In the late 1800s, physicists were making accurate measurements of the spectra (the intensities of light as a function of wavelength, or color) of the emissions from black bodies (objects which are opaque, or highly absorbing, to the light they emit). Good examples of these objects are the sun, the filament of an incandescent lamp, and the burner of an electric stove. The color of a black body depends on its temperature, a cool body emitting radiation of long wavelengths, i.e., in the radio frequency range or in the infrared which are invisible to the eye, a warmer body emitting radiation which includes shorter wavelengths and appearing deep red, a still warmer body emitting radiation which includes still shorter wavelengths and appearing yellow, and a hot body emitting even shorter wavelengths and appearing white. The emissions are always over a broad range of colors, or wavelengths, and the appearance is the net result of seeing all of the colors at once.

Classical physics could not explain the spectra of black bodies. It predicted that the intensity of emitted light should increase rapidly with decreasing wavelength without limit (the "ultraviolet catastrophe"). In the figure below, the curve labeled "Rayleigh-Jeans law" shows the classically expected behavior.          

However, the measured spectra showed an intensity maximum at a particular wavelength, while the intensity decreased at wavelengths both above and below the maximum. In order to explain the spectra, the German physicist Max Planck (1858 - 1947) in 1900 was forced to make a desperate assumption for which he had no physical explanation. As with classical physics, he assumed the body consisted of vibrating oscillators which were collections of atoms or molecules. However, in contrast with classical physics which assumed that each oscillator could absorb an arbitrary amount of energy from the radiation or emit an arbitrary amount of energy to it, Planck was forced to assume that each oscillator could receive or emit only discrete, quantized energies (E), such that

                                                                               E = hf

where h is an exceedingly small constant whose value we do not need to present here, and f is the frequency of vibration of the oscillator (the number of times it vibrates per second). Each oscillator is assumed to vibrate always at a fixed frequency (although different oscillators in general had different frequencies), so if it emitted some radiation, it would lose energy equal to hf, and if it absorbed some radiation, it would gain energy equal to hf. Planck did not understand how this could be, he merely made this empirical assumption in order to explain the spectra. The figure above shows Planck’s prediction, and this agreed with the measured spectra.

Also in the late 1800s, experimental physicists were measuring the emission of electrons from metallic objects when they shone light on the object. This is called the photoelectric effect. These experiments also could not be explained using classical concepts. These physicists observed that emission of electrons occurred only for light wavelengths shorter than a certain threshold value which depended on the metal. Classically, however, one expected that the emission should not depend on wavelength at all, but only on intensity, with greater intensities yielding more copious emission of electrons.

In one of a famous series of papers in 1905, Einstein explained the photoelectric effect by starting with Planck’s concept of quantized energy exchanges with light radiation, and making the startling assumption that these quantized exchanges were a direct result of the quantization of light itself, i.e. light consisted of discrete bundles of energy called photons, rather than the continuous waves which had always been assumed by classical physicists. However, these bundles still had a wave nature, and still could be characterized by a wavelength, which determined their color. He also used Planck’s relationship between energy and frequency to identify the energy of the photon, and he used the relationship between velocity, frequency, and wavelength that classical physics had always used. Einstein received the Nobel prize for this paper.

In addition to measuring the spectra of blackbody radiation in the 19th century, experimental physicists also were familiar with the spectra emitted by gases through which an electrical discharge (an electric current with enough energy to strip some of the electrons from the atoms of the gas) was passing. Examples of such discharges are the familiar neon sign, in which the gas is neon, and the fluorescent light, in which the gas is mercury vapor (the fluorescent light bulb has special coatings on the inner walls which change the spectrum of the light). The spectra of such light sources consist of emissions at discrete, separated wavelengths, rather than over a continuous band of wavelengths as in blackbody spectra. These spectra are called line spectra because of their appearance when they are viewed with a spectrometer, which is a device used to separate the different wavelengths.

