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Journal of Physics A Mathematical and General

A realization of the dynamical group for the square-well potential and its coherent states

R. Lemus, Alejandro Frank

Published Apr 14, 2003• 7 citations
Publisher

Abstract

A realization of the ladder operators for the infinitely deep well potential is presented. Both the asymmetric and symmetric wells are discussed. It is shown that these operators satisfy the commutation relations for the SU (1, 1) group. Closed analytical expressions for the matrix elements of some relevant functions are obtained. Perelomov coherent states for this system are explicitly constructed. The behaviour of the dispersion relation and the time evolution for such states are examined.

DOI
10.1088/0305-4470/36/17/316

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A realization of the ladder operators for the infinitely deep well potential is presented. Both the asymmetric and symmetric wells are discussed. It is shown that these operators satisfy the commutation relations for the SU (1, 1) group. Closed analytical expressions for the matrix elements of some relevant functions are obtained. Perelomov coherent states for this system are explicitly constructed. The behaviour of the dispersion relation and the time evolution for such states are examined.
The exact solutions of quantum system with a symmetric well potential V(x) = D tan 2(πx/a) are obtained. The ladder operators are constructed directly from the normalized eigenfunctions with the factorization method. It is shown that these ladder operators satisfy the commutation relations of the generators for an su(1, 1) algebra. The infinitely deep square well and harmonic limits of this potential are briefly studied. The controllability of this system is also investigated. It is demonstrated that this system with discrete bound states can be strongly completely controlled. This may be realized theoretically by acting the creation operator [Formula: see text] on the ground state.
We present exact solutions of the one-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square potential. The ladder operators are constructed by the factorization method. We find that these operators satisfy the commutation relations of the generators of the dynamical group SU(1, 1). Based on those ladder operators, we obtain the analytical expressions of matrix elements for some related functions ρ and [Formula: see text] with ρ=x2. Finally, we make some comments on the Barut–Girardello coherent states and the hidden symmetry between E(x) and E(ix) by substituting x→ix.
The eigenfunctions and eigenvalues of the three-dimensional Schrödinger equation with a harmonic oscillator plus an inverse square interaction are obtained. A realization of the ladder operators for the wave functions is studied. It is found that these operators satisfy the commutation relations of an SU(1,1) group. The closed analytical expressions for the matrix elements of different functions ρ and ρd/dρ with ρ = r 2 are evaluated. Another hidden symmetry explores the relations between the eigenvalues and eigenfunctions for substituting r→ ir. PACS number(s): 03. 65. Fd, 03. 65.Ge and 02. 20.Qs.
The exact solutions of the Schrödinger equation with a harmonic oscillator plus an inverse square potential are obtained in two dimensions. We construct the ladder operators directly from the radial wave functions and find that these operators satisfy the commutation relations of an SU (1, 1) group. We obtain the explicit expressions of the matrix elements for some related functions ρ and [Formula: see text] with ρ = r2. We also explore another symmetry between the eigenvalues E(r) and E(ir) by substituting r→ir.
A realization of the raising and lowering operators for the Morse potential is presented. We show that these operators satisfy the commutation relations for the SU(2) group. Closed analytical expressions are derived for the matrix elements of different functions such as 1/y and d/dy. The harmonic limit of the SU(2) operators is also studied. The transition probability between two eigenstates produced by a harmonic perturbation as a function of the operators [Formula: see text]±,0 is discussed. The average values of some observables in the coherent states |α > for the Morse potential are also calculated. PACS Nos.: 02.30+b, 03.65Fd, 42.50Ar, and 33.10Cs