The Eigenvalue Equation and the Evolution Operator

Abstract

The operators that represent variables of classical dynamics are built by analogy with classical analogues. (Since the classical description, as we know, can be changed by canonical transformation, this statement implies that in Quantum Mechanics, we must enjoy the same freedom. We shall see later how this arises) Sometimes, there are complications due to the fact that some operators fail to commute; such cases will be noted below. Therefore, for the simple case of one particle in a potential,

$$\begin{aligned} \hat{H} = - {\hbar ^{2} \over 2m} \nabla ^{2} +V(\hat{x}) \end{aligned}$$

is the Hamiltonian operator. By the same criterion, one can write down the Hamiltonian operator for systems of particles.

It is a strange theory that predicts particles that interfere with themselves and propagate through forbidden regions. But that is just what happens in reality.