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Operators with commutator ihbar

  1. Nov 20, 2013 #1
    I know that the commutator of the position and momentum operators is ihbar. Can any other combination of two different operators produce this same result, or is it unique to position and momentum only?
     
  2. jcsd
  3. Nov 20, 2013 #2

    jfizzix

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    Any position and its conjugate momentum have commutator [itex]i \hbar[/itex].

    It's not just linear position and momentum,
    [itex] [x,p_{x}]=i\hbar[/itex],
    but angular position and (the proper component of) the angular momentum (the component being the one parallel to the axis of rotation),
    [itex] [\theta,L_{\theta}]=i\hbar[/itex].

    Going into a bit more detail:
    According to Dirac, one way of getting quantum mechanics from classical mechanics is by substituting the Poisson bracket algebra with i hbar times the corresponding commutator algebra.
    [itex][\hat{q_{j}},\hat{p_{j}}] = i \hbar \{q_{j},p_{j}\} = i \hbar[/itex].
    Assuming this always works, then the commutator of any generalized coordinate with its conjugate momentum will always be i hbar.
     
  4. Nov 20, 2013 #3
    Thank you, jfizzix... That is a very helpful answer! Thanks for posting !!!
     
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