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Chapter 15 Time Evolution in Quantum Mechanics hysical systems are, in general, dynamical, i.e. they evolve in time. The dynamics of classical P mechanical systems are described by Newton’s laws of motion, while the dynamics of the classical electromagnetic field is determined by Maxwell’s equations. Either Newton’s equations or Maxwell’s equations constitute laws of physics whose form is particular to the kind of system that is being described, though some unification is provided in terms of Hamilton’s principle, a universal way of stating the laws of classical dynamics. What we require is a means by which the dynamical properties of quantum systems can be described, i.e. given the state |(0) of a quantum system at some initial time t = 0, what is required is a physical law that tells us what the state will be at some other time t, i.e. a quantum law of evolution. Given that a state vector is a repository of the information known about a system, what is required is a general physical law that tells us how this information evolves in time in response to the particular physical circumstances that the system of interest finds itself in. It is to be expected that the details of this law will vary from system to system, but it turns out that the law of evolution can be written in a way that holds true for all physical systems, in some sense similar to the way that Hamilton’s principle provides a way of stating classical dynamical laws in one compact form. Of course, the quantum law of evolution ¨ is Schrodinger’s equation. 15.1 Stationary States The idea of ‘stationary states’ was first introduced by Bohr as a name given to those states of a hydrogen atom for which the orbits that the electron occupied were stable, i.e. the electron remained in the same orbital state for all time, in contrast to the classical physics prediction that no orbiting electron could remain in orbit forever: it would always radiate aw