# SHO ladder operators & some hamiltonian commutator relations

## Homework Statement

For the SHO, find these commutators to their simplest form:
$[a_{-}, a_{-}a_{+}]$
$[a_{+},a_{-}a_{+}]$
$[x,H]$
$[p,H]$

## The Attempt at a Solution

I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first:
$[a_{-}, a_{-}a_{+}]\psi = a_{-}(n+1)\psi_{n} - a_{-}n\psi_{n} = a_{-}\psi_{n}$
$= \sqrt{n}\psi_{n-1}$

$[a_{+}, a_{-}a_{+}]\psi = a_{+}(n+1)\psi_{n} - a_{-}a_{+}\sqrt{n+1}\psi_{n+1}$
$= (n+1)^{3/2}\psi_{n+1} - (n+1)^{3/2}\psi_{n+1} = 0$

Now what I am confused about is the $\psi_{n-1}$ term in the first commutator. Surely there is a general form of the commutator without the test wavefunction? And I can't just drop this term and have root of n as the result. So did I do something wrong?

I tried the first part again using the explicit form of the ladder operators, in terms of H, p, x, with all of the constants. What I have gotten so far looks quite messy and involves $[p,H]$ and $[x,H]$, which I've yet to compute and are the last two parts of the problem... So it seems this route is not the easiest?

Any assistance is appreciated!

Last edited:

Duh! The first answer is just $a_{-}$