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Acta Physica Sinica

ON THE QUANTUM MECHANICAL TREATMENT OF A DAMPED HARMONIC OSILLATOR

Ru-Zeng Zhu, 中国科学院力学研究所

Published Jan 1, 1981• 2 citations
Publisher

Abstract

We analyse the method of direct quantization suggested in reference [1] for adamped harmonic osillator, in which the quantum condition xp-px=ihe(-(C/M)t) is introduced. It is pointed out that this method can not be generalized to treat the case in which C is a function of time. In order to treat this case in a general approach, Heiaenberg relation xp-px=in must be kept and the force acting upon the osillatormust be supposed to contain a component fR that does not commute with x and satisfies xfR-fRx=ih(C/M).An electronic osillator is analysed as an example to show that ourapproach is consistent with quantum mechanical Langevin theory.

DOI
10.7498/aps.30.1410

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We analyse the method of direct quantization suggested in reference [1] for adamped harmonic osillator, in which the quantum condition xp-px=ihe(-(C/M)t) is introduced. It is pointed out that this method can not be generalized to treat the case in which C is a function of time. In order to treat this case in a general approach, Heiaenberg relation xp-px=in must be kept and the force acting upon the osillatormust be supposed to contain a component fR that does not commute with x and satisfies xfR-fRx=ih(C/M).An electronic osillator is analysed as an example to show that ourapproach is consistent with quantum mechanical Langevin theory.
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].
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.
Abstract By selecting a right generalized coordinate X, which contains the general solutions of the classical motion equation of a forced damped harmonic oscillator, we obtain a simple Hamiltonian which does not contain time for the oscillator such that Schrödinger equation and its solutions can be directly written out in X representation. The wave functin in x representation are also given with the help of the eigenfunctions of the operator X^ \hat X X^ in x representation. The evolution of ⟨x^⟩ \left\langle {\hat x} \right\rangle ⟨x^⟩ is the same as in the classical mechanics, and the uncertainty in position is independent of an external influence; one part of energy mean is quantized and attenuated, and the other is equal to the classical energy.