- #1
raintrek
- 75
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[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!
[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!