Consider a particle in a harmonic pscillator potential V (x) is given by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]V = \frac{1}{2}m\omega^2[/tex]

Also [tex]\hat a = n^\frac{1}{2}|n-1>[/tex], and

[tex]\hat a\dagger = (n-1)^\frac{1}{2}|n-1>[/tex]

where

[tex]

\hat a = \frac{\beta}{\sqrt 2}(\hat x + \frac{i\hat p}{m\omega})

[/tex]

[tex]

\hat a\dagger = \frac{\beta}{\sqrt 2}(\hat x - \frac{i\hat p}{m\omega})

[/tex]

Construct a matrix representation for

[tex]\hat a[/tex]

[tex]\hat a\dagger[/tex]

[tex]\hat N = \hat a\hat \dagger[/tex]

and

[tex]\hat H[/tex]

where[tex]\hat H[/tex] is the Hamiltonian.

If someone can show me one and explain it to me, I will try the rest by myself before asking questions about them.

I am really desparate

Please help

James

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# Homework Help: Quantum mechanics in DIRAC notation

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