- #1
qualal
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Homework Statement
Hi, I'm currently studying for a quantum mechanics exam but I am stuck on a line in my notes:
[tex]Ha\left|\Psi\right\rangle =\hbar\omega\left(a^{t}a a + \frac{a}{2}\right)\left|\Psi\right\rangle[/tex][tex]Ha\left|\Psi\right\rangle =\hbar\omega\left(\left(a a^{t} - 1\right)a + \frac{a}{2}\right)\left|\Psi\right\rangle[/tex]
Where H is the harmonic oscillator hamiltonian
[tex]H = \frac{P^{2}}{2m} + \frac{1}{2}m\omega^{2} x^{2}[/tex]and [tex] a, a^{t}[/tex] are the lowering and raising operators respectively.
[tex] a^{t} = \frac{1}{\sqrt{2\hbar m \omega}}\left(m \omega x - i p\right)[/tex]
[tex] a= \frac{1}{\sqrt{2\hbar m \omega}}\left(m \omega x + i p\right)[/tex]
The part I don't understand is how they got from [tex] a^{t}a a[/tex] to [tex]\left(a a^{t} - 1\right)a [/tex]
I'm not afraid of doing a bit of calculating but I really need a hint as to what method I should use.
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