Line spectra are another example of phenomena which could not be explained by classical physics. Indeed, the explanation could not come until some developments in the understanding of the structure of atoms had been made by Ernest Rutherford and coworkers in 1911. By scattering alpha particles (consisting of two protons and two neutrons bound together), which were emitted by radioactive sources, from thin gold foils, they discovered that the gold atom consisted of a tiny (10meters) very dense positively charged nucleus surrounded by a much larger (10meters) cloud of negatively charged electrons. (Quantum mechanically, this picture is not correct, but for now it is adequate.)

When classical physics was applied to such a model of the atom, it predicted that the electrons could not remain in stable orbits about the nucleus, but would radiate away all of their energy and fall into the nucleus, much as an earth satellite falls into the earth when it loses its energy due to atmospheric friction. In 1913, after Niels Bohr had learned of these results, he constructed a model of the atom which made use of the quantum ideas of Planck and Einstein. He proposed that the electrons occupied discrete stable orbits without radiating their energy. The discreteness was a result of the quantization of the orbits, with each orbit corresponding to a quantized energy for the electron. The electron was required to have a certain minimum quantum of energy corresponding to a smallest orbit, and thus, the quantum rules did not permit the electron to fall into the nucleus. However, an electron could jump from a higher orbit to a lower orbit and emit a photon in the process. The energy of the photon could take on only the value corresponding to the difference between the energy of the electron in the higher and lower orbit. Bohr applied his theory to the simplest atom, the hydrogen atom, which consists of one electron orbiting a nucleus of one proton. The theory explained many of the properties of the observed line spectrum of hydrogen, but could not explain the next more complicated atom, that of helium which has two electrons. Nevertheless, the theory contained the basic idea of quantized orbits which was retained in the more correct theories that came later.

In the earliest days of the development of quantum theory, physicists, such as Bohr, tried to create physical pictures of the atom in the same way they had always created physical pictures in classical physics. However, although Bohr developed his initial model of the hydrogen atom by using an easily visualized model, it had features that were not understood, and it could not explain the more complicated two-electron atom. The theoretical breakthroughs came when some physicists who were highly sophisticated mathematically, such as Heisenberg, Pauli, and Dirac, largely abandoned physical pictures, and created highly mathematical theories which explained the detailed features of the hydrogen spectrum in terms of the energy levels and the radiative transitions from one level to another. The key feature of these theories was the use of matrices instead of ordinary numbers to describe physical quantities such as energy, position, and momentum. A matrix is an array of numbers that obeys rules of multiplication which are different from the rules obeyed by numbers.

The step of resorting to entirely mathematical theories which are not based on physical pictures was a radical departure in the early days of quantum theory, but today in developing the theories of elementary particles it is standard practice. Such theories have become so arcane that physical pictures have become difficult to create and to picture, and they are always developed to fit the mathematics rather than fitting the mathematics to the physical picture. Thus, adopting a positivist philosophy would prevent progress in developing models of reality, and the models that are intuited are more mathematical than physical.

Nevertheless, in the early 1920s some physicists continued to think in terms of physical rather than mathematical models. In 1923, de Broglie reasoned that if light could behave like particles, then particles like electrons could behave like waves, and he deduced the formula for the wavelength of the waves:

                                                                                            l =h/p

where p is the momentum (mass times velocity) of the electron. Experiments subsequently verified that electrons actually do behave like waves in experiments that are designed to reveal wave nature. We will say more about such experiments later.

In physics, if there is a wave then there must be an equation which describes the wave. De Broglie did not find that equation, but in 1926 Erwin Schrödinger discovered the celebrated equation which bears his name. He verified his equation by using it to calculate the line emission spectrum from hydrogen, which he could do without really understanding the significance of the waves. In fact, Schrödinger misinterpreted the waves and thought they represented the particles themselves. However, such an interpretation could not explain why experiments always showed that the photons emitted by an atom were emitted at random rather than predictable times, even though the average rate of emission could be predicted from both Heisenberg’s and Schrödinger’s theories. It also could not explain why, when a particle is detected, it always has a well defined position in space, rather than being spread out over space like a wave.

