Quantum Harmonic Oscillator Operator Commution

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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!
 
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  • #2
CompuChip
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I keep ending up with 2!
Then you're doing something wrong :P
For us to see what exactly, you could post your calculation (either scanned or - preferably - nicely TeXed)

But they have simply multiplied [tex]\widehat{a}[/tex][tex]\widehat{a}[/tex][tex]^{+}[/tex]
I don't see where they did that. I just see them using the bilinearity of the commutator operation, i.e.
[r x, y] = r [x, y] (when r is a real number and x, y are operators - sorry, don't feel like putting hats and stuff)
[x + y, z] = [x, z] + [y, z] (where x, y, z are operators)
together with anti-symmetry ([x, y] = -[y, x]).

The proof looks really straightforward to me, can you maybe try to explain which step exactly is giving the problem?
 
  • #3
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Thank you so much for the reply.:smile:
I believe I am wrong, because QM indeed works! I just "need" to see how.

What was written where this sentence is, had a very stupid mathematical mistake. :)

I cannot see where I am wrong.
 
Last edited:
  • #4
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Holy crap.
I see my giant fail.

For some reason I turned things into commutators that shouldn't be them.
Thank you so much!
 

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