Expectation Values of Observables in Time-Dependent Quantum Mechanics |
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| Authors: | J M Barbaroux A Joye |
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| Institution: | (1) Centre de Physique Théorique (Unité Propre de Recherche 7061), CNRS Luminy, Case 907, F-13288 Marseille, Cedex 9, France, and;(2) PHYMAT, Université de Toulon et du Var, B.P. 132, F-83957 La Garde Cedex, France;(3) Institut Fourier, Université de Grenoble 1, B.P. 74, 38402 Saint-Martin d'Hères Cedex, France |
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| Abstract: | Let U(t) be the evolution operator of the Schrödinger equation generated by a Hamiltonian of the form H
0(t) + W(t), where H
0(t) commutes for all twith a complete set of time-independent projectors
. Consider the observable A= j
P
j jwhere
j
 j
 , >0, for jlarge. Assuming that the  matrix elements of W(t) behave as for p>0 large enough, we prove estimates on the expectation value
for large times of the type where  >0 depends on pand . Typical applications concern the energy expectation  H0  (t) in case H
0(t)  H
0or the expectation of the position operator x2(t) on the lattice where W(t) is the discrete Laplacian or a variant of it and H
0(t) is a time-dependent multiplicative potential. |
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| Keywords: | Time-dependent Hamiltonians Schrö dinger operator quantum stability quantum dynamics |
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