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## 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!

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