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arXiv:quant-ph/0702213 · Modern Physics Letters A

WKB FORMALISM AND A LOWER LIMIT FOR THE ENERGY EIGENSTATES OF BOUND STATES FOR SOME POTENTIALS

Luis F. Barragán-Gil, Abel Camacho

Published Nov 9, 2007• 1 citation
arXivHTMLPDF

Abstract

In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.

DOI
10.1142/s0217732307022979

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In this work the conditions appearing in the so-called WKB approximation formalism of quantum mechanics are analyzed. It is shown that, in general, a careful definition of an approximation method requires the introduction of two length parameters, one of them always considered in the textbooks on quantum mechanics, whereas the other is usually neglected. Afterwards we define a particular family of potentials and prove, resorting to the aforementioned length parameters, that we may find an energy which is a lower bound to the ground energy of the system. The idea is applied to the case of a harmonic oscillator and also to a particle freely falling in a homogeneous gravitational field, and in both cases the consistency of our method is corroborated. This approach, together with the so-called Rayleigh–Ritz formalism, allows us to define an energy interval in which the ground energy of any potential, belonging to our family, must lie.
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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.
Generalizing a method used previously for three-body, four-body and very recently for five-body systems, we derive a lower bound for the ground state energy of an N-body Hamiltonian, with arbitrary N. Because the expression of the lower bound obtained in this way depends on a number of parameters, we obtain in fact a family of lower bounds, a lower bound for each set of values of these parameters. The best of these is of course obtained by maximizing over these parameters and is correspondingly named optimized lower bound. The set of values of the parameters corresponding to the optimized lower bound satisfy a number of relations, named universal dynamical constraints, which result from the application of a dynamical principle and are independent of the particular form of the potential. For N = 3, 4, 5, they can be worked out in the most general case. For N = 6 up, they can be worked out only for particular mass configurations. Furthermore, the optimized lower bound proves to be saturated in the harmonic oscillator case.
The well known and analytically solvable Lipkin–Meshkov–Glick model was devised to test several many-body approximation approaches used in nuclear physics. Here, we use this model to illustrate a method that sets lower bounds for the ground state energy of a quantum Hamiltonian. The idea of the method consists of dividing the Hamiltonian into identical groups of terms that represent subdivisions in the particle population of the system, and then to make an appropriate use of the ground energy of one of these groups. The bounds clearly improve as the size of these groups increases.
The determination of the Eigenwerte, and the normalization of the Eigenfunktionen of the one-dimensional wave equation, with a potential energy that has a single minimum, is customarily based on the use of the so-called phase integral and W.K.B. formulas. These formulas are asymptotic in character. As they have usually been applied, they accordingly lead to conclusions which can be regarded as established only when the respective quantum numbers are sufficiently large. This restriction has been both unfortunate and puzzling. For on the one hand, the cases of small or moderate quantum numbers are often peculiarly interesting while, on the other hand, the phase integral formulas have been found by trial, at least in a variety of simple cases, to give surprisingly good results even in the lower quantum number range.It is shown in this paper how the asymptotic method should be applied when small or moderate quantum numbers are in question. In the case of the Eigenwerte, the characteristic equation appropriate to the lower range is derived, and a comparison of this new equation with the familiar one based on the phase integral is made. It is thus theoretically established that under certain very liberal conditions upon the potential energy the older method does give good approximations. In the special case of the harmonic oscillator the two equations are, in fact, identical. For the normalization of the Eigenfunktionen a method which has been used in principle by W. H. Furry is applied. It is, however, framed in a manner appropriate to the smaller quantum number range. Formulas are given for the normalized Eigenfunktionen over the several intervals of the variable, over the critical interval which includes the minimum of the potential energy and the turning points, as well as over the intervals remote from the turning points.In the special case of the harmonic oscillator, the method here used gives certain exact---not merely asymptotic---results. These, therefore, supply a basis for the appraisal of the corresponding results that are obtainable by other means.