Symbolic computation and automatic FORTRAN code generation for eigenvalue determination by phase integral method (Q916321)

Let us have the Schrödinger equation \((1)\quad \psi ''(\phi)+[E- V(\phi)]\psi (\phi)=0.\) The phase integral quantization condition for a potential V(\(\phi\)) of periodicity of \(2\pi\) with m different wells and barriers is obtained by the following changes in the coefficients \((C_+,C_ -)\) in the phase integral wave function of the form \[ (2)\quad \psi (\phi)=[q(\phi)]^{-1/2}[C_+\exp (+i\int^{\phi}_{c}q(\phi)d\phi)+C_ -(- i\int^{\phi}_{c}q(\phi)d\phi)], \] where q(\(\phi\)) is the phase function and c the phase reference point, subject to the boundary conditions \(\phi (0)=\phi (2\pi)\), \(\psi '(0)=\psi '(2)\). For a potential well with classical turning points at c and b, equation (2) may also be written as \[ \psi (\phi)=[q(\phi)]^{-1/2}[B_+\exp (+i\int^{\phi}_{b}q(\phi)d\phi)+B_ -\exp (- i\int^{\phi}_{b}q(\phi)d\phi)] \] so that the connection from b to c is given by \(\left( \begin{matrix} C_+\\ C_ -\end{matrix} \right)=B\left( \begin{matrix} B_+\\ B_ -\end{matrix} \right),\text{ where } B=\left( \begin{matrix} e^{i\alpha}\\ 0\end{matrix} \begin{matrix} 0\\ e^{-i\alpha}\end{matrix} \right),\quad \alpha =i\int^{c}_{b}q(\phi)d\phi.\) A combination of a symbolic language MACSYMA and numerical FORTRAN software are used for determination of energy eigenvalues for (1) by the phase integral method.