Quantum Hamiltonian eigenstates for a free transverse field

TL;DR

This paper introduces a new set of eigenstates for the quantum Hamiltonian of a free transverse field, using an extended quadratic form of the transverse Laplace operator to construct spherical basis functions with singularities.

Contribution

It presents an alternative eigenstate basis for the quantum Hamiltonian of a free transverse field, expanding the mathematical framework for such systems.

Findings

New eigenstate basis for free transverse field Hamiltonian

Extension of quadratic form of transverse Laplace operator

Basis functions with singularity at the origin

Abstract

We demonstrate that quantum Hamiltonian operator for a free transverse field within the framework of the second quantization reveals an alternative set of states satisfying the eigenstate functional equations. The construction is based upon extensions of the quadratic form of the transverse Laplace operator which are used as a source of spherical basis functions with singularity at the origin. This basis then naturally takes place of the one of plane or spherical waves in the process of Fourrier or spherical variable separation.