7. Time Evolution

AbstractThe 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

packetsof 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.

- 7.1 The Schrödinger Equation
- 7.1.1 The equation
- 7.1.2 Solution of the equation
- 7.1.3 Energy conservation
- 7.1.4 Stationary states
- 7.1.5 The adiabatic approximation

- 7.2 Time Variation of Expectation Values

- 7.3 Conservation Laws and Symmetries
- 7.4 Conservation Laws in Emission
- 7.4.1 Conservation of energy
- 7.4.2 Combining angular momenta and parities
- 7.4.3 Transition types and their photons
- 7.4.4 Selection rules

- 7.5 Symmetric Two-State Systems

- 7.6 Asymmetric Two-State Systems

- 7.7 Absorption and Stimulated Emission

- 7.8 General Interaction with Radiation
- 7.9 Position and Linear Momentum

- 7.10 Wave Packets
- 7.10.1 Solution of the Schrödinger equation.
- 7.10.2 Component wave solutions
- 7.10.3 Wave packets
- 7.10.4 Group velocity
- 7.10.5 Electron motion through crystals

- 7.11 Almost Classical Motion
- 7.11.1 Motion through free space
- 7.11.2 Accelerated motion
- 7.11.3 Decelerated motion
- 7.11.4 The harmonic oscillator

- 7.12 Scattering

- 7.13 Reflection and Transmission Coefficients