The proper interpretation was discovered by Max Born in 1926, who suggested that the wave (actually, the square of the amplitude or height of the wave, at each point in space) represents the probability that the particle will appear at that specified point in space if an experiment is done to measure the location of the particle. This interpretation introduces two extremely important features of quantum mechanics: 1) from the theory we can calculate only probabilities, not certainties (the theory is probabilistic, not deterministic), and 2) the theory tells us the probability of finding something only if we look, not what is there if we do not look (there is no objective reality of matter, i.e., matter does not exist independent of observers and observations). The Schrödinger wave is a probability wave, not a wave that carries force, energy, and momentum like the electromagnetic wave. However, the Schrödinger equation allows us to calculate precisely the wave at all points in space at any future time if we know the wave at all points in space at an initial time. In this sense, even quantum theory is completely deterministic.

As Born proposed, quantum theory is intrinsically probabilistic in that in most cases it cannot predict the results of individual observations. However, it is deterministic in that it can exactly predict the probabilities that specific results will be obtained. Another way to say this is that it can exactly predict the average values of measured quantities, like position, velocity, energy, or number of photons emitted or absorbed per unit time, when a large number of measurements is made on identical systems. For a single measurement, it cannot predict the exact results except in special cases. This randomness is not a fault of the theory--it is an intrinsic property of nature. Nature is not deterministic as was thought in classical physics.

Another feature of the quantum world, the world of microscopic objects, is that it is intrinsically impossible to measure simultaneously both the position and momentum of a particle. This is the famous uncertainty principle of Heisenberg, who derived it using the multiplication rules for the matrices which he used for position and momentum. For example, an apparatus designed to measure the position of an electron with a certain accuracy is shown in the following diagram. The hole in the wall ensures that the positions of the electrons as they pass through the hole are within the hole, not outside of it.    

So far, this is not different from classical physics. However, quantum theory says that if we know the position of the electron to within an accuracy of D q (the diameter of the hole), then our knowledge of the momentum at that point is limited to an accuracy D p such that

                                                    (D p)(D q) h (Heisenberg uncertainty relation)

In other words, the more accurately we know the position of the electron (the smaller D q is), the less accurately we know the momentum (the larger D p is). Remember that momentum is (mass) x (velocity) so the uncertainty in momentum is equivalent to an uncertainty in velocity. The uncertainty in velocity is in the same direction as the uncertainty in position. In the drawing above, the uncertainty in position is a vertical uncertainty. This means that the uncertainty in velocity is also a vertical uncertainty. This is represented in the above drawing by the lines diverging (by an uncertain amount) after the electrons emerge from the hole (uncertain vertical position) rather than remaining parallel as they are on the left.

Likewise, an experiment designed to measure momentum with a certain accuracy will not be able to locate the position of the particle with better accuracy than the uncertainty relationship allows. Notice that in the uncertainty relationship, if the right side equals zero, then both D p and D q can be zero. This is the assumption of classical physics, which says that if the particles follow parallel trajectories on the left, they will not be disturbed by the hole, and they will follow parallel trajectories on the right.

If we divide both sides of the uncertainty relation by the mass m of the particle, we obtain

                                                                (D v)(D q) h/m

Here we see that the uncertainties in velocity v or position q are inversely proportional to the mass of the particle. Hence, one way to make the right side effectively zero is to make the mass very large. When numbers are put into this relationship, it turns out that the uncertainties are significant only when the mass is microscopic, and for a macroscopic mass the uncertainty is unmeasurably small. Thus, classical physics, which always dealt with macroscopic objects, was close to being correct in assuming that the position and velocity of all objects could be determined arbitrarily accurately.

The uncertainty relation is closely related to the complementarity principle, which was first enunciated by Bohr. The complementarity principle states that quantum objects have both a particle and a wave nature, and the attempt to measure a particle property precisely will tend to leave the wave property undefined, while the attempt to measure a wave property precisely will tend to leave the particle property undefined. In other words, particle properties and wave properties are complementary properties. Examples of particle properties are momentum and position. Examples of wave properties are wavelength and frequency. It can be shown that, if the wavelength of a wave is well-defined, the position of the wave is not, and vice versa. But the wavelength is related to the momentum by the de Broglie relation, so a well-defined wavelength implies a well-defined momentum. Thus, wave-particle complementarity is equivalent to momentum-position complementarity.