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The implications of the Ehrenfest theorem for systems with classically chaotic dynamics are discussed at Scholarpedia article Ehrenfest time and chaos. Due to exponential instability of classical trajectories the Ehrenfest time, on which there is a complete correspondence between quantum and classical evolution, is shown to be logarithmically ...
From mean recurrence theorem follows that even the expected time to going back to the initial state is finite, and it is . Using Stirling's approximation one finds that if we start at equilibrium (equal number of particles in the containers), the expected time to return to equilibrium is asymptotically equal to π N / 2 {\displaystyle ...
A similar equation describes the time evolution of the expectation values of observables, given by the Ehrenfest theorem. Corresponding to the trace-preserving property of the Schrödinger picture Lindblad equation, the Heisenberg picture equation is unital, i.e. it preserves the identity operator.
Taking expectation values automatically yields the Ehrenfest theorem, featured in the correspondence principle. By the Stone–von Neumann theorem , the Heisenberg picture and the Schrödinger picture are unitarily equivalent, just a basis change in Hilbert space .
So Newton's laws are exactly obeyed by the expected values of the operators in any given state. This is Ehrenfest's theorem, which is an obvious corollary of the Heisenberg equations of motion, but is less trivial in the Schrödinger picture, where Ehrenfest discovered it.
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity. In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity , [ 1 ] it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. [ 2 ]
Remarkably, this effect can drive a wide variety of novel physical phenomena, especially operating electron spins by electric fields, even when it is a small correction to the band structure of the two-dimensional metallic state. An example of a physical phenomenon that can be explained by Rashba model is the anisotropic magnetoresistance (AMR).
This is an example of Noether's theorem. If H is just the Hamiltonian for one particle, the total angular momentum of that one particle is conserved when the particle is in a central potential (i.e., when the potential energy function depends only on | r | {\displaystyle \left|\mathbf {r} \right|} ).