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Correspondence Principle

  1. Aug 9, 2007 #1
    Hello.

    I know about the correspondence principle which states that classical mechanics emerges from quantum mechanics as high quantum numbers are reached. I can see this for a particle in a box, but can this be mathematically justified for a general quantum system?

    Thanks in advance.
     
  2. jcsd
  3. Aug 9, 2007 #2

    Meir Achuz

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    The moon is in a Bohr orbit of huge N.
     
  4. Aug 9, 2007 #3
    Ok...I could see that. Perhaps I should rephrase my question. The correspondence principle as stated for simple quantum systems like a square well and a harmonic oscillator, in most books, is that for large quantum numbers, classical effects become apparent. I was just wondering whether we could generalize this principle mathematically (I know it sounds vague) or whether a more formal statement exists. But I guess not...its a principle after all and a physical principle.
     
  5. Aug 9, 2007 #4
    the most direct way is to use the Heisenberg operator picture:

    [tex]\frac{d <\Omega>}{dt} = \frac{i}{\hbar}<[\hat{H}, \hat{\Omega}]>[/tex]

    and show that if omega is the position or momentum operators, that you get back (as an average) a form of newton's second law.

    then construct the wavepacket. in the high temperature limit, the wavepacket approaches a delta function in position and momentum. the average above will then become exact.

    this all assumes that there is no _explicit_ time dependence in your potential


    yet an alternative demonstration of the bohr correspondence principle, is to consider the de Broglie thermal wavelength as a function of temperature. yet another is to look at the density matrix formulation of the partition function [tex]e^{- \beta \hat{H}}[/tex] and show that in the classical limit you get boltzmann statistics, etc.

    (cant get the beta above to show up in the partition function)
     
    Last edited: Aug 9, 2007
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