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International Journal of Modern Physics B

QUANTUM AND THERMAL STATE FOR EXPONENTIALLY DAMPED HARMONIC OSCILLATOR WITH AND WITHOUT INVERSE QUADRATIC POTENTIAL

J. R. CHOI

Published Apr 10, 2002• 17 citations
Publisher

Abstract

By taking advantage of dynamical invariant operator, we derived Schrödinger solution for exponentially damped harmonic oscillator with and without inverse quadratic potential. We investigated quantum mechanical energy expectation value, uncertainty relation, partition function and density operator of the system. The various expectation values in thermal state are calculated using the diagonal element of density operator.

DOI
10.1142/s021797920201018x

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By taking advantage of dynamical invariant operator, we derived Schrödinger solution for exponentially damped harmonic oscillator with and without inverse quadratic potential. We investigated quantum mechanical energy expectation value, uncertainty relation, partition function and density operator of the system. The various expectation values in thermal state are calculated using the diagonal element of density operator.
Taking advantage of the dynamical invariant method and the unitary transformation method together, we obtained quantum-mechanical solutions of the general time-dependent harmonic oscillator with a singular perturbation. The wave functions of the system are represented in terms of the associated Laguerre polynomial. We introduced a density operator satisfying the Leouville–von Neumann equation and used its diagonal elements to evaluate various expectation values in thermal state. We confirmed that the time-dependency of variables such as and 2 in thermal state are exactly the same as that in number state.
Assuming a damping force proportional to the velocity, it is shown that the usual quantum condition between the physical variables, namely the displacement and the momentum, has to be modified by the damping. However, it is possible to change to new variables which are canonical. Then we can apply the usual quantum mechanics and use the usual physical interpretation through the statistical average values. The case of a one-dimensional damped harmonic oscillator under the action of an arbitrary external force has been worked out as an example. No difficulty arises and the results are reasonable.
The solutions for a combination of the isotropic harmonic oscillator plus the inversely quadratic potentials and a combination of the pseudo-harmonic with inversely quadratic potentials has not been reported, though the individual potentials have been given attention. This study focuses on the solutions of the combination of the potentials, as stated above using the parametric Nikiforov–Uvarov (PNV) as the traditional technique to obtain the energy equations and their corresponding unnormalized radial wave functions. To deduce the application of these potentials, the expectation values, the uncertainty in the position and momentum, and the thermodynamic properties, such as the mean energy, entropy, heat capacity, and the free mean energy, are also calculated via the partition function. The result shows that the spectra for the PHIQ are higher than the spectra for the IHOIQ. It is also shown that the product of the uncertainties obeyed the Heisenberg uncertainty relation/principle. Finally, the thermal properties of the two potentials exhibit similar behaviours.
We have analysed the quantization schemes presented in references [1, 2] for a damped harmonic oscillator and found out that there are some defects in volved in these schemes. In order to remedy these defects, we assume the coordinate system fluctuating with a velocity g(t) in addition to damping force f(t) proportional to the velocity and suggest a method of quantization for this system based on the full quantum theory of lasers. It is proved that the usual Heisenberg relation xp-px=ih can be maintained, but the quantum term in statistical average value of physical variable does not damp as shown in reference [1].
Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.