Conjugates to one particle Hamiltonians in 1-dimension in differential form

Abstract

A time operator is a Hermitian operator that is canonically conjugate to a given Hamiltonian. We construct such operators in position representation for a 1-dimensional particle. The construction is first simplified by assuming a definite form for the kernel that is based on the free particle case and is justified by the correct classical limit of the operator. This leads to a family of Hamiltonian conjugates that can be derived by finding a twice-differentiable function using a hyperbolic second-order partial differential equation with appropriate boundary conditions. Additional conditions may be imposed to produce different Hamiltonian conjugates such as those corresponding to time of arrival operator. A larger solution space of Hamiltonian conjugates, like those that can arise from kernels involving Dirac Deltas, can be also constructed by removing the simplifying assumption and treating the operators as a distribution on some function space.