Introduction

So far, we have considered general dynamical variables represented by general linear operators acting in ket space. However, in classical mechanics the most important dynamical variables are those involving position and momentum. Let us investigate the role of such variables in quantum mechanics.

In classical mechanics, the position $q$ and momentum $p$ of some component of a dynamical system are represented as real numbers which, by definition, commute. In quantum mechanics, these quantities are represented as noncommuting linear Hermitian operators acting in a ket space which represents all of the possible states of the system. Our first task is to discover a quantum mechanical replacement for the classical result $q p-p q = 0$. Do the position and momentum operators commute? If not, what is the value of $q p-p q$?