The algebraic approach to the Morse oscillator (original) (raw)
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Several exact recursion relations connecting different Morse oscillator matrix elements associated with the operators q e~' and q e~'(d/dr} are derived. Matrix elements of the other useful operators may then be obtained easily. In particular, analytical expressions for (y"d/dr) and [y "d /dr +(d/dr)y "], matrix elements of interest in the study of the internuclear motion in polyatomic molecules, are obtained.
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2012
We intend to realize the step-up and step-down operators of the potential V(x)=V_1e^2β x+V_2e^β x. It is found that these operators satisfy the commutation relations for the SU(2) group. We find the eigenfunctions and the eigenvalues of the potential by using the Laplace transform approach to study the Lie algebra satisfied the ladder operators of the potential under consideration. Our results are similar to the ones obtained for the Morse potential (β→ -β).
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Annals of Physics, 2019
Ladder functions in classical mechanics are defined in a similar way as ladder operators in the context of quantum mechanics. In the present paper, we develop a new method for obtaining ladder functions of one dimensional systems by means of a product of two 'factor functions'. We apply this method to the curved Kepler-Coulomb and Rosen-Morse II systems whose ladder functions were not found yet. The ladder functions here obtained are applied to get the motion of the system.
ANALYTICAL CALCULATIONS FOR ROSEN-MORSE AND MANNING-ROSEN OSCILLATORS
It is well known that matrix elements of common operators play an important role in quantum mechanics. In this paper, analytic expressions for the matrix elements of some operators are obtained in the Rosen-Morse and Manning-Rosen basis. With such matrix elements, the Rosen-Morse and Manning- Rosen oscillators become a practical building block in molecular theory and a simultaneous treatment by introduction of the Jacobi polynomials.
An su (1, 1) dynamical algebra for the Morse potential
Journal of Physics A: Mathematical and General, 2004
An su(1, 1) dynamical algebra to describe both the discrete and the continuum part of the spectrum for the Morse potential is proposed. The space associated to this algebra is given in terms of a family of orthonormal functions {Φ σ n } characterized by the parameter σ. This set is constructed from polynomials which are orthogonal with respect to a weighting function related to a Morse ground state. An analysis of the associated algebra is investigated in detail. The functions are identified with Morse-like functions associated with different potential depths. We prove that for a particular choice of σ the discrete and the continuum part of the spectrum decouple. The connection of this treatment with the supersymmetric quantum mechanics approach is established. A closed expression for the Mecke dipole moment function is obtained.
An algebraic construction of the coherent states of the Morse potential based on SUSY QM
1999
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.
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The SO(2,1 ) algebraic approach is applied to the double ring-shaped oscillator for which the Schr6dinger equation separates in circular cylindrical coordinates. The energy spectrum and eigenfunctions obtained are identical to those obtained earlier using the path integral method.