On the adiabatic invariance method of calculating semiclassical eigenvalues

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Abstract
In this paper we analyze and further develop the adiabatic invariance method for computing semiclassical eigenvalues. This method, which was recently introduced by Solov’ev, is basically an application of the Ehrenfest adiabatic hypothesis. The eigenvalues are determined from a classical calculation of the energy as the time dependent Hamiltonian H(t)=H0+s(t)H1 is switched adiabatically from the separable reference Hamiltonian H0 to the system Hamiltonian H0+H1. A systematic study is carried out to determine the best form for the switching function, s(t), to maximize the rate of convergence of the energy to its adiabatic limit. Five switching functions, including the linear function used by Solov’ev, are defined and tested on three different systems. The linear function is found to have a very slow convergence rate compared to the others. The classical energy is shown to be a periodic function of the angle coordinates of H0. The coefficients of the Fourier series representation of this function are then shown to be related to the classical energy expectation value and classical energy uncertainty which we define and compare to their quantum mechanical counterparts. Four example problems are analyzed and solved in the course of this investigation. They are: (i) the forced harmonic oscillator, (ii) the harmonic oscillator with time dependent frequency, (iii) the nondegenerate, and (iv) the degenerate two dimensional coupled oscillator problems. In the degenerate oscillator case, we have discovered a correction, similar to the Langer correction in WKB theory, which significantly increases the accuracy of the semiclassical eigenvalues. States that are a linear combination of several energy eigenstates are discussed and it is demonstrated that the energy expectation value of these nonstationary states can also be computed semiclassically.