Uncertainty Relation for Pseudo-Hermitian Quantum Systems

TL;DR

This paper extends the uncertainty relation to pseudo-Hermitian quantum systems, showing its equivalence to standard cases and analyzing the real nature of uncertainties in non-Hermitian quantum mechanics.

Contribution

It introduces an extended uncertainty relation for pseudo-Hermitian systems and provides analytical solutions to their Schrödinger equation.

Findings

Uncertainty relation remains real and greater than 1/2 in pseudo-Hermitian systems

Extended the uncertainty principle to non-Hermitian quantum mechanics

Derived analytical solutions for time-dependent Schrödinger equation with linear potential

Abstract

This study investigates pseudo-Hermitian quantum mechanics, where the Hamiltonian satisfies a modified Hermiticity condition. We extend the uncertainty relation for such systems, demonstrating its equivalence to the standard Hermitian case within a pseudo-Hermitian inner product. Analytical solutions to the time-dependent Schr\"odinger equation with a linearly evolving potential are derived. Furthermore, we show that the uncertainty relation for position and momentum remains real and greater than 1/2, highlighting the significance of non-Hermitian systems in quantum mechanics.