THE QUARTIC ANHARMONIC-OSCILLATOR AND ITS ASSOCIATED NONCONSTANT MAGNETIC-FIELD

Quantum mechanical anharmonic oscillators and Hamiltonians for particl es in external magnetic fields are related to representations of nilpo tent groups. Using this connection the eigenfunctions of the quartic a nharmonic oscillator with potential V-alpha(x)=(alpha+(x(2)/2))(2) can be used to determine the eigenfunctions of a charged particle in a no nconstant magnetic field, of the form B-z=beta(2)+beta(3)x. The quarti c anharmonic oscillator eigenvalues for low-lying states are obtained numerically and a function which interpolates between alpha much less than 0 (a double harmonic oscillator) and alpha much greater than 0 (a harmonic oscillator) is shown to give a good fit to the numerical dat a. Approximate expressions for the quartic anharmonic oscillator eigen functions are then used to get the eigenfunctions for the magnetic fie ld Hamiltonian. (C) 1997 American Institute of Physics.