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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 that, by definition, commute. In quantum mechanics, these quantities are represented as non-commuting linear Hermitian operators acting in a ket space that 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$ .

Richard Fitzpatrick 2016-01-22