A new method to derive low-lying N-dimensional quantum wave functions by quadratures along a single trajectory

A new method to solve N-dimensional ground state quantum wave functions is developed based on quadratures along a single trajectory. By introducing a scaling factor g2 in the potential, expanding the wave function and energy in terms of 1/g, and employing the method of solving the classical Hamilton-Jacobi equation, the second order partial differential Schroedinger equation is cast into a series of first order ordinary differential equations. The ground state wave function can be expressed as a series of integrations along the trajectory and the energy as a series of differentials at the minimum of the potential. A new perturbation expansion is obtained based o this method. The corresponding N-dimensional Green function is derived. As an example, this method is applied to an attractive Coulomb potential and the Stark effect