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# Problem/QM-02001

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Let $$q,p$$ be the coordinate and momentum operators and define
$$q,q^\dagger$$ by $$a = {1\over \sqrt{2m\omega \hbar}}( p-im\omega q)$$ $$a^\dagger = {1\over \sqrt{2m\omega \hbar}}( p+im\omega q)$$ and $N= a^\dagger a$

1.  Compute the commutator $[ a, a^\dagger ]$
Use results in part (1)} and find the commutators $$[N,a] \mbox{ and } [ N,a^\dagger]$$
2.  Express the harmonic oscillator Hamiltonian \begin{equation*} H =
\frac{p^2}{2m} + \frac{1}{2}\, m \omega^2 q^2 \end{equation*} in terms of $a$
and $a^\dagger$.
3. Show that $$H = \hbar\omega\Big(a^\dagger a + \frac{1}{2}\Big)$$

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