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Journal of Molecular Structure THEOCHEM

Analytical results for the classical and semiclassical coordinate probability distribution function of anharmonic oscillators

F. Röscheisen, Wolfgang Witschel

Published May 1, 1982• 10 citations
Publisher
DOI
10.1016/0166-1280(82)80009-2

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The evaluation of the average value of the position coordinate, 〈x〉, of a particle moving in a harmonic oscillator potential (V(x)=kx2/2) with a small anharmonic piece (V(x)=−λkx3) is a standard calculation in classical Newtonian mechanics and statistical mechanics where the problem has relevance to thermal expansion. In each case, the calculation is most easily done using a perturbative expansion. In this note, we perform the same computation of 〈x〉 in quantum mechanics using time-independent perturbation theory and the ladder operator formalism to show how similar results are obtained. We also indicate how a semiclassical calculation using a classical probability distribution can also be used to obtain the same result.
We consider the quantum model of a driven anharmonic oscillator, in the presence of dissipation, and present an exact analytic solution for the corresponding Wigner function in the steady-state regime. This provides explicit phase-space images of the resulting state of the cavity mode, and allows us to understand how the quantum interference is built up into it. The photon number probability distribution is calculated and analysed as well. We monitor the transition from the semiclassical to extreme quantum regime of operation, and identify qualitative changes, where the conventional characteristics of the model, such as bistability or turning points, become meaningless.