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.
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 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 for obtaining the WKB wave function describing the particle tunneling outside of a two-dimensional potential well is suggested. The Cartesian coordinates
(x,y) are chosen in such a way that the
x axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by simultaneous expansion of the wave function in the coordinate
y and the parameter determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. It is shown that the method provides systematic approximation to the outgoing probability flux. Both the technical and conceptual advantages of this approach in comparison with the usual approach based on the solution of classical equations of motion are pointed out. The method is applied to the problem of the coupled anharmonic oscillators and verified through the dispersion relations.
Solution of the Schrödinger's equation in the zero order WKB approximation is analyzed. We observe and investigate several remarkable features of the WKB
0 method. Solution in the whole region is built with the help of simple connection formulas we derive from basic requirements of continuity and finiteness for the wave function in quantum mechanics. We show that, for conservative quantum systems, not only total energy, but also momentum is the constant of motion. We derive the quantization conditions for two and more turning point problems. Exact energy eigenvalues for solvable and some ``insoluble'' potentials are obtained. The eigenfunctions have the form of a standing wave,
Ancos(knx+δn), and are the asymptote of the exact solution.