Phase space eigenfunctions of multidimensional quadratic hamiltonians

V.V. Dodonov and V.I. Man'ko

Physica A: Statistical Mechanics and its Applications, 1986, vol. 137, issue 1, 306-316

Abstract: We obtain the explicit expressions for phase space eigenfunctions (PSE), i.e. Weyl's symbols of dyadic operators like |n > , |m > being the solution of the Schrödinger equation with the Hamiltonian which is a quite arbitrary multidimensional quadratic form of the operators of Cartesian coordinates and conjugated to them momenta with time-dependent coefficients. It is shown that for an arbitrary quadratic Hamiltonian one can always construct the set of completely factorized PSE which are products of N factors, each factor being dependent only on two arguments for n≠m and on a single argument for n=m. These arguments are nothing but constants of motion of the correspondent classical system. PSE are expressed in terms of the associated Laguerre polynomials in the case of a discrete spectrum and in terms of the Airy functions in the continuous spectrum case. Three examples are considered: a harmonic oscillator with a time-dependent frequency, a charged particle in a nonstationary uniform magnetic field, and a particle in a time-dependent uniform potential field.

Date: 1986
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:137:y:1986:i:1:p:306-316

DOI: 10.1016/0378-4371(86)90078-6

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