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arXiv:1612.07700 · Physics of Atomic Nuclei

A finite quantum oscillator model related to special sets of Racah polynomials

Roy Oste, Joris Van Der Jeugt

Published Jul 1, 2017• 4 citations
arXivHTMLPDF

Abstract

In [R. Oste and J. Van der Jeugt, arXiv: 1507.01821 [math-ph]] we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are referred to as (dual) Hahn doubles, and the same technique was then applied to obtain all Racah doubles. We now consider a special case concerning the doubles related to Racah polynomials. This gives rise to an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. Just as it was the case for (dual) Hahn doubles, the resulting two-diagonal matrix can be used to construct a finite oscillator model. We discuss some properties of this oscillator model, give its (discrete) position wavefunctions explicitly, and illustrate their behavior by means of some plots.

DOI
10.1134/s1063778817040196

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In [R. Oste and J. Van der Jeugt, arXiv: 1507.01821 [math-ph]] we classified all pairs of recurrence relations in which two (dual) Hahn polynomials with different parameters appear. Such pairs are referred to as (dual) Hahn doubles, and the same technique was then applied to obtain all Racah doubles. We now consider a special case concerning the doubles related to Racah polynomials. This gives rise to an interesting class of two-diagonal matrices with closed form expressions for the eigenvalues. Just as it was the case for (dual) Hahn doubles, the resulting two-diagonal matrix can be used to construct a finite oscillator model. We discuss some properties of this oscillator model, give its (discrete) position wavefunctions explicitly, and illustrate their behavior by means of some plots.
We consider dual polynomials of the multi-indexed (⁠|qqq|-)Racah orthogonal polynomials. The |MMM|-indexed (⁠|qqq|-)Racah polynomials satisfy the second-order difference equations and various |1+2L1+2L1+2L| (⁠|L≥M+1L\geq M+1L≥M+1|⁠) term recurrence relations with constant coefficients. Therefore their dual polynomials satisfy the three term recurrence relations and various |2L2L2L|th-order difference equations. This means that the dual multi-indexed (⁠|qqq|-)Racah polynomials are ordinary orthogonal polynomials of Krall type. We obtain new exactly solvable discrete quantum mechanics with real shifts, whose eigenvectors are described by the dual multi-indexed (⁠|qqq|-)Racah polynomials. These quantum systems satisfy the closure relations, from which the creation/annihilation operators are obtained, but they are not shape invariant.
In quantum mechanics the linear harmonic oscillator representatives of the canonical observables q and p are certain well-known infinite square matrices, qmn and pmn, with denumerable sets of rows and columns. It is pedagogically useful to investigate various problems which arise when one considers the finite matrices Qmn and Pmn which result when qmn and pmn are truncated after the Nth row and column, where N( 2≥2) is some preassigned positive integer. So that the enquiry may be more instructive, it begins with an abstract formulation of the problem of the “truncated harmonic oscillator.” Formally, the latter has the usual Hamiltonian H = 12( Q2+P2 ) , but Q and P are not canonical—their commutator [Q,P] now differing from i by a certain projection operator. The spectrum of H and the (finite) basis { |n〉} generated by H are found by standard methods, though there are now some complications which are absent from the theory of the linear harmonic oscillator. The diagonalization of Q and P presents no difficulties if full use is made of the classical theory of Hermite polynomials. The spectrum of Q is the set of N zeros hs of the Nth Hermite polynomial. Although the wavefunctions 〈 Q′|n〉 are therefore functions of the discrete variable hs, one can formally contemplate them as functions of a continuous real variable h; and some curious intimations of Schrödinger representation appear. The analog of a component of orbital angular momentum is considered, and its spectrum obtained. Finally, the asymptotic limit N → ∞ is briefly discussed, in order thus explicitly to dispel the impression that this limit might somehow represent the “usual” linear harmonic oscillator.
A superintegrable finite model of the quantum isotropic oscillator in two dimensions is introduced. It is defined on a uniform lattice of triangular shape. The constants of the motion for the model form an SU (2) symmetry algebra. It is found that the dynamical difference eigenvalue equation can be written in terms of creation and annihilation operators. The wavefunctions of the Hamiltonian are expressed in terms of two known families of bivariate Krawtchouk polynomials, those of Rahman and those of Tratnik. These polynomials form bases for SU (2) irreducible representations. It is further shown that the pair of eigenvalue equations for each of these families is related to each other by an SU (2) automorphism. A finite model of the anisotropic oscillator that has wavefunctions expressed in terms of the same Rahman polynomials is also introduced. In the continuum limit, when the number of grid points tends to infinity, standard two-dimensional harmonic oscillators are obtained. The analysis provides the N → ∞ limit of the bivariate Krawtchouk polynomials as a product of one-variable Hermite polynomials.
Abstract A formalism is presented to obtain approximate analytic expressions for the eigenstates and eigenvalues of a quantum double oscillator (QDO). The matrix elements of a large class of operators with respect to states of different double oscillators result as finite sums of explicit functions of the respective parameters. Matrix elements between states of a harmonic oscillator and a double oscillator are also determined. The analytic expressions were used to calculate Franck-Condon factors for electronic transitions including double oscillator anharmonicities.
We study a linear spin chain which was originally introduced by Shi et al. [Phys.Rev. A 71 (2005), 032309, 5 pages], for which the coupling strength contains a parameter α and depends on the parity of the chain site.Extending the model by a second parameter β, it is shown that the single fermion eigenstates of the Hamiltonian can be computed in explicit form.The components of these eigenvectors turn out to be Hahn polynomials with parameters (α, β) and (α + 1, β -1).The construction of the eigenvectors relies on two new difference equations for Hahn polynomials.The explicit knowledge of the eigenstates leads to a closed form expression for the correlation function of the spin chain.We also discuss some aspects of a q-extension of this model.
Multi-indexed orthogonal polynomials (the Meixner, little q-Jacobi (Laguerre), (q-)Racah, Wilson, and Askey-Wilson types) satisfying second-order difference equations were constructed in discrete quantum mechanics.They are polynomials in sinusoidal coordinates η(x) (x is the coordinate of the quantum system) and are expressed in terms of Casorati determinants whose matrix elements are functions of x at various points.By using shape-invariance properties, we derive various equivalent determinant expressions, especially those whose matrix elements are functions of the same point x.Except for the (q-)Racah case, they can be expressed in terms of η only, without explicit x-dependence...........