- #1
sa1988
- 222
- 23
Homework Statement
Firstly, I'm looking at this:
I'm confused because my understanding is that the commutator should be treated like so:
$$[a,a^{\dagger}] = aa^{\dagger} - a^{\dagger}a$$
but the working in the above image looks like it only goes as far as
$$aa^{\dagger}$$
This surely isn't the case, so I think I may have got my understanding wrong regarding how to take the correct steps when working out a commutation relation
Secondly, I have this question, which I think may further shed light on where I'm going wrong, if anyone may be kind enough to have a look for me:
I work through it like so:
$$[a^2,a^{\dagger}] = \Bigg[\frac{m}{2\hbar\omega}(\omega \hat{x} + \frac{i}{m}\hat{p})^2, \sqrt{\frac{m}{2\hbar\omega}}(\omega \hat{x} - \frac{i}{m}\hat{p}) \Bigg]$$
$$ = \frac{m}{2\hbar\omega}(\omega \hat{x} + \frac{i}{m}\hat{p})^2 \sqrt{\frac{m}{2\hbar\omega}}(\omega \hat{x} - \frac{i}{m}\hat{p}) - \frac{m}{2\hbar\omega}(\omega \hat{x} + \frac{i}{m}\hat{p})^2 \sqrt{\frac{m}{2\hbar\omega}}(\omega \hat{x} - \frac{i}{m}\hat{p})$$
taking this part alone:
$$(\omega \hat{x} + \frac{i}{m}\hat{p})^2$$
I'm getting:
$$\omega^2[\hat{x},\hat{x}] + \frac{\omega i}{m}[\hat{x},\hat{p}] + \frac{\omega i}{m}[\hat{p},\hat{x}] - \frac{1}{m}[\hat{p},\hat{p}]$$
$$ = 0 + \frac{\omega i}{m}(i \hbar) + \frac{\omega i}{m}(-i \hbar) + 0 $$
$$ = 0$$
Which would give a final result:
$$[a^2,a^{\dagger}] = 0 $$
So I clearly don't know how to go about this the right way!
The only error I can think of is that I'm supposed to put a dummy function alongside the operators i.e. I'm supposed to actually show that:
$$[a^2,a^{\dagger}]\Psi(x) = 2a\Psi(x) $$
This worked in a previous exercise where I demonstrated
$$[\hat{x},\hat{p}]\Psi(x) = i\hbar \Psi(x)$$
But this isn't how it's done in the first image I've embedded in this post where there isn't any kind of "dummy function" involved.
Or perhaps I'm not expanding the brackets properly? This is probably more likely...
Any help much appreciated, thank you.