Energy raising/lowering operators, algebra

raintrek

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

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

I'm trying to demonstrate that

[tex]\hat{H} = (\hat{a}^{+}\hat{a} + \frac{1}{2})\hbar w[/tex]

where [tex]\hat{H} = \frac{1}{2m} \hat{p}^{2} + \frac{mw^{2}}{2} \hat{x}^{2}[/tex]

Given the commutation relation:

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

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

[tex]\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] [/tex]

[tex]\hat{H} = \frac{\hbar w}{4} (1 + 2\hat{a}^{+}\hat{a})[/tex]

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

malawi_glenn

[tex]\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] [/tex]

is not [tex]\hat{H} = \frac{\hbar w}{4} (1 + 2\hat{a}^{+}\hat{a})[/tex]

but:
[tex] \frac{\hbar \omega}{2}(aa^+ + a^+a) [/tex]

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

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 [tex]aa^{+} + a^{+}a[/tex] from the factorising! Thanks malawi! Been a long day hehe

malawi_glenn

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