Conjugates to One Particle Hamiltonians in 1-Dimension in Differential Form

TL;DR

This paper explores the construction of conjugate operators to one-particle Hamiltonians in 1D using differential equations, expanding the solution space beyond the traditional time of arrival operator.

Contribution

It introduces a modified time kernel equation to find additional Hamiltonian conjugate operators, broadening the understanding of time operators in quantum mechanics.

Findings

Derived new solutions to the time kernel equation

Expanded the set of possible conjugate operators

Provided insights into boundary condition effects on solutions

Abstract

A time operator is a Hermitian operator that is canonically conjugate to a given Hamiltonian. For a particle in 1-dimension, a Hamiltonian conjugate operator in position representation can be obtained by solving a hyperbolic second-order partial differential equation, known as the time kernel equation, with some boundary conditions. One possible solution is the time of arrival operator. Here, we are interested in finding other Hamiltonian conjugates by further studying the boundary conditions. A modified form of the time kernel equation is also considered which gives an even bigger solution space.