Energy raising/lowering operators, algebra

[raintrek]

$$\hat{x} = \left(\frac{\hbar}{2wm}\right)^{1/2}(\hat{a} + \hat{a}^{+})$$

$$\hat{p} = -i\left(\frac{\hbar wm}{2}\right)^{1/2}(\hat{a} - \hat{a}^{+})$$

I'm trying to demonstrate that

$$\hat{H} = (\hat{a}^{+}\hat{a} + \frac{1}{2})\hbar w$$

where $$\hat{H} = \frac{1}{2m} \hat{p}^{2} + \frac{mw^{2}}{2} \hat{x}^{2}$$

Given the commutation relation:

$$[\hat{a},\hat{a}^{+}]=1$$

However I seem to have too many twos! Here's my attempt:

$$\hat{H} = \left[\frac{1}{2m} \frac{\hbar wm}{2} (-\hat{a}^{2} + \hat{a}\hat{a}^{+} + \hat{a}^{+}\hat{a} - \hat{a}^{+2})\right] + \left[\frac{mw^{2}}{2} \frac{\hbar}{2wm} (\hat{a}^{2} + \hat{a}\hat{a}^{+} + \hat{a}^{+}\hat{a} + \hat{a}^{+2})\right]$$

$$\hat{H} = \frac{\hbar w}{4} (1 + 2\hat{a}^{+}\hat{a})$$

Can anyone point out where I've gone wrong? Many thanks!

 

[malawi_glenn]

$$\hat{H} = \left[\frac{1}{2m} \frac{\hbar wm}{2} (-\hat{a}^{2} + \hat{a}\hat{a}^{+} + \hat{a}^{+}\hat{a} - \hat{a}^{+2})\right] + \left[\frac{mw^{2}}{2} \frac{\hbar}{2wm} (\hat{a}^{2} + \hat{a}\hat{a}^{+} + \hat{a}^{+}\hat{a} + \hat{a}^{+2})\right]$$

is not $$\hat{H} = \frac{\hbar w}{4} (1 + 2\hat{a}^{+}\hat{a})$$

but:
$$\frac{\hbar \omega}{2}(aa^+ + a^+a)$$

you know that $$aa^+ - a^+a = 1$$, how can you manipulate $$aa^+ + a^+a$$ to become what you are looking for? ($$\hat{H} = (\hat{a}^{+}\hat{a} + \frac{1}{2})\hbar w$$
)

HINT: Try adding and substract the same entity, 3 = 3 +1 -1

 

[raintrek]

Ha, my own stupid fault. I'd only taken one lot of $$aa^{+} + a^{+}a$$ from the factorising! Thanks malawi! Been a long day hehe

 

[malawi_glenn]

Ha, my own stupid fault. I'd only taken one lot of $$aa^{+} + a^{+}a$$ from the factorising! Thanks malawi! Been a long day hehe

I've been there myself 1000times ;) Good luck!