Construction of time of arrival operators for the inverse square potential

Abstract

We demonstrate the construction of time of arrival operators for the inverse square potential of the form q̂^-2. The construction was done through the use of the works of Bender and Dunne, wherein the prescribed method is to write the time operators as a sum of symmetric Hermitian operators T̂_m,n and obtain minimal solutions from equations that arise from the canonical commutation relation with the Hamiltonian. In the classical limit, the resulting operators were not able to replicate the classical time of arrival. Thus, an additional process is presented to be able to fully replicate the expression of the classical time of arrival.