A Semiclassical Approach to Level Crossing in Supersymmetric Quantum Mechanics

Much use has been made of the techniques of supersymmetric quantum mechanics (SUSY QM) for studying bound-state problems characterized by a superpotential φ(x). Under the analytic continuation φ(x)→ iφ(x), a pair of superpartner bound-state problems is transformed into a two-state level-crossing problem in the continuum. The description of matter-enhanced neutrino flavor oscillations involves a level-crossing problem. We treat this with the techniques of supersymmetric quantum mechanics. For the benefit of those not familiar with neutrino oscillations and their description, enough details are given to make the rest of the paper understandable. Many other level-crossing problems in physics are of exactly the same form. Particular attention is given to the fact that different semiclassical techniques yield different results. The best result is obtained with a uniform approximation that explicitly recognizes the supersymmetric nature of the system. 1 SUSY QM and the Bound-State Problem Starting with a superpotential φ(x), one can generate two superpartner potentials V± = φ (x)± h̄ √ 2m φ′(x) . (1) For these two potentials the two corresponding Schrödinger equations are [ − h̄ 2 2m ∂ ∂2x + V± ] Ψ±(x) = EΨ±(x) . (2) It can be shown that the eigenspectrum of the “+” system can be obtained by shifting the quantum numbers n of the “−” system by n → n−1, with the ground state of the “−” system discarded. That is, the spectra of the two systems are identical except for a single state. In applications, this property is exploited in the following way. Given a potential V (x), one attempts to find a superpotential φ(x) that will generate V (x) via Eq. (1), with one or the other sign. If this can be done, one can immediately generate the superpartner potential by choosing the opposite sign in Eq. (1). In some fortunate circumstances, the equations of motion for the second system are much easier to solve than the first. See Schwabl ⋆ Speaker. Current address: Physics 161-33, Caltech, Pasadena, CA 91125, USA. [email protected] ⋆⋆ [email protected] 2 J. F. Beacom and A. B. Balantekin (1995) for more introductory material, and Cooper et al. (1995) and references therein for active areas of research. The application of supersymmetric quantum mechanics to the solution of bound-state problems has been extensively developed. There has been particular interest in semiclassical techniques. A direct primitive semiclassical (WKB) approach to the Schrödinger equation yields the usual Bohr-Sommerfeld quantization condition: √ 2m ∫ x2 x1 dx √ E − V (x) = (