Simple solution of the generalized Schrödinger equations

Jochen Pade

Jochen Pade

Hasan Hüseyin Erbil

By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).

Bruce W. Shore, Michael Austin Johnson

We demonstrate the manner in which solutions of the (coherent) Schrödinger equation for multistep excitation approach those of an (incoherent) rate equation. For a two-level system, coherence produces oscillations about the mean rate-equation values. In more general N-level sequences the two types of solutions agree in the limit described by Ackerhalt and Eberly. We point out instances in which a simple rate equation cannot duplicate results of the Schrödinger equation.