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 $
- Compute the commutator $ [ a, a^\dagger ]$ Use results in part (1)} and find the commutators $$[N,a] \mbox{ and } [ N,a^\dagger] $$
- 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 $a^\dagger$.
- Show that \(H = \hbar\omega\Big(a^\dagger a + \frac{1}{2}\Big)\)
Exclude node summary :
n
Exclude node links:
0
4727:Diamond Point
0