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International Journal of Quantum Chemistry

Bound states of oscillators in infinite and finite domains: A semiclassical study

S. Mukhopadhyay, K. Bhattacharyya

Published Oct 22, 2001• 4 citations
Publisher

Abstract

Abstract By involving the uncertainty product along with a semiclassical requirement for observables, a simple variational scheme is employed to extract major features of bound states of systems in (−∞,∞) and the same of systems confined in (−L,L). Special attention is paid to perturbative studies on the asymptotic behavior of energies for oscillators in infinite domains and dependence of energy spectra of oscillators in finite domains on various system parameters. A corrected form of the virial theorem is obtained in the latter case. The governing equations for quantum isothermal and adiabatic processes, derived recently, are also shown to be modified for general confined oscillator systems and closed‐form expressions are found. These results are useful in dealing with the quantum Carnot cycles. Advantages of the present route over other semiclassical strategies are stressed. Pilot calculations demonstrate nicely the efficacy of the endeavor under various situations. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001

DOI
10.1002/qua.1114

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Abstract By involving the uncertainty product along with a semiclassical requirement for observables, a simple variational scheme is employed to extract major features of bound states of systems in (−∞,∞) and the same of systems confined in (−L,L). Special attention is paid to perturbative studies on the asymptotic behavior of energies for oscillators in infinite domains and dependence of energy spectra of oscillators in finite domains on various system parameters. A corrected form of the virial theorem is obtained in the latter case. The governing equations for quantum isothermal and adiabatic processes, derived recently, are also shown to be modified for general confined oscillator systems and closed‐form expressions are found. These results are useful in dealing with the quantum Carnot cycles. Advantages of the present route over other semiclassical strategies are stressed. Pilot calculations demonstrate nicely the efficacy of the endeavor under various situations. © 2001 John Wiley & Sons, Inc. Int J Quantum Chem, 2001
Confinement of atoms inside impenetrable (hard) and penetrable (soft) cavities has been studied for nearly eight decades. However, a unified virial theorem for such systems has not yet been found. Here we provide a general virial-like equation in terms of mean square and expectation values of potential and kinetic energy operators. It appears to be applicable in both free and confined situations. Apart from that, we have derived an equation using the time-independent Schr\"odinger equation, which can be treated as a sufficient condition for a given stationary quantum state. A change of boundary condition does not affect these virial equations. In the hard confining condition, the perturbing (confining potential) does not affect the expression; it merely shifts the boundary from infinity to a finite region. In the soft case, on the contrary, the final expression includes contributions from the perturbing term. These are demonstrated numerically for several representative enclosed systems like harmonic oscillators (one-dimensional and three-dimensional) and hydrogen atoms. Its applicability in various other confinements (including angular) has been discussed. In essence, a virial equation has been proposed for free and confined quantum systems, from simple arguments.
We explore a few advantages of studying the change in information entropy of a bound quantum state with energy. It is known that the property generally increases with the quantum number for stationary states, in spite of the concomitant gradual increase in the number of constraints for higher levels. A simple semiclassical proof of this observation is presented via the Wilson-Sommerfeld quantization scheme. In the small quantum number regime, we numerically demonstrate how far the semiclassical predictions are valid for a few systems, some of which are exactly solvable and some not so. Our findings appear to be significant in a number of ways. We observe that, for most problems, information entropy tends to a maximum as the quantum number tends to infinity. This sheds some light on the Bohr limit as a classical limit. Noting that the dependence of energy on the quantum number governs the rate of increase of information entropy with the degree of excitation, we extend our analysis to include the role of the kinetic energy. The endeavor yields a relation that possesses a universal character for any one-dimensional problem. Relevance of information entropy in studying the goodness of approximate stationary states obtained from finite-basis linear variational calculations is also delineated. Finally, we expound how this property behaves in situations where shape resonances show up. A typical variation is indeed observed in such cases when we proceed to detect Siegert states via the stabilization method.
We review various methods of deriving expressions for quantum-mechanical quantities in the limit when hslash is small (in comparison with the relevant classical action functions). To start with we treat one-dimensional problems and discuss the derivation of WKB connection formulae (and their reversibility), reflection coefficients, phase shifts, bound state criteria and resonance formulae, employing first the complex method in which the classical turning points are avoided, and secondly the method of comparison equations with the aid of which uniform approximations are derived, which are valid right through the turningpoint regions. The special problems associated with radial equations are also considered. Next we examine semiclassical potential scattering, both for its own sake and also as an example of the three-stage approximation method which must generally be employed when dealing with eigenfunction expansions under semiclassical conditions, when they converge very slowly. Finally, we discuss the derivation of semiclassical expressions for Green functions and energy level densities in very general cases, employing Feynman's path-integral technique and emphasizing the limitations of the results obtained. Throughout the article we stress the fact that all the expressions obtained involve quantities characterizing the families of orbits in the corresponding purely classical problems, while the analytic forms of the quantal expressions depend on the topological properties of these families. This review was completed in February 1972.
Employing the path integral formalism, we study a quantum (test) system coupled to an environment consisting of infinitely many harmonic oscillators. A simple semiclassical approxi- mation, in which only real classical trajectories are required, is used to derive the propagator of the reduced density matrix for Ohmic dissipation and high temperatures. The inclusion of a summation over trajectories with the correct Maslov phases in the final expression for the propagator permits investigation of nonharmonic test systems. In a numerical example for a Morse oscillator interacting with a high temperature heat bath, the semiclassical result, obtained by the method described here, correctly displays the suppression of quantum behavior as observed in the exact calculation.
A fresh look at the origin of the Wilson–Sommerfeld quantization rule has been pursued to gain new insight. The rule is shown to provide states that satisfy several well-known theorems of standard quantum mechanics. A few other useful results and scaling relations are also derived. They emerge to act as nice guiding rules of thumb in the course of rigorous computations. Certain features of true excited-state densities can be understood. Goodness of approximate densities can be assessed. Compressed systems can be studied profitably. A route is also sketched that allows one to retrieve classical trajectories from near-exact energy eigenfunctions for both bound and resonant states by exploiting this rule. Additionally, a discussion on semiclassical perturbation theory is presented emphasizing the asymptotic behavior. Pilot calculations demonstrate the success of the present endeavor under various circumstances. © 2001 John Wiley & Sons, Inc. Int J Quant Chem 82: 113–125, 2001