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Umbral harmonic oscillator and its symmetries

J. E. López-Sendino

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The umbral calculus is an old mathematical tool that began its first steps in the XVII century [1]. Since the second half of XIX [2, 3] it was applied in a systematical way although until the second half of XX [4, 5] there was not a formal theory. Recently it has been used to provide discrete representations of canonical commutation relations [6]–[12]. In this approach it can be used to map equations and solutions from a continuous frame to a discrete one. The theory works very well for polynomials, however this is not the case for its extension to non-polynomial functions. A complicated, and then interesting, problem in this formalism is the harmonic oscillator. Due to the non preservation of the complete set of symmetries after discretization, some divergences appear. To second physical solutions of the discrete harmonic oscillator we use a non standard form of the umbral calculus discretization. Some surprising results are obtained.

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The umbral calculus is an old mathematical tool that began its first steps in the XVII century [1]. Since the second half of XIX [2, 3] it was applied in a systematical way although until the second half of XX [4, 5] there was not a formal theory. Recently it has been used to provide discrete representations of canonical commutation relations [6]–[12]. In this approach it can be used to map equations and solutions from a continuous frame to a discrete one. The theory works very well for polynomials, however this is not the case for its extension to non-polynomial functions. A complicated, and then interesting, problem in this formalism is the harmonic oscillator. Due to the non preservation of the complete set of symmetries after discretization, some divergences appear. To second physical solutions of the discrete harmonic oscillator we use a non standard form of the umbral calculus discretization. Some surprising results are obtained.
We consider a quantum system consisting of a one-dimensional chain of M identical harmonic oscillators with natural frequency ω, coupled by means of springs. Such systems have been studied before, and appear in various models. In this paper, we approach the system as a Wigner quantum system, not imposing the canonical commutation relations, but using instead weaker relations following from the compatibility of Hamilton’s equations and the Heisenberg equations. In such a setting, the quantum system allows solutions in a finite-dimensional Hilbert space, with a discrete spectrum for all physical operators. We show that a class of solutions can be obtained using generators of the Lie superalgebra gl(1∣M). Then we study – from a mathematical point of view – the properties and spectra of the physical operators in a class of unitary representations of gl(1∣M). These properties are both interesting and intriguing. In particular, we can give a complete analysis of the eigenvalues of the Hamiltonian and of the position and momentum operators (including multiplicities). We also study probability distributions of position operators when the quantum system is in a stationary state, and the effect of the position of one oscillator on the positions of the remaining oscillators in the chain.
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.
Using the algebraic second quantization formalism, the general recurrence relations for some two-center harmonic oscillator integrals are obtained. The various known special cases for one-center integrals are evident in the formalism. Generating functions for these integrals are obtained and most of them are solved to find explicit expressions for polynomial, exponential, and Gaussian operator integrals. For these cases, all the generating integrals needed for utilizing the recurrence relations are also calculated in their simplest forms.
The simple harmonic oscillator is solved by geometry on the two-ring instrument and the Fibonacci 60-ring of the companion papers, with every step checked as it is made, in lengths a student can draw. The route it replaces is the operator mechanics that Freericks and Doughty (arXiv:2305.00062, 2023) propose for teaching: trade the differential equations of wave mechanics for abstraction and run every solvable problem on four operator identities, with the oscillator as the worked case, solved three ways, the operator way resting on the canonical commutation relation, the factorization of the Hamiltonian, a second singular factorization that finds the odd states, and wavefunctions built from a translation operator and an operator form of the Rodrigues formula. This paper is the fourth way. The oscillator is a point on a ring: one setting reads its position between the centers and its momentum between the crossings. Velocity sits a quarter turn ahead and acceleration a half turn ahead, and the ring's digit laws, times seven and sum to ten, check each step. The energy is the radius squared. The commutation relation, the postulate their h-bar enters by, is the size of one cell of phase space: the ground orbit encloses h/2, one triangle, and every annulus between orbits encloses h, one diamond, so h enters the system as a unit, where m and omega do, and nowhere else. On the ring the area per step is the Casoratian of the wave, one constant at all sixty positions, with Cassini's identity as its digit shadow. The stationary states are rings themselves: by Wigner's phase-space formulation, which is exact for the oscillator, time evolution is a rigid rotation, a stationary state is a function of the radius alone, the position density is that ring projected on the center line and the momentum density the same ring projected on the crossing line, and the paper verifies that these projections are the standard densities. The wavefunctions themselves come from one slide: slide the ground blob along the center line, and the powers of the slide are the Hermite functions, with the explicit formula, the recurrence and the matrix elements falling out of two multiplied series where their paper needs a Gaussian similarity transform, nested commutators and an induction. The quarter turn is the Fourier transform, which is how momentum is measured in a trap. Their singular factorization sorts the states by the sign of the ring at its center, the parity. The spectrum E = h-bar omega (n + 1/2), counted in the equilateral triangles of the Tetryonics model, is the odd numbers: the ground state is one unsplittable triangle, each rung is one diamond, one photon, the totals are the squares, hydrogen's n squared, and the three-ring oscillator's degeneracies are the triangular numbers, one triangle read three ways. Their four identities, at the oscillator, are the addition of positions plus the area term. Every formula used is standard and cited, and every number is verified by the script in the appendix.
In this paper we present the first steps for obtaining a discrete Quantum Mechanics making use of the Umbral Calculus. The idea is to discretize the continuous Schroedinger equation substituting the continuous derivatives by discrete ones and the space-time continuous variables by well determined operators that verify some Umbral Calculus conditions. In this way we assure that some properties of integrability and symmetries of the continuous equation are preserved and also the solutions of the continuous case can be recovered discretized in a simple way. The case of the Schroedinger equation with a potential depending only in the space variable is discussed.
The harmonic oscillator is a useful model for the interpretation of many physical phenomena. It is used as a first approximation to the vibration of molecules and crystal lattices and for the quantization of electromagnetic fields, among other applications. One can solve the time-independent Schrodinger equation for the harmonic oscillator in many different ways; operator methods give rise to recurrence relations and generating functions for the stationary states. The coordinate representation renders the coordinate operator diagonal. In this particular representation the eigenvectors of the harmonic oscillator are functions of the coordinate that are commonly expressed in terms of Hermite polynomials. The chapter shows that calculation with the eigenfunctions of the harmonic oscillator is remarkably simple in the realm of the Lie algebras. The application of the bosonic algebra in theoretical physics and chemistry is so vast that it is impossible even to give a reasonable summary. An important application of the bosonic algebra is the modeling of simple quantum-mechanical systems.