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Molecular Physics

Quasiclassical quantization of simple two-level systems. I

S. P. Karkach

Published Feb 20, 1986• 4 citations
Publisher

Abstract

The quantization problem of a system of two coupled parabolic electronic potentials with equal frequencies is reduced to a search for periodic solutions of two coupled first-order differential equations. The quantization condition obtained involves the classical actions and a set of quantum corrections. It is tested numerically over a wide parameter range.

DOI
10.1080/00268978600100361

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The quantization problem of a system of two coupled parabolic electronic potentials with equal frequencies is reduced to a search for periodic solutions of two coupled first-order differential equations. The quantization condition obtained involves the classical actions and a set of quantum corrections. It is tested numerically over a wide parameter range.
The Laplace method is considered in application to one-dimensional two-state eigenvalue problems in the example of simple models, including that of coupled parabolic terms and radial oscillators with quadratic, linear and constant coupling (basic models of Renner, pseudo-Jahn-Teller effects etc.). In all cases, the solution can be presented either as a two-state analogue of the Bohr-Sommerfeld quasiclassical quantization rule, or in more general form. Compared with the semiclassical picture of inelastic collisions in the momentum-space representation, the contour of integration of 'trajectory equations' for the case of bound states is finite, and any desired degree of accuracy can easily be achieved. Because no convergence problems arise, the need to compute the equation possessing 'physical sense' (e.g. for adiabatic amplitudes) is circumvented, suggesting that the method would be applicable also to complicated potentials and to many-state problems. Numerical tests for the energy level positions and calculations of eigenfunctions are presented; practical applications and generalizations are outlined.
The relation between the solutions of the time-independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half-width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well-defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.
Both unitary evolution and the effects of dissipation and decoherence for a general three-level system are of wide interest in quantum optics, molecular physics, and elsewhere. A previous paper presented a technique for solving the time-dependent operator equations involved but under certain restrictive conditions. We now extend our results to a general three-level system with arbitrary time-dependent Hamiltonians and Lindblad operators. Analytical handling of the SU(3) algebra of the eight operators involved leaves behind a set of coupled first-order differential equations for classical functions. Solution of this set gives a complete solution of the quantum problem, without having to invoke rotating-wave or other approximations. Numerical illustrations for multiphoton couplings and quantum control are given.
The eigenvalue problem in quantum mechanics is reduced to quantization of the classical action of the physical system. State function of the system, ψ0(ϕ)ψ_0(ϕ)ψ0​(ϕ), is written in the form of superposition of two plane waves in the phase space. Quantization condition is derived from the basic requirements of continuity and finiteness for ψ0(ϕ)ψ_0(ϕ)ψ0​(ϕ) in the whole region.