Transformation Theory for Systems with Continuous Spectra

Abstract

So far, we have studied transformations from one basis to another only for basis vectors that are the eigenvectors of a set of commuting hermitian operators whose spectra are discrete. Moreover, we have studied finite-dimensional subspaces of these vector spaces, so our unitary transformation matrices were finite-dimensional. Commuting operators, however, such as the operators x, y, z or the operators p _x , p _y , p _z exist with continuous spectra. Still other operators have both discrete and continuous spectra. We need to study the transformation theory for the base vectors of this type. In particular, the coordinate and momentum representations are of great importance.