Expectation values of observables in time-dependent quantum mechanics
JM Barbaroux, A Joye - Journal of statistical physics, 1998 - Springer
Journal of statistical physics, 1998•Springer
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 t with a complete set of time-
independent projectors {P_j\} _ j= 1^ ∞. Consider the observable A=∑ j P j λ j where λ j≃ j μ,
μ> 0, for j large. Assuming that the “matrix elements” of W (t) behave as for p> 0 large
enough, we prove estimates on the expectation value ⟨ U (t) ϕ| AU (t) ϕ ⟩ ≡ ⟨ A\rangle_ ϕ
(t) for large times of the type where δ> 0 depends on p and μ. Typical applications concern …
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^ ∞. Consider the observable A=∑ j P j λ j where λ j≃ j μ,
μ> 0, for j large. Assuming that the “matrix elements” of W (t) behave as for p> 0 large
enough, we prove estimates on the expectation value ⟨ U (t) ϕ| AU (t) ϕ ⟩ ≡ ⟨ A\rangle_ ϕ
(t) for large times of the type where δ> 0 depends on p and μ. Typical applications concern …
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 $$\{ P_j \} _{j = 1}^\infty $$ . Consider the observable A=∑j P jλjwhere λ j ≃ j μ, μ>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 $$\langle U(t)\phi|AU(t)\phi\rangle\equiv\langle A\rangle_\phi(t)$$ for large times of the type where δ>0 depends on pand μ. Typical applications concern the energy expectation <H0>ϕ(t) in case H 0(t) ≡ H 0or the expectation of the position operator <x2>ϕ(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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