Abstract
.We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.
Highlights
Standard quantum mechanics allows for many equivalent variants to describe the same physical observables
In [1] we demonstrated that the time-dependent Schrodinger equation (TDSE) for a time-dependent Hermitian Hamiltonian, h(t) = h†(t), and the easier TDSE for a time-independent Hermitian Hamiltonian, H = H†, h(t)φ(t) = i ∂tφ(t) and HΨ (t) = i ∂tΨ (t), (1)
We have demonstrated that metric representations lead to consistent descriptions equivalent to the operator representation by providing further solutions to the time-dependent quasi-Hermiticity relation (4) and the time-dependent Dyson relation (5)
Summary
Standard quantum mechanics allows for many equivalent variants to describe the same physical observables. We will refer to the former as the observable operator representation and the latter as the metric representation by indicating the time-dependent object in the name of the representation These physically equivalent representations are made possible in this setting as it always involves nontrivial metric operators on the non-Hermitian side. The metric operator in (2) and the Dyson operator in (3) are related as ρ(t) := η†(t)η(t) In this picture the time-dependence has been moved from the Hamiltonian in the Hermitian system to the metric operator in the non-Hermitian system. In [1] we pursued the following process: Starting from a given a non-Hermitian Hamiltonian H we solved the time-dependent quasi-Hermiticity relation (5) first, which seems most natural as it only involves one unknown quantity, namely ρ(t). We will solve the time-dependent Dyson relation (5) and the time-dependent quasi-Hermiticity relation (4) in more detail and compare the advantages of one approach over the other
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