is very easy if you exploit the orthonormality of and 2. Recall that, having normalized at t = 0, you can rest assured that it stays normalized-if you doubt this, check it explicitly after doing part b.) (b) Find V (x, t) and | (x, t)|2. (Express the latter in terms of sinusoidal functions of time, eliminating the exponentials with the help of Euler's formula: ei cos e +i sin 0.) Let w = n'h/2ma?. (c) Compute (x). Notice that it oscillates in time. What is the 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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