QUASICLASSICAL ANALYSIS OF THREE-DIMENSIONAL SCHRÖDINGER'S EQUATION AND ITS SOLUTION

M. N. SERGEENKO
2000 Modern Physics Letters A  
The three-dimensional Schredinger'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 $\vec M^2=(l+\frac 12)^2\hbar^2$ and the
more » ... stence 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.
doi:10.1142/s0217732300000104 fatcat:ufzxerpygvephbr73jdgpiiqlm