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Annalen der Physik

A Relativistic Model of the Isotropic Oscillator

Natig M. Atakishiyev, R. M. Mir‐Kasimov, Sh. M. Nagiyev

Published Jan 1, 1985• 24 citations
Publisher

Abstract

Abstract For a three‐dimensional model of the harmonic oscillator in the relativistic configurational r‐representation wave functions in the spherical coordinates r = (r, Θ, ϕ) are found. The generating function, orthogonality and various recurrence relations for the radial part of the wave function are obtained. The radial and orbital quantum numbers raising and lowering operators are defined and the dynamical symmetry group is constructed by means of the Infeld‐Hull factorization method.

DOI
10.1002/andp.19854970104

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Abstract For a three‐dimensional model of the harmonic oscillator in the relativistic configurational r‐representation wave functions in the spherical coordinates r = (r, Θ, ϕ) are found. The generating function, orthogonality and various recurrence relations for the radial part of the wave function are obtained. The radial and orbital quantum numbers raising and lowering operators are defined and the dynamical symmetry group is constructed by means of the Infeld‐Hull factorization method.
We show that the supersymmetric radial ladder operators of the three-dimensional isotropic harmonic oscillator are contained in the spherical components of the creation and annihilation operators of the system. Also, we show that the constants of motion of the problem, written in terms of these spherical components, lead us to second-order radial operators. Further, we show that these operators change the orbital angular momentum quantum number by two units and are equal to those obtained by the Infeld–Hull factorization method.
Exactly solvable NNN-dimensional model of the quantum isotropic singular oscillator in the relativistic configurational r⃗N\vec r_NrN​-space is proposed. It is shown that through the simple substitutions the finite-difference equation for the NNN-dimensional singular oscillator can be reduced to the similar finite-difference equation for the relativistic isotropic three-dimensional singular oscillator. We have found the radial wavefunctions and energy spectrum of the problem and constructed a dynamical symmetry algebra.
The factorization of four vector operators, D±(ω) and D±(−ω), which occur in a representation-independent, spectrum-generating algebra for the three-dimensional, isotropic harmonic oscillator in an angular momentum basis, is considered (ω is the angular frequency of the oscillator). The D±(ω) are quantum-mechanical analogs of the classical vectors (1∓iL̂×)Fc (ω), where Fc(ω)=−Mωr×L+pL is constant in a frame rotating with angular velocity ωL̂. It is shown that these four vector operators can be factorized in two different ways to yield operators that, apart from their dependence on a constant of the motion (L2), are linear in either p or r. In this way 20 abstract operators are obtained. The properties of these operators are discussed: (i) Twelve are ladder operators for the quantum numbers l, and l and m, in the eigenkets ‖lm〉 of L2 and Lz. In linearized, differential form six of these operators are ladder operators for the spherical harmonics in the coordinate representation, while the other six are the corresponding operators in the momentum representation. (ii) The remaining eight operators factorize linear combinations of the Hamiltonian and the dimension operator. In linearized, differential form four of these operators are ladder operators for energy and angular momentum in the radial part of the coordinate-space wave functions, while the other four are the corresponding operators in the momentum representation.
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.