Relações de incerteza entrópicas de sistemas hamiltonianos dependentes do tempo

Abstract

In this work by using the Lewis and Riesenfeld invariant method we obtain the exact wave functions for two time-dependent systems: (i) a spinless charged particle in the presence of a time-dependent magnetic field , and (ii) a spinless charged particle in a time-dependent Penning trap. By considering the quantum solutions in the lowest-lying states, we were able to obtain the expressions of Fisher infrmation (Fᵣ and Fp), Shannon entropies (Sᵣ and Sp) and uncertainties in terms of a c-number quantity, ρ (and ρz in the system (ii)), which must be a real solution of the Milne–Pinney equation. In the system (i) we analyze three different configurations of time-varying magnetic fields. We observe that the inequality FᵣFᵣ ≤ 16 holds for the systems considered. We also observed squeezing phenomenon in momentum or/and coordinate spaces with increasing time. In the system (ii) we obtain the analytical expressions for the uncertainties in terms of two c-number functions satisfying a Milne-Pinney-like equation. We analyze the static and the time-dependent cases. For the former, where B(t) = B₀K and 𝚅(t) = 𝚅₀, we observe that the Heisenberg and Robertson-Schrödinger uncertainty relations are fulfilled and the behavior of the uncertainties ∆x, y e ∆pₓ, py and when 𝐵₀ changes indicates the occurrence of a squeezing phenomenon. For the later, where B(t) = (B₀² + B' cos² ( vt ) )¹′² K and 𝚅(t) = 𝚅₀ + 𝚅' cos² (vt), we observe that ∆x,y oscillate in time exhibiting a squeezing phenomenon. Relations among the uncertainties, Shannon entropies and Fisher lengths were stablished. We observe for both cases that the Shannon entropy in position, Sr, and in momentum,Sp, satisfy the relation Sr + Sp ≥ 3 (1 + 𝒍nπ), while the product of Fisher lengths δrδp exhibits a lower bound than the product of uncertainties ∆r∆p.