Crypto-unitary forms of quantum evolution operators

TL;DR

This paper explores the equivalence between time-dependent and time-independent quantum Hamiltonians for describing quantum evolution, recommending the simpler time-independent form while ensuring unitarity in an appropriate Hilbert space representation.

Contribution

It demonstrates the equivalence of time-dependent and time-independent quantum evolution operators and proposes a tradeoff analysis favoring the simpler, time-independent approach.

Findings

Time-independent Hamiltonians can effectively replace time-dependent ones in quantum evolution.

Physical unitarity is maintained in a specific Hilbert space representation.

The study provides a framework for simplifying quantum evolution descriptions.

Abstract

For the description of quantum evolution, the use of a manifestly time-dependent quantum Hamiltonian $\mathfrak{h}(t) =\mathfrak{h}^\dagger(t)$ is shown equivalent to the work with its simplified, time-independent alternative $G\neq G^\dagger$. A tradeoff analysis is performed recommending the latter option. The physical unitarity requirement is shown fulfilled in a suitable ad hoc representation of Hilbert space.