The wave function for the harmonic oscillator with $v=1$ is given by:
\[\psi_{1} = \sqrt{\frac{a}{\pi}} x e^{-ax^{2}/2}\]
where $a = \sqrt{\frac{k\mu}{\hbar^{2}}}$, $k$ is the force constant, $\mu$ is the reduced mass, and $\hbar$ is the reduced Planck's constant.
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