Quantum

One of the striking consequences of the quantum theory is the noncommutation of pairs of "operators" representing physical quantities, such as position and momentum (whether regarded as matrices, derivatives acting upon the psi-function or, as Dirac would put it, abstract algebraic quantities). If p and q are two such operators, their commutation relations read: p q − q p = h/2πi, where i is the square root of −1. From this condition Heisenberg proved that q and p cannot be simultaneously measured with arbitrary accuracy, but that each must have an error, the product of the errors exceeding h /2. Since an exact knowledge of the initial conditions is necessary in order to make predictions of the future behavior of the system in question, and since such exactness is in principle impossible, physics is no longer a deterministic science. Even in classical physics, exact knowledge is not usually possible, but the important difference is that quantum mechanics forbids exactness "in principle." Heisenberg's uncertainty relations and other "acausal" predictions of quantum mechanics have given rise to an enormous amount of philosophical debate, and Einstein and other prominent older physicists never accepted the so-called Copenhagen interpretation, developed by the school of Bohr and Heisenberg. Bohr's principle of complementarity states that the world (or the part being studied) reveals itself to experimental probing in many different guises, but no one of these pictures is complete. Only the assembly of all possible pictures can reveal the truth.

One of the great aims of the Bohr-Sommerfeld quantum theory was to explain the periodic table of the chemical elements in terms of atomic structure, especially the periods 2, 8, 18, 32, and so on, having the general form 2 n², where n is an integer. Bohr and others made great progress in this direction, based on the quantum numbers of the allowed stationary states, except for the "2" in the formula. Wolfgang Pauli (1900–1958), who had worked with Born and Bohr and was studying the classification of atomic states in the presence of a strong magnetic field, suggested in 1925 that the electron must possess a new nonclassical property, or quantum number, that could take on two values. With this assumption he could describe the atomic structures in terms of shells of electrons by requiring that no two electrons in the atom could have identical quantum numbers. This became known as Pauli's "exclusion principle."

A short time after this, two young Dutch physicists, Samuel Goudsmit and George Uhlenbeck, found Pauli's new property in the form of spin angular momentum. This is the analog for an elementary particle of the rotation of the earth on its axis as it revolves around the sun, except that elementary particles are pointlike and there is no axis. However, they do possess a property that behaves as and combines with orbital angular momentum and that can take on values of the form s(h/2π), where s can be zero, integer, or half-integer. That was a totally unexpected development, especially the half-integer quantum numbers. The electron is one of several elementary particles that has ½ unit of spin, while the photon has spin of one unit.

The subject of quantum statistics, as opposed to classical statistical mechanics, is an important field. (Recall that h entered physics through Planck's analysis of the statistics of radiation oscillators.) It is found that identical stable elementary particles of half-integer spin (electron, protons) form shells, obey the exclusion principle, and follow a kind of statistics called Fermi-Dirac, worked out by Dirac and the Italian-American physicist Enrico Fermi (1901–1954). Other Fermi-Dirac systems are the conduction electrons in a metal and the neutrons in a neutron star. On the other hand, identical elementary particles of integer spin, such as photons of the same frequency, tend to occupy the same state when possible, and obey Bose-Einstein statistics, worked out by the Indian physicist Satyendranath Bose (1894–1974) and by Einstein. This results in phenomena that characterize lasers, superfluid liquid helium at low temperature, and other Bose-Einstein condensates that have been studied recently.