The calculation of multidimensional semiclassical wave functions in the forbidden region using real valued...

A method that uses only real valued coordinates is presented for integrating the many dimensional semiclassical wave function into the forbidden region. The procedure first determines a surface of caustic points by running the set of trajectories that define the wave function in the allowed region. In the forbidden region, the momentum and the action integral are both complex functions of position, and their imaginary parts vanish on the caustic surface. The direction of the imaginary part of the momentum p(I) can be chosen to the perpendicular to the caustic surface at all points on that surface. Equations are derived for integrating the values of the real and imaginary parts of the momentum along the curves that follow the direction of p(I). The equations for the change in the action integral and the prefactor for the semiclassical wave function along these curves are also obtained, allowing for the determination of the semiclassical wave function in the forbidden region. Calculations are performed for a two dimensional problem, and the semiclassical wave function is found to be is excellent agreement with the results of exact quantum calculations.