The Schrödinger Equation

Abstract

Schrödinger’s equation

$$\begin{array}{lll} & i\rlap{--} h \frac{{\partial \psi }} {{\partial t}}(x,t)\, = \,\hat H\psi (x,t)\end{array}$$

is considered by physicists as a postulate that cannot be rigorously derived from Hamiltonian mechanics. It is however well known that in the case of linear Hamiltonian flows, the Schrödinger equation is obtained using the metaplectic representation: this will be proven in the first part of this chapter (Section 15.1). In the second part of this chapter (Section 15.2), which is somewhat tentative in the sense that we pay little attention to domain questions, we will show that the Schrödinger equation can be derived using Stone’s theorem on strongly continuous one-parameter groups of unitary operators, if one requires in addition that quantum states satisfy a certain covariance property. This derivation is made possible thanks to Theorem 356 (essentially due to Wong [163]) which says that Weyl calculus is the only symplectically covariant pseudo-differential theory. We mention that a more physical version of these results is to appear in de Gosson and Hiley [73] (not surprisingly leading to negative emotional reactions from some physicists); for those interested in the physical aspects (including the scientific ontology) of quantum mechanics, we recommend the texts [92, 93, 94, 95] by Hiley and collaborators.