February 8, 2009

PHY 745 - Problem Set #10

This homework is due Wednesday, February 11, 2009.

Continue reading Chapter 4 in Tinkham.

1. Consider a quantum mechanical free particle of mass $m$ confined within a rectangular box of dimensions $-a \le x \le a$, $-a \le y \le a$, and $-b \le z \le b$.
   1. Check that the eigenstates of the particle all vanish on the 6 planes that bound the box and take the form:
      
      $$\Psi_{lmn}(x,y,z) = \frac{1}{\sqrt{a^2b}} \; w_l\left( \fra... ...{m \pi y}{2a}\right) \; w_n\left( \frac{n \pi y}{2b}\right),$$
      
      
      where
      
      $$w_n(u) = \left\{ \begin{array}{ll} \cos(u) & {\mbox{\rm {if... ...rm {if }}} n \equiv {\mbox{\rm { even}}} \end{array} \right.$$
      
      

   2. Now consider these states with reference to the $D_4$ point group discussed in your text book and in the Notes for Lecture 9 - http://www.wfu.edu/~natalie/s09phy745/lecturenote/ . From the character table for this point group, for each irreducible representation, find at least one example of a basis function from the $\Psi_{lmn}(x,y,z)$ eigenstates.

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