Boyer, C.P., Kalnins, E.G. & Miller, W., Jr. (1975). Lie theory and separation of variables. 6. The equation iUt + ∆2U = 0. Journal of Mathematical Physics, 16, 499.
Permanent Research Commons link: http://hdl.handle.net/10289/1244
This paper constitutes a detailed study of the nine−parameter symmetry group of the time−dependent free particle Schrödinger equation in two space dimensions. It is shown that this equation separates in exactly 26 coordinate systems and that each system corresponds to an orbit consisting of a commuting pair of first− and second−order symmetry operators. The study yields a unified treatment of the (attractive and repulsive) harmonic oscillator, linear potential and free particle Hamiltonians in a time−dependent formalism. Use of representation theory for the symmetry group permits simple derivations of addition and expansion theorems relating various solutions of the Schrödinger equation, many of which are new.
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