- #1
TupoyVolk
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Quantum Harmonic Oscillator Operator Commution (solved)
EDIT
This was solved thanks to CompuChip! The entire post is also not very interesting as it was a basic mistake :P No need to waste time
This is not homework (I am not currently in college :P), but it is a mathematical question I'm stuck and I would greatly appreciate help.
The Quantum harmonic oscillator Operator method uses:
[tex]\widehat{a}[/tex] = [tex]\sqrt{\frac{m\omega}{2\hbar}}[/tex]([tex]\widehat{x}[/tex] + [tex]\frac{i\widehat{p}}{m\omega}[/tex])
and
[tex]\widehat{a}[/tex][tex]^{+}[/tex] = [tex]\sqrt{\frac{m\omega}{2\hbar}}[/tex]([tex]\widehat{x}[/tex] - [tex]\frac{i\widehat{p}}{m\omega}[/tex])
It also says that:
[[tex]\widehat{a}[/tex],[tex]\widehat{a}[/tex][tex]^{+}[/tex]] = 1
[[tex]\widehat{a}[/tex],[tex]\widehat{a}[/tex][tex]^{+}[/tex]] = [tex]\widehat{a}[/tex][tex]\widehat{a}[/tex][tex]^{+}[/tex] - [tex]\widehat{a}[/tex][tex]^{+}[/tex][tex]\widehat{a}[/tex]
I keep ending up with 2!
Here is a "proof"
http://quantummechanics.ucsd.edu/ph130a/130_notes/node169.html
But they have simply multiplied [tex]\widehat{a}[/tex][tex]\widehat{a}[/tex][tex]^{+}[/tex]
I feel like I cannot continue(self-study) until I see how I'm wrong. Please help!
EDIT
This was solved thanks to CompuChip! The entire post is also not very interesting as it was a basic mistake :P No need to waste time
This is not homework (I am not currently in college :P), but it is a mathematical question I'm stuck and I would greatly appreciate help.
The Quantum harmonic oscillator Operator method uses:
[tex]\widehat{a}[/tex] = [tex]\sqrt{\frac{m\omega}{2\hbar}}[/tex]([tex]\widehat{x}[/tex] + [tex]\frac{i\widehat{p}}{m\omega}[/tex])
and
[tex]\widehat{a}[/tex][tex]^{+}[/tex] = [tex]\sqrt{\frac{m\omega}{2\hbar}}[/tex]([tex]\widehat{x}[/tex] - [tex]\frac{i\widehat{p}}{m\omega}[/tex])
It also says that:
[[tex]\widehat{a}[/tex],[tex]\widehat{a}[/tex][tex]^{+}[/tex]] = 1
[[tex]\widehat{a}[/tex],[tex]\widehat{a}[/tex][tex]^{+}[/tex]] = [tex]\widehat{a}[/tex][tex]\widehat{a}[/tex][tex]^{+}[/tex] - [tex]\widehat{a}[/tex][tex]^{+}[/tex][tex]\widehat{a}[/tex]
I keep ending up with 2!
Here is a "proof"
http://quantummechanics.ucsd.edu/ph130a/130_notes/node169.html
But they have simply multiplied [tex]\widehat{a}[/tex][tex]\widehat{a}[/tex][tex]^{+}[/tex]
I feel like I cannot continue(self-study) until I see how I'm wrong. Please help!
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