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March 1, 2007
7.35 The wavefunction that we are given depends only on x and t. So the only nonzero component of J will be Jx . We are given: ¥÷(x, t) = Aei¥õ(x,t) , where ¥õ(x, t) is real. Use equation (7.106) in Libo? to ?nd: Jx = h ? ¡Ó¥÷ ? ¡Ó¥÷ ¥÷? ?¥÷ 2mi ¡Óx ¡Óx
=
h ? 2mi
A? e?i¥õ(x,t)
Aiei¥õ(x,t)
¡Ó¥õ ¡Óx
? Aei¥õ(x,t)
A? (?i)e?i¥õ(x,t)
¡Ó¥õ ¡Óx
=
h ? ¡Ó¥õ |A|2 . m ¡Óx
7.38 a.According to Eq. (7.107), J= h ? ¥÷? ¥÷ ? ¥÷ ¥÷? . 2mi
If ¥÷ is real, we have ¥÷ ? = ¥÷, and then clearly J = 0. b.From the continuity equation (equation (7.105) in Libo?), J = 0 implies ¡Ó¥ñ ¡Ót = 0, where ¥ñ is the probability density or alternatively the particle density in scattering problems. So, we see that a real state function implies that the probability density (aka particle density) remains constant.
1
Ph12b Solutions
Problem Set 7
March 1, 2007
7.41 Consider the two con?gurations in Figure 7.21. Libo? has calculated the re?e
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