On the quantal treatment of the double-well potential problem by means of certain phase-integral approximations

✍ Scribed by N. Fröman; P.O. Fröman; U. Myhrman; R. Paulsson

• Publisher: Elsevier Science
• Year: 1972
• Tongue: English
• Weight: 517 KB
• Volume: 74
• Category: Article
• ISSN: 0003-4916
• DOI: 10.1016/0003-4916(72)90143-1

✦ Synopsis

In the formulas given in the previous papers by FrGman and by Frtiman and Myhrman on the eigenvalue problem of the double oscillator, a certain quantity D occurs, which is negligible for energies lying far from the top of the barrier but which is important for energy levels in the immediate neighborhood of the top of the barrier. For the higherorder phase-integral approximations the quantity o, in fact, cancels a singularity appearing in certain contour integrals when the energy approaches the top of the barrier. The numerical results reported in the papers mentioned were obtained disregarding o, since convenient approximate expressions for O, pertaining to the higher order phaseintegral approximations used, were not known when the papers were published. As a consequence, the energy levels in the immediate neighborhood of the top of the barrier were not good. This deficiency is now remedied, and the purpose of the present paper is to give definitive formulas obtained by the quanta1 treatment of the energy eigenvalue problem of the double oscillator by means of the phase-integral approximations in question, up to the fifth order. The great accuracy of these formulas, also for energies close to the top of the barrier, is illustrated numerically.

1. THE ASYMMETRIC DOUBLE OSCILLATOR

Consider a smooth double-well potential V(X) of general shape. If the independent variable x is allowed to assume also complex values z, the Schrbdinger equation can be written s: + Q'(z) 1cI

where, with obvious notations,