Elements of Classical and Quantum Physics pp 213-226 | Cite as

# Postulate 2

## Abstract

\(D \equiv \frac{d}{dx}\), and also \(\hat{x}\) (which multiplies by *x*), are examples of linear operators \(\hat{O}\): \(\hat{O}(\varPhi +\varPsi )=\hat{O}\varPhi + \hat{O}\varPsi \). Coordinates and momenta of Classical Mechanics wear a hat and become quantum operators, and we shall meet more. The scalar product of \( \hat{Q} |\varPsi \rangle \) with \( |\varPhi \rangle \), which is \(\langle \varPhi |\hat{Q} |\varPsi \rangle , \) may be regarded as the element \(M_{ij}\) of some matrix *M* with *i* and *j* replaced by the indices \(\varPhi \) and \(\varPsi ,\) which are functions \(\in L^{2}\); it is called a matrix element of \(\hat{Q}\); the only real difference with a conventional matrix \(M_{ij}\) is that the indices are often continuous and the matrix most often has an infinity of rows and columns. Werner Heisenberg initially formulated the theory in terms of matrices.