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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 ...
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.
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.
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 ...
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 ]
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 Timoshenko–Ehrenfest beam theory was developed by Stephen Timoshenko and Paul Ehrenfest [1] [2] [3] early in the 20th century. [ 4 ] [ 5 ] The model takes into account shear deformation and rotational bending effects, making it suitable for describing the behaviour of thick beams, sandwich composite beams , or beams subject to high ...
In Riemannian geometry, Schur's lemma is a result that says, heuristically, whenever certain curvatures are pointwise constant then they are forced to be globally constant. The proof is essentially a one-step calculation, which has only one input: the second Bianchi identity.