Quantum Theory

Quantum Theory was the theory introduced by Max Planck and Albert Einstein at the beginning of the 20^th century that showed that light really has a dual nature: it can interact with matter either as a particle (photon), exchanging energy and momentum, or as a wave, producing interference and diffraction phenomena. In 1924 Louis de Broglie introduced the concept that all forms of matter possess the dual behavior of particles and waves. The de Broglie wavelength λ_d of a particle is given by λ_d p = h, where p is the momentum of the particle and h is the Planck constant. Clinton J. Davisson and Lester Germer experimentally confirmed the wave/particle duality of particles when they showed that electrons exhibit interference effects. Erwin Schrödinger constructed a set of differential equations that describe the wave structure of atomic particles, and the laws that govern their interactions. Later developments in quantum mechanics, principally by Paul Dirac and Werner Heisenberg, provided the backbone of the quantum theory of matter.

The quantum-mechanical approach pictures the electron as a wave structure that resides in energy levels where the amplitude of the wave is greatest in the vicinity of the classical Bohr electron orbit. In the quantum-mechanical picture, there is a finite probability that the electron can be anywhere in the Universe, but the probability density is highest at the classical orbit or energy level. The quantization of electron orbits emerges in a very straightforward manner in Schrödinger’s wave formulation: the ‘orbit’ corresponds to the region where there are an integral number of waves around the atom. This three dimensional pattern can also be thought of as the probability density of a cloud of negative charge (integrating to the charge on a single electron) around the atom.

The nucleus of the atom, being constructed of individual protons and neutrons, is also described by quantum mechanics. Its greater mass results in a more concentrated probability distribution than for an electron. Schrödinger’s equation can be written down for any atomic system. It will include the effects of the electrostatic and magnetic interactions of all the constituent particles. The solution of this equation gives an exact description of the energy states of the system. Only in the simplest of atoms (hydrogen and helium) can accurate solutions – called wave functions or Eigen functions – of the Schrödinger equation be obtained.

For complex structures (atoms containing many electrons, or the atomic lattice structure of crystals) only approximate solutions to the Schrödinger equation are, in general, possible. Modern high-speed computers are capable of furnishing solutions of wave equations for quite complex systems. In the area of astrophysics, such solutions can give, for example, the energy levels and hence emission-line wavelengths of highly ionized species or complicated molecules not observable in the laboratory. The principal structures of atomic nuclei and the ways in which they interact are also governed by quantum mechanics. Some of the nuclear reactions of interest in the construction of theoretical models of stars cannot be measured in the laboratory because they occur at too low a rate. For such reactions, calculations of rates must be made, which involve solutions of the wave equations for the two nuclei as they approach, collide and recede from one another.

The probabilistic nature of quantum mechanics provides an explanation for the ability of particles to ‘tunnel’ from one state to another (of equal energy) without having to acquire the extra energy to pass over the potential barrier that separates the two states. Such tunneling, for example, allows nuclear reactions to occur at lower energies (and hence lower temperatures) than they would otherwise require.

On a somewhat grander scale, attempts are currently being made to examine the properties of wave functions that represent the entire Universe, to see if the Big Bang could be the result of tunneling from a different state.

Questions to Ponder

• What is the significance of Heisenberg's Uncertainty Principle?
• What is the significance of Pauli’s Exclusion Principle?
• Under what conditions are the wave-like behaviors of a particle observed?