On the diagonalization of quadratic Hamiltonians

TL;DR

This paper introduces a novel method for diagonalizing quadratic Hamiltonians using a unitary transformation that preserves original observables, providing a general approach and detailed examples.

Contribution

It presents a new procedure for diagonalizing quadratic Hamiltonians that maintains the original position and momentum observables.

Findings

A general method for diagonalization is developed.

Explicit solutions for simple cases are provided.

The approach preserves the original observables in the transformed Hamiltonian.

Abstract

A new procedure to diagonalize quadratic Hamiltonians is introduced. We show that one can find a unitary transformation such that the transformed quadratic Hamiltonian is diagonal but still written in terms of the original position and momentum observables. We give a general method to diagonalize an arbitrary quadratic Hamiltonian and derive a few of the simplest special cases in detail.