Ehrenfest Theorem: Significance & Relation to Space-Time

[kehler]

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?

 

[malawi_glenn]

But quantum mech has common ground with classical mech by construction, so what is there to prove?

 

[kehler]

I don't know. I was just wondering what the theorem does...

 

[malawi_glenn]

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.

 

[Dahaka14]

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.

 

[kehler]

^ Thanks :). That makes it clear

 

[malawi_glenn]

one can say that one would like to derive classical mechanics from quantum mechanics.

 

[kehler]

Is it possible though to derive the whole of classical mechanics from quantum mechanics?

 

[malawi_glenn]

if you consider the path-integral formalism then it is straight forward to do so.

Also consider the correspondence principle

http://hyperphysics.phy-astr.gsu.edu/Hbase/hframe.html