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Communications in Theoretical Physics

Morse Potential in the Momentum Representation

Guo-Hua Sun, Shi‐Hai Dong

Published Dec 1, 2012• 6 citations
Publisher

Abstract

The momentum representation of the Morse potential is presented analytically by hypergeometric function. The properties with respect to the momentum p and potential parameter β are studied. Note that |Ψ(p)| is a nodeless function and the mutual orthogonality of functions is ensured by the phase functions arg[Ψ(p)]. It is interesting to see that the |Ψ(p)| is symmetric with respect to the axis p = 0 and the number of wave crest of |Ψ(p)| is equal to n + 1. We also study the variation of |Ψ(p)| with respect to β. The amplitude of |Ψ(p)| first increases with the quantum number n and then deceases. Finally, we notice that the discontinuity in phase occurs at some points of the momentum p and the position and momentum probability densities are symmetric with respect to their arguments.

DOI
10.1088/0253-6102/58/6/05

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The momentum representation of the Morse potential is presented analytically by hypergeometric function. The properties with respect to the momentum p and potential parameter β are studied. Note that |Ψ(p)| is a nodeless function and the mutual orthogonality of functions is ensured by the phase functions arg[Ψ(p)]. It is interesting to see that the |Ψ(p)| is symmetric with respect to the axis p = 0 and the number of wave crest of |Ψ(p)| is equal to n + 1. We also study the variation of |Ψ(p)| with respect to β. The amplitude of |Ψ(p)| first increases with the quantum number n and then deceases. Finally, we notice that the discontinuity in phase occurs at some points of the momentum p and the position and momentum probability densities are symmetric with respect to their arguments.
Abstract The Morse potential is an important problem to examine due to its applications in describing vibrations and bond breaking in molecules. It also shares some properties with the simpler harmonic oscillator, at the same time displaying differences, allowing for an interesting contrast to its well-studied counterpart. The solution of the Morse potential is not usually taught in a quantum mechanics class, since using differential equations makes it very tedious. Here, we illustrate how to solve the Morse potential using the Schrödinger factorization method. This operator method is a powerful tool to find the energy eigenvalues, eigenstates, and wavefunctions without using differential equations in position space, allowing us to solve more problems without requiring a discussion of hypergeometric or confluent hypergeometric functions.
We develop a systematic approach to determine the large |p| behavior of the momentum-space wave function, ϕ(p), of a one-dimensional quantum system for which the position-space wave function, ψ(x), has a discontinuous derivative at any order. We find that if the kth derivative of the potential energy function for the system has a discontinuity, there is a corresponding discontinuity in ψ(k+2)(x) at the same point. This discontinuity leads directly to a power-law tail in the momentum-space wave function proportional to 1/pk+3. A number of familiar pedagogical examples are examined in this context, leading to a general derivation of the result.
The Hypergeometric method was used to describe radial equations by probeneus approaximation at a singular regular point. Schrodinger equations radial part was reduced into differential equations with variable substitution. By using q-deformed momentum parameter and Coulomb potential as well as variable substitution, the second order of Schrodinger equation with centrifugal approach for q-deformed momentum in Coulomb potential can be solved with Hypergeometric method. The analytical result of energy spectrum and wave function with quantum number and q-deformed variations, while the numerical result for energy spectrum can be calculated by using Matlab 2013 software.
We investigate the quantum-information properties of a particle confined by the double Morse potential in position and momentum spaces. The quasi-exact solvability of the model gives analytical expressions for the first two bound states, allowing the corresponding probability densities to be analyzed directly. Shannon entropy, Onicescu energy, Fisher information, statistical complexity, and Fisher-Shannon products are evaluated as functions of the parameter AAA, which controls the transition from a well-separated double well to a merged single-well profile. The position distribution is more delocalized and structurally complex when the double-well character is pronounced, whereas the momentum distribution exhibits the complementary trend. As the wells merge, the ground-state Fisher--Shannon product approaches its Gaussian reference value, whereas the excited state retains stronger non-Gaussian structure.
Information entropies, such as Shannon, Fisher and Rényi entropies, with their associated quantities, are calculated and discussed for the Morse potential in the position ( r ) and momentum ( p ) spaces using the J-matrix method. The entropies, for the s-wave, are numerically obtained for the diatomic molecules H 2 and LiH. It is the first information entropy calculation for real diatomic molecules using the Morse potential. Interesting characteristic features of the entropy densities are shown, where the position and momentum space information entropies satisfy the uncertainty relations.