Does Hamilton's principle follow from the Feynman formulation of QM?

[Jano L.]

Do you think Hamilton's principle from classical mechanics can be deduced from Feynman's path integral in quantum mechanics?

(We get across this question in another discussion:

https://www.physicsforums.com/showthread.php?t=609087&page=5)

Of course, there is a loose connection, since main contribution to the Feynman integral is from the classical path.

The problem is, the Feynman integral in QM gives merely Green's function of Schr. equation; this needs to be integrated with the initial condition to produce probability density |\psi|^2.

However, Hamilton's principle says something about trajectory q(t). How to get this from the Feynman integral?

Jano

 

[bhobba]

Hi Jano

Thanks for starting another thread - much appreciated

You mentioned Ballentine didn't derive it in pages 116-123 I mentioned.

Actually he does on pages 120-121 in the section Classical Limit Of The Path Integral. However its the heuristic derivation I gave and not the more rigorous one using the method of steepest decent. That can be found in more detail (but still not in full detail) in the link I gave previously:
http://www.phys.vt.edu/~ersharpe/6455/ch1.pdf

Trouble is even the accounts that give more detail (at least the ones I have seen) use a dubious change of variable - really contour integration is required:
http://www.maths.manchester.ac.uk/~gajjar/MATH44011/notes/44011_note4.pdf

Thanks
Bill

 

[Jano L.]

Hmm, still I do not think the Hamilton principle is derived in the above pdf. It merely shows that the main contribution to the Green function comes from the classical trajectory. But in Hamilton's principle you do not have any Green function. You say something about the trajectory q(t).