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Acta Physica Sinica

Near classical states of oscillator in square potential well with infinitely high walls

Xinghua Li, Yatian Yang, Gong-ou Xu, (1)福建师范大学物理与光电信息科技学院,福州 350007; (2)南京大学物理系,南京 210093；兰州重离子加速器国家实验室理论核物理中心，兰州 730000

Published Jan 1, 2009• 1 citation
Publisher

Abstract

We define such quantum state as near classical state （NCS), in which the mean values of coordinates equal to the classical solution in macroscopic scale. We obtained the NCS for oscillator in square potential well with infinitely high walls as example.
DOI
10.7498/aps.58.7466

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One can easily understand the transition from special relativity to Newton mechanics under the condition of v/c 1. But it is not so easy to understand the transition from quantum representation to classical representation from the point of view of wave mechanics. We define such a quantum state as near classical state (NCS), in which the mean value of coordinates equals the classical solution on a macroscopic scale. We take the NCS for three-dimensional isotropic harmonic oscillator in a spherical coordinate system for example. We take and choose cnl =(1/(2N+1))(1/(2lM+1)). The mean values of coordinates are r2 =(Ecl)/(2)(1+1-((2Lcl2)/(Ecl2)cos(2t)) and tg = (Ecl/lcl)[1-1-((Lcl)/(Ecl)2]tg(t)) in this NCS, which are in agreement with the classical solution on a macroscopic scale, where N/N1, lM/lM1. N and lM are determined by the macroscopic state. N =[(Ecl)/(ħ)], Ecl = 1/22(a2+ b2) , lM= [Lcl}/ħ], and Lcl = ab. Here , Ecl and Lcl respectively denote the mass, the energy and the angular momentum of harmonic oscillator. And the bracket [c] means taking the integer part of the number c, for example [2.78]=2. It is also emphasized that for a definite macro state, there are many NCS corresponding to a macro state; just like the case in statistical physics, many micro dynamical states correspond to a macro thermodynamic state. Thus the transition from quantum representation to classical representation is a coarse-graining process and also an information losing process.
The harmonic oscillator is defined by a quadratic potential. Many important problems in physics reduce to that of a quantum harmonic oscillator because the leading term of a potential near its minimum is quadratic. It is a general and noteworthy feature of quantum systems that the lowest energy is not zero. Calculate the probability that the quantum oscillator is found beyond the classical limit. Agreement with the uncertainty principle may be checked by calculating the expectation values of x2 and p2 and then taking the square root of the product. For large quantum numbers, the solution must approach that of the classical harmonic oscillator.
Othman and Yevick [Int. J. Theor. Phys. 57, 2293 (2018)] introduced a new class of states defined as “near” coherent states attached to the simple harmonic oscillator. Such states can be expressed as superposition of a standard coherent state and a derivative state, which are neither completely quantum nor completely classical. Here, we introduce photon-added (-depleted) near coherent states [PA(D)NCS] through “m” times application of creation (annihilation) operators â†(â) to the near coherent state. A general analysis of nonclassical properties of the PA(D)NCS, such as sub-Poissonian statistics and squeezing effect, is given analytically and numerically in the context of the conventional quantum optics. We also derive the Wigner distribution function of the PA(D)NCS over phase space which may bear negative values, which is a good indication of their nonclassical properties. Finally, an experimental procedure for generating the PA(D)NCSs is established.