PHYS 307:  Computational Physics Lab (QM)

Winter 2008

Homework #2
(Due:  January 22, 2008)

1. In class we studied the finite potential well problem--that is, the solution to Schrödinger's equation with potential
   
   $$V(x) = \left\{ \begin{array}{ll} 0~~~ & (\vert x\vert < a)\,,\\ V_0~~~ & (\vert x\vert > a)\,. \end{array} \right.$$
   
   
   Analytical solution entails solving the equations
   
   $$\xi\tan\xi = \eta$$
   
   
   or
   
   $$-\xi\cot\xi = \eta$$
   
   
   subject to the condition
   
   $$\xi^2 + \eta^2 = \frac{2 m V_0 a^2}{\hbar^2} \equiv \Phi_0\,,$$
   
   
   where $m$ is the mass of the particle, $\xi^2 = 2mEa^2/\hbar^2$, and $\eta^2 = 2m(V_0-E)a^2/\hbar^2$. The energy $E$ lies in the range $0 < E < V_0$. The ``tan'' and ``cot'' equations correspond, respectively, to even and odd solutions for the wavefunction.

   Modify the programs discussed in class to solve these equations using the bisection/false-position technique, making sure that all solutions are found. Then, by looping over $\Phi_0$, plot your solutions for scaled energy $\xi^2$ as functions of scaled potential $\Phi_0$, for $\Phi_0 = 0,\ldots,100$ in steps of 0.1. Turn in your final program and a table of results, as well as a graph of $\xi^2(\Phi_0)$.