Probability Amplitude phases

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Discussion Overview

The discussion centers on the concept of probability amplitude phases in quantum mechanics, specifically relating to Feynman's path integral formulation. Participants explore the physical reasoning behind the phase being proportional to the classical action and its implications in quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant questions the physical reasoning behind the phase of probability amplitude being proportional to the action in units of the quantum of action, \hbar.
  • Another participant suggests that plugging the wavefunction e^{iS/\hbar} into the Schrödinger equation leads to the classical Hamilton-Jacobi equation, providing strong evidence for Feynman's arguments.
  • This participant notes that while there is no explicit proof for using classical action in this manner, the consistency of the equations with experimental results supports the approach.
  • They also mention that the Schrödinger equation can be derived from the path integral formalism and that there is a classical limit as \hbar approaches zero.
  • A different participant expresses enthusiasm for the path integral formulation after reading introductory material from "QFT in a Nutshell" by Zee, highlighting its intuitive nature.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and enthusiasm about the path integral formulation, but no consensus is reached on the physical reasoning behind the phase of probability amplitude or the implications of the classical action.

Contextual Notes

Some assumptions regarding the interpretations of quantum mechanics and the derivations involved may not be fully articulated, and the discussion does not resolve the foundational questions raised.

Frankww
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Quesion in approaching to Path Integral

I've just read "Quantum Mechanics and Path Integral" book which was written by Feynman, he said the phase of probability amplitude is proportional to "the ACTION [tex]S[/tex] in units of Quantum of action [tex]\hbar[/tex]. What is the reason to be that? Can anybody explain it to me physically? Thank you for all replies.
 
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There might be a better answer than this, but one way to see it is to plug the wavefunction [itex]e^{iS/\hbar}[/itex] into the Schrödinger equation and get an equation for S. The equation you get (to leading order in an expansion in [itex]\hbar[/itex]) is just the (classical) Hamilton-Jacobi equation, where S plays the role of the classical action. This is very strong evidence in favor of Feynman's arguments.

In the end, I don't think there's an explicit "proof" that you should use the classical action in this way, just like there's no "proof" that the Schrödinger equation is correct - at the end of the day, you just ask if the equations are consistent with experiment (and the other descriptions of QM). And in this case, they are. In particular:

1. You can derive the Schrödinger equation from the path integral formalism.
2. There is a nice classical limit as [itex]\hbar\rightarrow 0[/itex].
 
Thank you very much for your answer. I'll review them, then.
 
Read the first few pages of QFT in Nutshell by ZEE

The motivation for introducing path integral formulation of QM is startted as
a doubt that feynmann had regarding the inteference of photon as it goes through a doble slit...
please read further

I cannot describe the hapiness that i fealt after reading those few pages.
I mean after reading it youll feel the basic idea of path integrals is so intuitive
and that anyone could think about that idea if just only he had looked beyond the box
 

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