# 3.7: The Morse Oscillator

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

The Morse oscillator model is often used to go beyond the harmonic oscillator approximation. In this model, the potential $$E_j(R)$$ is expressed in terms of the bond dissociation energy $$D_e$$ and a parameter a related to the second derivative k of $$E_j(R)$$ at $$R_e k = \frac{d^2E_j}{dR^2} = 2a^2D_e$$ as follows:

$E_j(R) - E_j(R_e) = D_e \left[ 1 - e^{-a(R-R_e)} \right]^2.$

The Morse oscillator energy levels are given by

$E^0_{j,v} = E_j(R_e) + \hbar \dfrac{\sqrt{k}}{\mu}\left( v+\dfrac{1}{2} \right) - \dfrac{\hbar^2}{4}\left( \dfrac{k}{\mu D_e} \right) \left( v + \dfrac{1}{2} \right)^2$

the corresponding eigenfunctions are also known analytically in terms of hypergeometric functions (see, for example, Handbook of Mathematical Functions , M. Abramowitz and I. A. Stegun, Dover, Inc. New York, N. Y. (1964)). Clearly, the Morse solutions display anharmonicity as reflected in the negative term proportional to $$(v+\dfrac{1}{2})^2.$$