Ehrenfest Theorem: Significance & Relation to Space-Time

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Discussion Overview

The discussion revolves around the Ehrenfest theorem, its physical significance, and its relationship to classical mechanics and space-time. Participants explore whether the theorem provides insights into the connection between quantum mechanics and classical mechanics, as well as the implications for understanding observables in both frameworks.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants question the physical significance of the Ehrenfest theorem and whether it leads to conclusions about space and time.
  • Others argue that quantum mechanics inherently has common ground with classical mechanics, suggesting that the theorem may not prove anything new.
  • A participant provides the mathematical formulation of the theorem, indicating its role in describing the time evolution of expectation values of observables.
  • It is noted that the theorem allows for the evaluation of observables that have classical correspondences, potentially yielding classical equations under certain conditions.
  • One participant expresses a desire to derive classical mechanics from quantum mechanics, raising the question of whether this is feasible.
  • Another participant suggests that using path-integral formalism could facilitate deriving classical mechanics from quantum mechanics, referencing the correspondence principle.

Areas of Agreement / Disagreement

Participants express differing views on the significance of the Ehrenfest theorem and whether it serves as a bridge between quantum and classical mechanics. The discussion remains unresolved regarding the extent to which classical mechanics can be derived from quantum mechanics.

Contextual Notes

Some claims depend on specific interpretations of quantum mechanics and classical mechanics, and the discussion includes unresolved assumptions about the applicability of the theorem in various contexts.

kehler
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Is there any physical significance of this theorem? Can we make some kind of conclusion about space and time because the derivative of the expectation value of momentum with respect to time is equal to the negative of the expectation value of the derivative of potential energy w.r.t. space (d<p>/dt = -<dV/dx>)?? Or does it just prove to us that quantum mech and classical mech have some common ground?
 
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But quantum mech has common ground with classical mech by construction, so what is there to prove?
 
I don't know. I was just wondering what the theorem does...
 
the theorem says:

\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\langle [A,H] \rangle + \left\langle \frac{\partial A}{\partial t}\right\rangle

for any operator A.
 
The equation simply shows how the expectation value of an observable evolves with time. If you know of Heisenberg's equation of motion, the Ehrenfest theorem is simply the expectation value of the operator (observable) in question. The theorem itself is significant in the fact that you should be able to evaluate for observables and receive a classical equation, if the observable happens to have a classical correspondence. This theorem is a way to check that quantum mechanics is still consistent with classical mechanics in certain limits, because we do not want to lose Newton's law on the macroscopic scale now do we.

Basically, if some observable in quantum mechanics can also be measured classically, like momentum, you should get a familiar classical equation when making appropriate operator substitutions in the Ehrenfest theorem.
 
^ Thanks :). That makes it clear
 
one can say that one would like to derive classical mechanics from quantum mechanics.
 
Is it possible though to derive the whole of classical mechanics from quantum mechanics?
 

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