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arXiv:1101.3075

Exact solution of Schroedinger equation in the case of reduction to Riccati type of ODE

Sergey V. Ershkov

Published Jan 16, 2011• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.1101.3075

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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.

Abstract In this chapter we consider the time-independent Schrödinger equation for three spatial dimensions. Making use of the orbital angular momentum observables, including the Laplacian in spherical polar coordinates, and introducing an effective potential, we recast the three-dimensional Schrödinger as a one-dimensional differential equation for the reduced radial wave function. In this context we discuss conditions for the existence of bound states and an upper bound on the ground-state energy obtained from the Rayleigh-Ritz variational principle. As an example we then consider the bound-state solutions for the spherical potential well and discuss the conditions for it admitting a ground state. We then turn to the Coulomb potential representing the hydrogen atom and discuss methods for obtaining upper and lower bounds on the ground-state energy to explain the stability of matter. Finally, obtain the Laguerre-polynomial solutions of the radial Schrödinger equation for the hydrogen atom, determine their energies and discuss their properties

The time dependent Schroedinger equation with a time dependent delta function potential is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation of the Volterra type. It is shown that by knowing the wave function at the origin, one may derive the wave function everywhere. Thus, the problem is reduced from a PDE in two variables to an integral equation in one. These results are used to compare adiabatic versus sudden changes in the potential. It is shown that adiabatic changes in the p otential lead to conservation of the normalization of the probability density.

The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.

In this paper, we solve exactly the Schrödinger equation for the free-particle, the pseudo-harmonic oscillator and the Mie-type potential in three dimensions with the Dunkl derivative. The equations for the radial and angular parts are obtained by using spherical coordinates and separation of variables. The wave functions and the energy spectrum for these potentials are derived in an analytical way and it is shown that our results are adequately reduced to those previously reported when we remove the Dunkl derivative parameters.