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Physical Review A

qanalogs of the radial Schrödinger equation inNspace dimensions

É. Papp

Published Jul 1, 1995• 5 citations
Publisher

Abstract

Transparent closed forms of the radial ${\mathrm{SO}}_{\mathit{q}}$(N) Schr\"odinger equation acting on the noncommutative quantum Euclidean space are written both for l=0 and l\ensuremath{\ne}0. Starting from the s-wave equation leads to the derivation of an alternative q analog of the radial Schr\"odinger equation in N space dimensions exhibiting a modified symmetry, now in terms of wave functions depending on the radial coordinate only. The harmonic oscillator and the Coulomb system are treated as concrete examples. These results open the way to the derivation of q-deformed (1/N)-energy formulas for arbitrary spherically symmetrical potentials. Comparisons with exact ${\mathrm{SO}}_{\mathit{q}}$(N) results, as well as further interpretations, have also been made.

DOI
10.1103/physreva.52.101

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Transparent closed forms of the radial SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) Schr"odinger equation acting on the noncommutative quantum Euclidean space are written both for l=0 and l\ensuremath{\ne}0. Starting from the s-wave equation leads to the derivation of an alternative q analog of the radial Schr"odinger equation in N space dimensions exhibiting a modified symmetry, now in terms of wave functions depending on the radial coordinate only. The harmonic oscillator and the Coulomb system are treated as concrete examples. These results open the way to the derivation of q-deformed (1/N)-energy formulas for arbitrary spherically symmetrical potentials. Comparisons with exact SOq{\mathrm{SO}}_{\mathit{q}}SOq​(N) results, as well as further interpretations, have also been made.
There is much discussion in the mathematical physics literature as well as in quantum mechanics textbooks on spherically symmetric potentials. Nevertheless, there is no consensus about the behavior of the radial function at the origin, particularly for singular potentials. A careful derivation of the radial Schrödinger equation leads to the appearance of a delta function term when the Laplace operator is written in spherical coordinates. As a result, regardless of the behavior of the potential, an additional constraint is imposed on the radial wave function in the form of a vanishing boundary condition at the origin.
We consider the q-deformed Schrödinger equation of the harmonic oscillator on the N-dimensional quantum Euclidean space. The creation and annihilation operators are found, which systematically produce all energy levels and eigenfunctions of the Schrödinger equation. In order to get the q series representation of the eigenfunction, we also give an alternative way to solve the Schrödinger equation which is based on the q analysis. We represent the Schrödinger equation by the q difference equation and solve it by using q polynomials and q exponential functions.
Radial power-like potentials: from the Bohr-Sommerfeld ℓ≠0\ell \neq 0ℓ=0 state energies to the exact ones
Two new Bohr-Sommerfeld (B-S) type quantization conditions for ddd-dimensional radial potentials are proposed for non-singular and singular potentials, respectively. The power-like potentials \hbox{V=sign⁡(m) rmV=\operatorname{sign}(m)\,r^mV=sign(m)rm} for m=−3/2,−1,−1/2,1,3,4,6m=-3/2,-1,-1/2, 1,3,4,6m=−3/2,−1,−1/2,1,3,4,6 and V=log⁡rV=\log rV=logr (that corresponds to m=0m=0m=0) are studied in detail for radial quantum numbers nr∈[0,100]n_r \in [0,100]nr​∈[0,100] and for angular momenta ℓ=1,2,3,4,5\ell=1,2,3,4,5ℓ=1,2,3,4,5 in dimensions d=2,3d=2,3d=2,3. It is shown that for large nrn_rnr​ the B-S energies EBSE_{BS}EBS​ always tend to the exact ones. For m=2m=2m=2 (the radial harmonic oscillator) and m=−1m=-1m=−1 (the ddd-dimensional analogue of the 3D3D3D Coulomb potential) the B-S spectrum coincides with the exact one for the first and second quantization conditions, respectively. In general, up to a small parametric shift of the radial quantum number, the analytic formulas for the energy spectrum can be written for arbitrary power-like potential. A comparison of EBSE_{BS}EBS​ with the energies EBS(L)E^{(L)}_{BS}EBS(L)​, obtained for the radial potentials modified by the Langer correction for d=3d=3d=3, are in good agreement in general: in detail, it was checked for the quartic oscillator V=r4V=r^4V=r4. It is demonstrated that introducing a dimensionless {\it WKB correction} γγγ into the r.h.s. of the proposed B-S type quantization condition allows us to reproduce the exact energies with γγγ being very small for positive degrees m≥0m \geq 0m≥0 and sufficiently small for non-positive ones m<0m < 0m<0 for all nr,ℓn_r, \ellnr​,ℓ studied.
By using a simple procedure the general solution of the time-independent radial Schrodinger Equation for spherical symmetric potentials was made without making any approximation. The wave functions are always periodic. It appears two diffucilties: one of them is the solution of the equation E= U(r), where E and U(r) are the total an effective potential energies, respectively, and the other is the calculation of the integral of the square root of U(r). If analytical calculations are not possible, one must apply numerical methods. To find the energy wave function of the ground state, there is no need for the calculation of this integral, it is sufficient to find the classical turning points, that is to solve the equation E=U(r).
The spherically symmetric potential is considered whose dependence on the distance r is described by the smooth composition of Coulomb at r < r0 and oscillator at r > r0 potentials. The boundary distance r0 is determined by the parameters of these potentials. The exact continuous solution of the radial Schrödinger equation is expressed in terms of the confluent hypergeometric functions. The discrete energy levels are obtained. The graphic illustrations for the energy spectrum and the radial wave functions are presented.
Synopsis: Direct product spaces are encountered in the generalization to higher dimensions, and the connection to wave function separability is discussed. The spherical coordinate basis is introduced in preparation for discussing spherically symmetric potentials. Angular momentum operators and properties are examined, and a complete set of compatible observables is determined. The angular momentum eigenvalues and eigenstates are determined using ladder operators, and the connection to spherical harmonics is made. The stage is set for understanding the Schrödinger radial equation.