Exact Solutions of the Schroedinger Equation.

Anatoly Georgievich Meshkov, N.V. Sukhomlin, V. N. Shapovalov

Boris V. Alekseev, A. I. Аbakumov

Sergey V. Ershkov

A new type of solution for the full 3+1 dimensional space-time Schroedinger equation is presented here. We consider elegant presentation of the exact solution in a spherical coordinate system, along with the assuming of separation of the two angular co-ordinates from the radial and time variables. The separation of variables follows from an assumed product form of the full potential function, which should allow us to reduce the Schroedinger Eq. to Riccati ODE in stationary case. If the angular dependence is constant, then the azimuthal dependence is linear and the polar dependence is logarithmic. With this reduction the remaining partial differential equation links the radial and time dependence which is, in effect, the standard time-dependent spherical radial Schroedinger equation. Besides, we obtain that the time-depended 2-angles solutions exist if the axis of preferential direction of wave propagation is similar to the time arrow in mechanical processes. The other possibility is the appropriate phase-shifting of the time-depended solutions. A paraxial type approximation is obtained for the solution in the limit in which the polar angle approaches zero.

CAIChang-Ying, RENZhong-Zhou, JUGuo-Xing

For an exponentially position-dependent mass, we obtain the exact solutions of the three-dimensional Schroedinger equation by using coordinate transformation method for the reference problems with Coulomb potential, Kratzer potential, and spherically square potential well of infinite depth, respectively. The explicit expressions for the energy eigenvalues and the corresponding eigenfunctions of the three systems are presented.