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arXiv:quant-ph/9812041

An algebraic construction of the coherent states of the Morse potential based on SUSY QM

Balázs Molnár, M. G. Benedict

Published Dec 16, 1998• 0 citations
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Abstract

By introducing the shape invariant Lie algebra spanned by the SUSY ladder operators plus the unity operator, a new basis is presented for the quantum treatment of the one-dimensional Morse potential. In this discrete, complete orthonormal set, which we call the pseudo number states, the Morse Hamiltonian is tridiagonal. By using this basis we construct coherent states algebraically for the Morse potential, in a close analogy with the harmonic oscillator. We also show that there exists an unitary displacement operator creating these coherent states from the ground state. We show that our coherent states form a continuous and overcomplete set of states. They coincide with a class of states constructed earlier by Nieto and Simmons by using the coordinate representation. \pacs{3.65.Fd, 02.20.Sv, 42.50.-p}

DOI
10.48550/arxiv.quant-ph/9812041

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By introducing the shape invariant Lie algebra spanned by the SUSY ladder operators plus the unity operator, a new basis is presented for the quantum treatment of the one-dimensional Morse potential. In this discrete, complete orthonormal set, which we call the pseudo number states, the Morse Hamiltonian is tridiagonal. By using this basis we construct coherent states algebraically for the Morse potential, in a close analogy with the harmonic oscillator. We also show that there exists an unitary displacement operator creating these coherent states from the ground state. We show that our coherent states form a continuous and overcomplete set of states. They coincide with a class of states constructed earlier by Nieto and Simmons by using the coordinate representation. \pacs{3.65.Fd, 02.20.Sv, 42.50.-p}
By introducing the shape-invariant Lie algebra spanned by the supersymmetric ladder operators plus the identity operator, we generate a discrete complete orthonormal basis for the quantum treatment of the one-dimensional Morse potential. In this basis, which we call the pseudo-number-states, the Morse Hamiltonian is tridiagonal. Then we construct algebraically the continuous overcomplete set of coherent states for the Morse potential in close analogy with the harmonic oscillator. These states coincide with a class of states constructed earlier by Nieto and Simmons [Phys. Rev. D 20, 1342 (1979)] by using the coordinate representation. We also give the unitary displacement operator creating these coherent states from the ground state.
While dealing with the J -matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L 2 ( R ) , we rederive a set of generalized coherent states (GCS) of Perelomov type labelled by points z of the complex plane C and depending on an integer number m ∈ Z + . The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials and whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian, associated with the m th Landau level. We further generalize this construction to the Morse oscillator by introducing a new set of Glauber-type GCS.
While dealing with the J-Matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L2(R); we rederive a set of generalized coherent states (GCS) of Perelomov type labeled by points z of the complex plane C and depending on a positive integer number m The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian associated with the mth Landau level. We extend this procedure to the Morse oscillator by constructing a new set of GCS of Glauber type.
The coherent states of the Morse potential that have been obtained earlier from supersymmetric quantum mechanics, are shown to be connected with the representations of the affine group of the real line and some of its extensions. This relation is similar to the one between the Heisenberg-Weyl group and the coherent states of the harmonic oscillator. The states that minimize the uncertainty product of the generators of the affine Lie algebra are shown to contain all the coherent states of the Morse oscillator plus the intelligent states of the Morse Hamiltonians with different shape parameter s . The representations of the central extension of the affine group denoted by G 0 and its further extension will be shown to define the phase space relevant to the problem by choosing an appropriate orbit of the coadjoint representation of . This allows one to construct a generalized Wigner function on this phase space, which is again essentially in the same relation with the affine group, as the ordinary Wigner function with the Heisenberg-Weyl group.
Using the associated Bessel functions, a shape-invariant Lie algebra spanned by ladder operators plus the identity operator, is realized. The Hilbert space of the associated Bessel functions, representing the Lie algebra, are established and two kinds of generalized coherent states as an appropriate superposition of these functions are constructed. By implying appropriate similarity transformation on the constructed coherent states, the generalized coherent states for the Morse potential are obtained. By considering some statistical characteristics, it is revealed that the constructed coherent states indeed possess nonclassical features, such as squeezing and sub-Poissonian statistics.
The Morse potential one-dimensional quantum system is a realistic model for studying vibrations of atoms in a diatomic molecule. This system is very close to the harmonic oscillator one. We thus propose a construction of squeezed coherent states similar to the one of the harmonic oscillator using ladder operators. The properties of these states are analysed with respect to the localization in position, minimal Heisenberg uncertainty relation, the statistical properties and illustrated with examples using the finite number of states in a well-known diatomic molecule. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.