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{\bf Question}

The energy equation for a vibrating diatomic molecule is:

$$\frac{1}{2}mv^2 + V(y) = E,$$

where $y$ is the bond extension, $v=\frac{dy}{dx}$, m is the
reduced mass, $V(y)$ is the potential energy, E is the total
energy.  An approximation to $V(y)$ is given by the Morse
potential. $$Y(y)=d(1-e^{-by})^2$$ where $d$ and $b$ are positive
constants.  Sketch a rough graph of this function.  By expanding
$V(y)$ in powers of $y$ show that in the harmonic approximation,
when $y$ is regarded as a small quantity, the molecule vibrates
with a frequency:

$$f=\frac{1}{2\pi}\sqrt{\frac{2db^2}{m}}.$$



{\bf Answer}



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