PROOFS PROGRAMME

qm/que/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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