Quasiclassical solution of the three-dimensional Schrödinger's equation is given. It is shown apparently that the exitence of nonzero minimal angular momentum M0=ℏ/2 corresponds to the quantum fluctuations of the angular momentum and contributes to the energy of the ground state.
Three-dimensional Schrödinger's equation is analyzed with the help of the correspondence principle between classical and quantum-mechanical quantities. Separation is performed after reduction of the original equation to the form of the classical Hamilton–Jacobi equation. Each one-dimensional equation obtained after separation is solved by the conventional WKB method. Quasiclassical solution of the angular equation results in the integral of motion [Formula: see text] and the existence of nontrivial solution for the angular quantum number l = 0. Generalization of the WKB method for multi-turning-point problems is given. Exact eigenvalues for solvable and some "insoluble" spherically symmetric potentials are obtained. Quasiclassical eigenfunctions are written in terms of elementary functions in the form of a standing wave.
A discrete wave mechanical treatment of the hydrogen atom is extended to deal with states involving nonzero angular momentum. Only the radial portions of the wave vectors are covered. It is predicted that there is a nonzero minimum distance between the electron and the nucleus; this threshold distance increases with increasing angular momentum. Appropriate finite difference equations are formulated. The states with angular momentum exhibit the same degeneracy as do corresponding energy levels obtained from solutions of Schrödinger's equation.
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