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What does canonical mean?

  1. Aug 6, 2007 #1
    what does it mean in quantum mechanics when they say canonical variables or canonical momentum? what is the difference from regular momentum?
  2. jcsd
  3. Aug 6, 2007 #2
    Read the bottom of this page under physics:


    To paraphrase: Any two variables whose Poisson Bracket (or Commutator in Quantum) that give a delta are canonical. A symmetry in one canonical variable implies the other is conserved. For example, translational invariance and conservation of momentum.

  4. Aug 6, 2007 #3


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    Not exactly. For example, in single particle quantum mechanics in the presence of an external electromagnetic field the canonical variables are the position [tex]\vec x[/tex] and the *canonical* momentum [tex]\vec p[/tex] which satisfy:

    But, if the above holds, it should also be obvious that the *mechanical* momentum [tex]\vec \pi =\vec p-e\vec A[/tex] where [tex]e[/tex] is the charge also satisfies:

    Thus, both sets of variables satisfy canonical communtation relations even though only the set (x,p) are called "canonical variables."
  5. Aug 7, 2007 #4
  6. Aug 7, 2007 #5
    thanks for your help
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