7. Time Evolution

Abstract

The evolution of systems in time is less important in quantum mechanics than in classical physics, since in quantum mechanics so much can be learned from the energy eigen­values and eigen­functions. Still, time evolution is needed for such important physical processes as the creation and absorption of light and other radiation. And many other physical processes of practical importance are simplest to understand in terms of classical physics. To translate a typical rough classical description into correct quantum mechanics requires an under­standing of unsteady quantum mechanics.

The chapter starts with the introduction of the Schrödinger equation. This equation is as important for quantum mechanics as Newton’s second law is for classical mechanics. A formal solution to the equation can be written immediately down for most systems of inter­est.

One direct consequence of the Schrödinger equation is energy conserv­ation. Systems that have a definite value for their energy conserve that energy in the simplest possible way: they just do not change at all. They are stationary states. Systems that have uncertainty in energy do evolve in a non­trivial way. But such systems do still conserve the proba­bility of each of their possible energy values.

Of course, the energy of a system is only conserved if no devious external agent is adding or removing energy. In quantum mechanics that usually boils down to the condition that the Hamiltonian must be independent of time. If there is a nasty external agent that does mess things up, analysis may still be possible if that agent is a slowpoke. Since physicists do not know how to spell slowpoke, they call this the adiabatic approxi­mation. More precisely, they call it adiabatic because they know how to spell adiabatic, but not what it means.

The Schrödinger equation is readily used to describe the evolution of expec­tation values of physical quantities. This makes it possible to show that Newton’s equations are really an approxi­mation of quantum mechanics valid for macroscopic systems. It also makes it possible to formulate the popular energy-time uncertainty relationship.

Next, the Schrödinger equation does not just explain energy conserv­ation. It also explains where other conserv­ation laws such as conserv­ation of linear and angular momentum come from. For example, angular momentum conserv­ation is a direct consequence of the fact that space has no preferred direction.

It is then shown how these various conserv­ation laws can be used to better understand the emission of electro­magnetic radiation by say an hydrogen atom. In particular, they provide conditions on the emission process that are called selection rules.

Next, the Schrödinger equation is used to describe the detailed time evolution of a simple quantum system. The system alternates between two physi­cally equivalent states. That provides a model for how the fundamental forces of nature arise. It also provides a model for the emission of radiation by an atom or an atomic nucleus.

Unfortunately, the model for emission of radiation turns out to have some problems. These require the consid­eration of quantum systems involving two states that are not physi­cally equivalent. That analysis then finally allows a comprehensive description of the inter­action between atoms and the electro­magnetic field. It turns out that emission of radiation can be stimulated by radiation that already exists. That allows for the operation of masers and lasers that dump out macroscopic amounts of monochromatic, coherent radiation.

The final sections discuss examples of the non­trivial evolution of simple quantum systems with infinite numbers of states. Before that can be done, first the so-far neglected eigen­functions of position and linear momentum must be discussed. Position eigen­functions turn out to be spikes, while linear momentum eigen­functions turn out to be waves. Particles that have significant and sustained spatial local­ization can be identified as “packets” of waves. These ideas can be gener­alized to the motion of conduction electrons in crystals.

The motion of such wave packets is then examined. If the forces change slowly on quantum scales, wave packets move approximately like classical particles do. Under such conditions, a simple theory called the WKB approxi­mation applies.

If the forces vary more rapidly on quantum scales, more weird effects are observed. For example, wave packets may be repelled by attractive forces. On the other hand, wave packets can penetrate through barriers even though classi­cally speaking, they do not have enough energy to do so. That is called tunneling. It is important for various appli­cations. A simple estimate for the proba­bility that a particle will tunnel through a barrier can be obtained from the WKB approxi­mation.

Normally, a wave packet will be partially transmitted and partially reflected by a finite barrier. That produces the weird quantum situation that the same particle is going in two different directions at the same time. From a more practical point of view, scattering particles from objects is a primary technique that physicists use to examine nature.