(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 dont 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.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Raising and lowering operators on a simple harmonic oscillator

**Physics Forums | Science Articles, Homework Help, Discussion**