Word Count : 408

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### i) Show that the Hamiltonian for a simple harmonic oscillator can be written in terms of the raising and lowering operators as: The Hamiltonian for a simple harmonic oscillator is given by: \[ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2\hat{x}^2 \] To express this in terms of the raising (\(\hat{a}^\dagger\)) and lowering (\(\hat{a}\)) operators, we first define these operators: \[ \hat{a} = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x} + \frac{i\hat{p}}{m\omega}\right), \quad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}}\left(\hat{x} ___ __________ ______ _________ __________ _______ ____ ____ _______ ________.
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