Does the Correspondence Principle Apply to All Quantum Systems?

[maverick280857]

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.

 

[Meir Achuz]

The moon is in a Bohr orbit of huge N.

 

[maverick280857]

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.

 

[quetzalcoatl9]

the most direct way is to use the Heisenberg operator picture:

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

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 potentialyet 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 $$e^{- \beta \hat{H}}$$ and show that in the classical limit you get Boltzmann statistics, etc.

(cant get the beta above to show up in the partition function)