Keller, Joseph B. Semiclassical mechanics. (English) Zbl 0581.70012 SIAM Rev. 27, 485-504 (1985). Classical mechanics and the quantum conditions of Planck, Bohr, Sommerfeld, Wilson and Einstein are presented. The virtues and defects of this ”old quantum theory” are pointed out. Its replacement by quantum mechanics is described, leading to the Schrödinger equation for the wave function and the corresponding energy eigenvalues. For separable systems, the reduction of this equation to ordinary differential equations and their asymptotic solution by the WKB method are described, as well as the resulting corrected quantum conditions with integer or half-integer quantum numbers. For nonseparable systems, the analogous asymptotic solution constructed by the author is described, together with the corrected quantum conditions to which it leads. Examples of the use of these conditions in the solution of eigenvalue problems are presented. It is explained that difficulties arise in using this method when the classical motion is stochastic or chaotic. Suggestions for overcoming these difficulties are mentioned. Cited in 21 Documents MSC: 70H05 Hamilton’s equations 81Q15 Perturbation theories for operators and differential equations in quantum theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics Keywords:stochastic classical motion; eigenvalue problems; bound state problems; quantum conditions; Schrödinger equation; energy eigenvalues; separable systems; asymptotic solution; WKB method; corrected quantum conditions PDFBibTeX XMLCite \textit{J. B. Keller}, SIAM Rev. 27, 485--504 (1985; Zbl 0581.70012) Full Text: DOI