# 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.

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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