Quantum Harmonic Oscillator Excited State

Find the energy of the state given at some time by

\[\psi(x) = \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \sqrt{\frac{2m\omega}{\hbar}} xe^{-\frac{m\omega x^2}{2\hbar}}\]

in the harmonic oscillator potential \(V = \frac12 m \omega^2 x^2\).

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