SHO ladder operators & some hamiltonian commutator relations

[Hakkinen]

Homework Statement

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

Homework Equations

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:
[itex]
[a_{-}, a_{-}a_{+}]\psi = a_{-}(n+1)\psi_{n} - a_{-}n\psi_{n} = a_{-}\psi_{n}
[/itex]
[itex]
= \sqrt{n}\psi_{n-1}
[/itex]

[itex]
[a_{+}, a_{-}a_{+}]\psi = a_{+}(n+1)\psi_{n} - a_{-}a_{+}\sqrt{n+1}\psi_{n+1}
[/itex]
[itex]
= (n+1)^{3/2}\psi_{n+1} - (n+1)^{3/2}\psi_{n+1} = 0
[/itex]Now what I am confused about is the [itex]\psi_{n-1}[/itex] 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 [itex] [p,H] [/itex] and [itex] [x,H] [/itex], 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!

 

[Hakkinen]

Duh! The first answer is just [itex] a_{-}[/itex]

And I didnt realize you could just use the "product rule" for commutators to simplify the algebra a bit!

All set now