Word Count : 222

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The Hamiltonian for a quantum harmonic oscillator is [ \hat{H} = \hbar \omega \left( \hat{a}^\dagger \hat{a} + \frac{1}{2} \right), ] where (\hat{a}) and (\hat{a}^\dagger) are the annihilation and creation operators, and ([\hat{a}, \hat{a}^\dagger] = 1). i) To show ([\hat{H}, \hat{a}] = -\hbar \omega \hat{a}): [ [\hat{H}, \hat{a}] = \left[ \hbar \omega \left( \hat{a}^\dagger \hat{a} + _________ ___ ___ ______ __________.
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