hw8 - 6-35 d Applying the momentum operator px to each of...

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6-35 Applying the momentum operator p x   i d dx to each of the candidate functions yields (a) p x   A sin kx i k A cos kx (b) p x   A sin kx A cos kx i k A cos kx A sin kx (c) p x   A cos kx iA sin kx i k A sin kx iA cos kx (d) p x   e ik x a i ik e ik x a In case (c), the result is a multiple of the original function, since A sin kx iA cos kx i A cos kx iA sin kx . The multiple is i ik   k and is the eigenvalue. Likewise for (d), the operation p x   returns the original function with the multiplier k . Thus, (c) and (d) are eigenfunctions of p x   with eigenvalue k , whereas (a) and (b) are not eigenfunctions of this operator. 6-37 (a) Normalization requires 1 2 dx  C 2 1 * 2 * 1 2 dx  C 2 1 2 dx 2 2 dx 2 * 1 dx 1 * 2 dx . The first two integrals on the right are unity, while the last two are, in fact, the same integral since 1 and 2 are both real. Using the waveforms for the infinite square well, we find 2 1 dx 2 L sin x L sin 2 x L dx 0 L 1 L cos x L   cos 3 x L dx 0 L where, in writing the last line, we have used the trigonometric exponential identities of sine and cosine. Both of the integrals remaining are readily evaluated, and are zero. Thus, 1 C 2 1 0 0 0 2 C 2 , or C 1 2 . Since 1,2 are stationary states, they develop in time according to their respective energies E 1,2 as e iEt . Then x , t C 1 e iE 1 t 2 e iE 2 t .
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