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Journal of Physics A Mathematical and General

Quasiclassical and exact quantization of simple two-level model systems

S. P. Karkach

Published Nov 21, 1993• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/26/22/031

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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.
An algebraic reformulation of the Bohr - Sommerfeld (BS) quantization rule is suggested and applied to the study of bound states in one-dimensional quantum wells. The energies obtained with the present quantization rule are compared to those obtained with the usual BS and WKB quantization rules and with the exact solution of the Schrödinger equation. We find that, in diverse cases of physical interest in molecular physics, the present quantization rule not only yields a good approximation to the exact solution of the Schrödinger equation, but yields more precise energies than those obtained with the usual BS and/or WKB quantization rules. Among the examples considered numerically are the Poeschl - Teller potential and several anharmonic oscillator potentials, which simulate molecular vibrational spectra and the problem of an isolated quantum well structure subject to an external electric field.
With the use of the Weyl correspondence between quantum mechanical operators and classical dynamical functions, an exact quantization rule is derived for a system with one degree of freedom and arbitrary Hamiltonian. In the semiclassical limit of small Planck's constant, this reduces to the Bohr-Sommerfeld quantization rule as derived by the WKB approximation. Higher order terms are also derived. It is further proven that for a simple harmonic oscillator the Bohr-Sommerfeld rule must give the exact energy eigenvalues for all states.
Abstract. – An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom. In the development of quantum mechanics, the Bohr-Sommerfeld quantization rules of the old quantum theory [1] occupy a position intermediate between classical and quantum mechanics. The WKB approximation [2–4] is a method for the approximate treatment of the Schrödinger wave function with another quantization rule [1]. Both quantization rules are approximate. We are going to derive an exact quantization rule for the one-dimensional Schrödinger equation: d2 ψ(x) = −2M dx2 2 [E − V (x)] ψ(x), (1) ¯h where M is the mass of the particle, and the potential V (x) is a piecewise continuous real function of x satisfying for definiteness V (x)> E, − ∞ < x < xA or xB < x < ∞, V (x) = E, x = xA or x = xB, V (x) < E, xA < x < xB, (2) where xA and xB are two turning points. Between two turning points, the momentum is k(x) = √ 2M [E − V (x)]/¯h.
We demonstrate that the Bargmann representation of quantum mechanics is ideally suited for semiclassical analysis, using as an example the WKB method applied to the bound-state problem in a single well of one degree of freedom. While the WKB expansion formulas are basically the usual ones, in this representation they describe approximations that are uniform and nonsingular in the classically allowed region of phase space because no turning points appear there. The quantization of energy levels relies on a complex contour integral that tests the eigenfunction for analyticity. For the harmonic oscillator, this WKB method trivially gives the exact eigenfunctions in addition to the exact eigenvalues. For an anharmonic well, a self-consistent variational choice of the representation greatly improves the accuracy of the semiclassical ground state. Also, a simple change of scale illuminates the relationship of semiclassical versus linear perturbative expansions, allowing a variety of multidimensional extensions. All in all, the Bargmann representation appears to combine the advantages of a linear description and of a phase-space representation of the quantum state vectors.
Extensions of the Bohr–Sommerfeld quantization formula to smooth double wells are established which preserve the formula’s generality and simplicity as well as its accuracy and the ease of use. Moreover, the proposed quantization equations for double wells, while being free from the limitations of the Wentzel–Kramers–Brillouin approximation which is restricted to deep energy levels alone, are equally well suited for determining higher energy levels lying in the vicinity of the barrier’s top as well as above the latter, and they have the correct connection to the conventional Bohr–Sommerfeld formula as the particle’s energy becomes large enough. As a particular application, a simple quantization equation for the pure quartic oscillator is derived which yields correct energy eigenvalues for all eigenstates of the oscillator including the ground one, with relative errors being all less than 1%. Conditions for the validity of the extended quantization equations are discussed and their numerical verification is made.