Raising and lowering operators for semiclassical wave packets. (English) Zbl 0929.34067
The author uses raising and lowering operators to give simplified constructions of certain bases of \(L^2\) which can be used to develop asymptotic expansions for solutions to the time dependent Schrödinger equations in the semiclassical limit. These raising and lowering operators generalize the familiar raising and lowering operators of the harmonic oscillator, in fact in one dimension the lowering operators are of the form \((2\hbar)^{-1/2}(B(x-a)+iA(-i\hbar d/dx-\eta))\) where \(a,\eta\) are real and \(A,B\) are complex numbers. Their knowledge allows one to simplify the derivation of properties of the associated bases. Furthermore, the paper contains simplified proofs of several results in semiclassical quantum mechanics, and thus summarizes previous work of the author in this field.
Reviewer: Marcel Griesemer (Birmingham / Alabama)
MSC:
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 34E05 | Asymptotic expansions of solutions to ordinary differential equations |
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