January 23, 2009

PHY 745 - Problem Set #3

This homework is due Monday, January 26, 2009.

Continue reading Chapter 3 in Tinkham.

1. Consider the following $2 \times 2$ ``normal" matrix $(N N^{\dagger} = N^{\dagger} N)$ in terms of real constants $a$, $b$, $\beta$, and $\gamma$.
   
   $$N = \left( \begin{array}{cc} a & b {\rm {e}}^{i \beta} \\ b {\rm {e}}^{i \gamma} & a \end{array} \right).$$
   
   
   1. Find the eigenvalues $\lambda_i$ and eigenvectors $v_i$
      
      $$N v_i = \lambda_i v_i.$$
      
      

   2. Show that
      
      $$N^{\dagger} v_i = \lambda^*_i v_i.$$
      
      

   3. Find the relationships between the constants for the case that N is Hermitian.
   4. Find the relationships between the constants for the case that N is unitary.

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