Exact wave functions and uncertainties for a spinless charged particle in a time-dependent Penning trap

- We obtain the wave functions for a charged particle in a time-dependent Penning trap.
- We analyze the static and the time-dependent (oscillating fields) cases.
- The uncertainties are established in the lowest-lying state.
- The Shannon entropy and Fisher information for the system are also established.

We use the dynamical invariant method and two unitary transformations to obtain the exact Schrödinger wave functions, ψnmk(r,t), for a spinless charged particle in a time-dependent Penning trap. For the special case n=m=k=0, 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 Δx,y and Δpx,py when B0 changes indicates the occurrence of a squeezing phenomenon. For the later, where B(t)=(B02+B'cos2(νt))1/2k andV(t)=V0+V'Cos2(νt), we observe that Δx,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 Fisher lengths δr and δp exhibit a lower bound than the uncertainties Δr,p.

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