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Conjugate momentum in the hamiltonian

  1. May 25, 2010 #1

    I just have a quick question that I haven't quite been able to find a definitive answer to, regarding conjugate momenta in the Hamiltonian.

    Ok, so it regards the following term for the hamiltonian in a magnetic field:

    [tex]H=\frac{1}{2m}(p-qA)^2 [/tex]

    I'd like to ask whether [tex]p[/tex] is the conjugate momentum or if [tex]p_c=p-qA[/tex] is the conjugate momentum. As a guess, I would say that [tex]p_c=p-qA[/tex] is the conjugate momentum, as it seems to me that the hamiltonian should take into account the magnetic field. Would this then mean that the hamiltonian could be written as [tex]H=\frac{1}{2m}(p_c)^2 [/tex]

    Also, very important, does [tex] -i\hbar\nabla[/tex] represent the canonical momentum operator or the classical/mechanical momentum operatpor?

    Im sure the answers are around somewhere on the internet, but it strikes me that there are some conflicting statements and a tendency to just go "oh yeah, now we swap the canonical momentum, [tex]p[/tex] for mechanical momentum [tex]p[/tex]" and the like, so it would be nice to get a definitive answer.

    Thanks for your help!
  2. jcsd
  3. May 25, 2010 #2
    p is the conjugate/canonical momentum. p-eA is the mechanical momentum.
  4. May 25, 2010 #3
    Ah, brilliant! Thanks for your quick response!

    So that would make [tex] i\hbar\nabla[/tex] the conjugate/canonical momentum operator?
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