6-35 Applying the momentum operator px iddxto each of the candidate functions yields (a) px Asinkxik Acoskx(b) px AsinkxAcoskxik AcoskxAsinkx(c) px AcoskxiAsinkxikAsinkxiAcoskx(d) px eik xaiik eik xaIn case (c), the result is a multiple of the original function, since AsinkxiAcoskxi AcoskxiAsinkx. The multiple is iik kand is the eigenvalue. Likewise for (d), the operation px returns the original function with the multiplier k. Thus, (c) and (d) are eigenfunctions of px with eigenvalue k, whereas (a) and (b) are not eigenfunctions of this operator. 6-37 (a) Normalization requires 12dxC21*2*12dxC212dx22dx2*1dx1*2dx. The first two integrals on the right are unity, while the last two are, in fact, the same integral since 1and 2are both real. Using the waveforms for the infinite square well, we find 21dx2LsinxLsin2xLdx0L1LcosxL cos3xLdx0Lwhere, 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, 1C210002C2, or C12. Since 1,2are stationary states, they develop in time according to their respective energies E1,2as eiEt. Then x,tC1eiE1t2eiE2t.
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