# Morse potential and chaos

I have heard that the Morse potential equation

$$\tag{1} -\frac{\hbar ^{2}}{2m} \frac{d^{2}}{dx^{2}}y(x)+ae^{bx}y(x)-E_{n}y(x)=0$$

is related to the two dimensional equation on the Poincare half plane with a constant magnetic field

$$\tag{2} -\frac{y^{2}}{2m}( \partial _{x}^{2}+\partial _{y}^{2})f(x,y)+B\partial_{y}f(x,y) = 0$$

by means of a substitution that turns (2) into (1) but i do not know where to find some free avaliable info.

• 1. Don't you want to write the potential as $V(x) = D_e (1-e^{-a(x-x_e)})^2$, as for example in this wiki article? The form you wrote, $V(x) = a e^{bx}$, is different, and I don't think it is the Morse potential any more. 2. Could you maybe explain the relation of this problem to chaos? Or maybe the question title should be "Equivalence of solutions of Morse potential and a particle in constant magnetc field" (or something similar)? – au700 Nov 29 '12 at 17:43

## 1 Answer

Relevant article is: The path integral on the Poincaré upper half-plane with a magnetic field and for the Morse potential by Christian Grosche http://www.sciencedirect.com/science/article/pii/0003491688902837

Abstract Rigorous path integral treatments on the Poincaré upper half-plane with a magnetic field and for the Morse potential are presented. The calculation starts with the path integral on the Poincaré upper half-plane with a magnetic field. By a Fourier expansion and a non-linear transformation this problem is reformulated in terms of the path integral for the Morse potential. This latter problem can be reduced by an appropriate space-time transformation to the path integral for the harmonic oscillator with generalised angular momentum, a technique which has been developed in recent years. The well-known solution for the last problem enables one to give explicit expressions for the Feynman kernels for the Morse potential and for the Poincaré upper half-plane with magnetic field, respectively. The wavefunctions and the energy spectrum for the bound and scattering states are given, respectively.

However, it is not freely available. For some information you can consult http://arxiv.org/abs/hep-th/9302053 - Classification of Solvable Feynman Path Integrals co-authored by C. Grosche - in the table at the end of article the authors claim that Schroedinger equation for both Morse potential and particle in constant magnetic field can be viewed as generalisations of radial harmonic oscillator problem. Unfortunately they don't give much details.

• A free pdf file of Grosche's article is available here. – Qmechanic Nov 27 '12 at 0:47