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Quantum Harmonic Oscillator Operator Commution

  1. Mar 28, 2010 #1
    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!
     
    Last edited: Mar 28, 2010
  2. jcsd
  3. Mar 28, 2010 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    Then you're doing something wrong :P
    For us to see what exactly, you could post your calculation (either scanned or - preferably - nicely TeXed)

    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?
     
  4. Mar 28, 2010 #3
    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: Mar 28, 2010
  5. Mar 28, 2010 #4
    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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