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arXiv:quant-ph/9601030 · Physical Review A

Universal superpositions of coherent states and self-similar potentials

Vyacheslav Pavlovich Spiridonov

Published Sep 1, 1995• 125 citations
arXivHTMLPDF

Abstract

A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A=P(d/dx+x)/ \ensuremath{\surd}2 , where P is the parity operator. Such A arises naturally in the q\ensuremath{\rightarrow}-1 limit for a symmetry operator of a specific self-similar potential obeying the q-Weyl algebra ${\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$-${\mathit{q}}^{2}$${\mathit{A}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states \ensuremath{\Vert}${\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}$\ensuremath{\alpha}〉, k=0,1,...,N-1, where \ensuremath{\epsilon} is a primitive Nth root of unity, ${\mathrm{\ensuremath{\epsilon}}}^{\mathit{N}}$=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.

DOI
10.1103/physreva.52.1909

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A variety of coherent states of the harmonic oscillator is considered. It is formed by a particular superposition of canonical coherent states. In the simplest case, these superpositions are eigenfunctions of the annihilation operator A=P(d/dx+x)/ \ensuremath{\surd}2 , where P is the parity operator. Such A arises naturally in the q\ensuremath{\rightarrow}-1 limit for a symmetry operator of a specific self-similar potential obeying the q-Weyl algebra AA\ifmmode∘\else\textdegree\fi{\mathit{AA}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}AA\ifmmode∘\else\textdegree\fi-{\mathit{q}}^{2}$${\mathit{A}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}A=1. Coherent states for this and other reflectionless potentials whose discrete spectra consist of N geometric series are analyzed. In the harmonic oscillator limit, the surviving part of these states takes the form of orthonormal superpositions of N canonical coherent states \ensuremath{\Vert}\ensuremathϵk{\mathrm{\ensuremath{\epsilon}}}^{\mathit{k}}\ensuremathϵk\ensuremath{\alpha}〉, k=0,1,...,N-1, where \ensuremath{\epsilon} is a primitive Nth root of unity, \ensuremathϵN{\mathrm{\ensuremath{\epsilon}}}^{\mathit{N}}\ensuremathϵN=1. A class of q-coherent states related to the bilateral q-hypergeometric series and Ramanujan type integrals is described. It includes an unusual set of coherent states of the free nonrelativistic particle, which is interpreted as a q-algebraic system without a discrete spectrum. A special degenerate form of the symmetry algebras of self-similar potentials is found to provide a natural q analog of the Floquet theory. Some properties of the factorization method, which is used throughout the paper, are discussed from the differential Galois theory point of view.
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.
Commuting translations in the phase plane are shown to be observables that are canonically conjugate to the annihilation operator of the harmonic oscillator states. The eigenfunctions and the eigenvalues of these commuting operators form the kq-representation. The eigenfunctions of the annihilation operator are the coherent states. The kq-eigenfunctions and the coherent states are the limiting cases of quantum-mechanical and classical wavefunctions, respectively, and it is satisfying to discover that they relate to canonically conjugate operators.
Angle and action operators (w, j) for the simple harmonic oscillator are treated as resulting from a canonical transformation of coordinate and momentum operators (q, k) generated by a one-sided unitary operator U such that U†U = 1 and UU† commutes with k but not with q. From the discrete spectrum of the number operator n, eigenvectors |η〉 are constructed for every real value of η; the set {|η〉} is complete and orthogonal. Another complete set {|W〉} is obtained, consisting of the Fourier transforms of the kets in the set {|η〉}. The angle operator is w=U†qU=∫dW |W〉 W 〈W|. The set {|W〉} is not orthogonal; |W〉 is not an eigenvector of w. If v is defined as ∫dW|W〉exp (−2πiW) 〈W|, then the creation and destruction operators are given by a = vn½, a† = n½v†. v is a one-sided unitary operator such that vv† = 1, but v†v = 1 − |0〉 〈0|, where |0〉 is the ground state of the oscillator; v and v† are similar to the operators E- and E+ of Carruthers and Nieto. The Weyl transforms of w and j = 2πℏ(n + ½1) are the classical angle and action variables of the oscillator. The Weyl transform is formulated in terms of the coherent states of the oscillator. A time operator canonical to the Hamiltonian is defined as t = 2πw/ω (ω/2π = frequency). The observables for the oscillator are also given in the Heisenberg picture and their classical limits are considered.
This study presents a complete algebraic formulation of the quantum harmonic oscillator, demonstrating that the Hermite polynomials, H n ( x ) , emerge as the fundamental solution when the power-series method is applied to the Schrödinger equation. The physical requirement of square integrability enforces the truncation of the series, leading precisely to these polynomials, which constitute the polynomial part of the energy eigenfunctions, ψ n ( x )= N n H n ( αx ) e − α 2 x 2 /2 , where n is the quantum number. This quantum number directly determines the quantized energy levels through E n =ℏω( n+ 1 2 ) . The analysis based on creation and annihilation operators further reveals that the action of the creation operator on the ground state generates the excited states along with their corresponding Hermite functions. The orthogonality of these polynomials ensures the orthonormality of the complete set of eigenfunctions. Additionally, the geometric Berry phase emerges as a natural extension when the system undergoes adiabatic cyclic changes, revealing a deeper topological structure that complements the conventional algebraic and analytic descriptions. Therefore, Hermite polynomials encapsulate both the mathematical structure and the physical essence of quantization in this fundamental system.