A procedure for obtaining quantum mechanical transformation of diagonalization from the classical

Abstract

Using as an example two coupled harmonic oscillators, a transformation to normal coordinates is made using the classical-type simultaneous diagonalization of quadratic forms, and this is then used to develop a procedure for constructing the corresponding quantum mechanical transformation to normal coordinates. The total classical transformation is nonunitary, whereas the quantum mechanical is unitary as it has to be in order to satisfy Von Neumann's theorem. Since the classical transformation has definite steps and is a very straightforward procedure, this could be a very useful procedure for constructing the quantum mechanical transformation in many models, and/or an alternative method for many models.