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Acta Physica Sinica

Near classical states of three-dimensional isotropic harmonic oscillator in spherical coordinate system

Xinghua Li, Yatian Yang

Published Jan 1, 2015• 0 citations

Abstract

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.

DOI: 10.7498/aps.64.080301

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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 wave function for a state of definite angular momentum of the three-dimensional isotropic harmonic oscillator is expressed exactly in terms of the corresponding classical trajectories. In particular, the three-dimensional wave function as well as the radial wave function and the spherical harmonics are obtained as integrals over quantities determined entirely by the classical motion. The expression for the harmonic-oscillator radial wave function is shown also to yield the exact radial wave function for the free particle. The expressions are cast in forms suitable for use as uniform semiclassical approximations for wave functions of other systems. Numerical examples confirm that such wave functions obey boundary conditions appropriate for spherical coordinates and that they are free of caustic singularities. The wave functions obtained by this technique can be quite accurate even for low-energy states where semiclassical approximations are expected to be poor.

In this paper, we solve the rigorous solutions of the stationary Schrödinger equations for the harmonic oscillator within the framework of the quantum phase-space representation established by Torres-Vega and Frederick. We obtain the phase-space eigenfunctions of the three-dimensional isotropic harmonic oscillator in rectangular and spherical coordinate systems, and discuss the relationship between the eigenfunctions of the two coordinate systems. We also extend the "Fourier-like" projection transformations of the phase-space wave function to three-dimensional systems, and obtain the eigenfunctions in position and momentum representation through the transformations of the eigenfunction in the phase-space representation.

The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.

Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.

Physical consequences of the theorem of equivalence among harmonic oscillators of different frequencies, including a free particle as a special case, are explored. A new pictorial way of understanding time-dependent coherent states and squeezed states are presented and contrasted with the standard view. Three-dimensional spherical analogues of squeezed coherent states are constructed. It is also shown as a corollary of the theorem that a free wave packet which coincides initially with any one of the eigenfunctions of a harmonic oscillator maintains its shape, that is, the probability density of position does not change apart from a time-dependent scaling of spatial coordinates. Most of the results are derived without use of any sophisticated techniques but with the knowledge of free wave packets alone.