We present a method of obtaining WKB-type solutions for generalized Schr\"odinger equations for which the Hamiltonian is an arbitrary matrix function of any number of pairs of canonical operators. Our solution reduces the problem to that of finding the matrix which diagonalizes the classical Hamiltonian and determining the scalar WKB wave functions for the diagonalized Hamiltonian's entries (presented explicitly in terms of classical quantities). If the classical Hamiltonian has degenerate eigenvalues, the solution contains a vector in the classically degenerate subspace. This vector satisfies a classical equation and is given explicitly in terms of the classical Hamiltonian as a Dyson series. As an example, we obtain, from the Dirac equation for an electron with anomalous magnetic moment, the relativistic spin-precession equation.
The WKB wavefunctions for one-dimensional nonrelativistic problems are derived by a simple transformation method. By applying the Liouville substitution to Schrödinger's equation and then neglecting certain “small” terms in the transformed equation, one arrives at the WKB solutions. The terms neglected in obtaining the WKB solutions are explicit. No asymptotic expansions are used in the derivation. Use of a known criterion from the theory of differential equations gives assurance that the WKB solutions are asymptotic to exact solutions of Schrödinger's equation in certain cases.
A method of approximating solutions of the one-dimensional Schr\"odinger equation is presented in this paper. The method closely resembles the usual WKB approximation. Whereas in the ordinary WKB method the exponential function is used as the basis of the approximation, in this paper the solutions of an arbitrary Schr\"odinger equation are used. The general advantage is that by proper choice of the arbitrary equation an improved approximation can be obtained. The method is illustrated by treating the potential well and potential barrier problems when there are two turning points. The approximations to the wave functions are continuous even across the turning points. The barrier transmission problem is treated uniformly for energies above and below the peak of the barrier.
The equation of motion in the uniform WKB theory for a two-channel problem is reinvestigated in order to obtain improved approximate solutions that would perform better than the first order Magnus approximation solution previously used in the analysis of two-channel inelastic collision problems. We have developed two approximate solutions and investigated their variants in the paper. The two main improved approximate solutions are obtained by properly resumming the Magnus expansion in a novel way and also by appropriately modifying the equation of motion such that the Magnus approximation used on it would not damage the accuracy of the solution thus obtained. The solutions obtained give rise to transition probabilities well comparable over a wide range of the coupling constant with those obtained by the numerical solution of the coupled Schrödinger equations.
The method of Froman and Froman for proving exact quantization conditions is reviewed. This formalism, unlike the usual WKB approximation to which it bears a close resemblance, requires consideration of the behavior of the potential everywhere it is defined. This approach leads to proofs that certain quantization conditions are exact without having to compare the results to solutions of the Schrödinger equation obtained by other means. Using the formalism, we prove the correctness of all previously known exact quantization rules for the one-dimensional and radial cases. Furthermore, it is shown that exact quantization rules can be proved for two other potentials. For one of these, no analytic solutions to the Schrödinger equation are known. For the latter case, the proof is checked by numerical integration of the Schrödinger equation for a special case.
In a recent paper by Eu and Tsien on the uniform WKB theory of inelastic collision, the Magnus-approximation formula for the transition probability was applied to the
He+-Ne inelastic-scattering process. In this Comment, we report that for the system studied, i.e., the
He+-Ne system, the uniform WKB Magnus approximation underestimates the cross sections by about 50% and thus overestimates the coupling constant
a4 by about 30% above what it should be, while the distorted-wave theory underestimates the coupling constant by about 5%. The uniform-WKB Magnus-approximation formula for the transition probability is further assessed by making comparisons with the results of the exact theory, uniform-WKB numerical-solution method, and the Landau-Zener-St\"uckelberg theory, and is found to behave correctly in the strong-coupling regime as
a4 increases. The uniform-WKB Magnus approximation therefore appears to be complementary to the distorted-wave theory which is better in the weak-coupling regime, but breaks down in the strong-coupling regime. The uniform-WKB Magnus approximation is generally better behaved than the Landau-Zener-St\"uckelberg theory over all values of
a4 for the system studied.