Expectation values of observables in time-dependent quantum mechanics

摘要

Let U(t) be the evolution operator of the Schrodinger equation generated by a Hamiltonian of the form H-0(t)+ W(t), where H-0(t) commutes for all t with a complete set of time-independent projectors {P-j}(j=1)(infinity). Consider the observable A=Sigma(j)P(j) lambda(j), where lambda(j) similar or equal to j(mu), mu>0, for j large. Assuming that the "matrix elements" of W(t) behave as //PjW(t)P-k// similar or equal to 1//j-k/(p), j not equal k, forp>0 large enough, we prove estimates on the expectation value (U(t)phi / AU(t)phi)= [A](phi)(t) for large times of the type 0 depends on p and mu. Typical applications concern the energy expectation (H-0)(phi)(t) in case H-0(t) = H-0 or the expectation of the position operator [x(2)](phi)(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. [References: 20]