Third harmonic generation as a third-order quasi-exactly solvable system
Authors: Álvarez G.1; Álvarez-Estrada R.F.2
Source: Journal of Physics A: Mathematical and General, Volume 34, Number 47, 2001 , pp. 10045-10056(12)
Publisher: Institute of Physics Publishing
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Abstract:
The two degrees of freedom in the usual third harmonic generation effective Hamiltonian can be separated, both classically and quantum mechanically, into a harmonic oscillator and a nonlinear oscillator. In turn, the quantum Hamiltonian of the nonlinear oscillator can be written as a cubic polynomial in the generators of a quasi-exactly solvable Lie algebra, and the physically relevant eigenstates are precisely those that can be determined exactly (although, in general, not explicitly). Since the standard Jeffreys-Wentzel-Kramers-Brillouin methods are not easily applicable to general third-order differential equations, we use a Bohr-Sommerfeld quantization of the classical orbits to obtain approximate explicit formulas for the corresponding quantum eigenvalues.
Language: English
Document Type: Miscellaneous
Affiliations: 1: Departamento de Física Teórica II, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain 2: Departamento de Física Teórica I, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain
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