Quantum mechanics is the physicist's theory of the very small. It describes how forces and fundamental particles act for objects that are the sizes of molecules and smaller. In quantum mechanics matter (such as protons, neutrons, and electrons) and light (photons) can be thought of as both particles and waves. Mathematically, matter and light are represented by waveforms. When particles interact, or equivalently, when someone tries to measure the particle, the waveform is reduced to a state. When this idea is formulated as equations, it has several consequences.
First, because matter and light behave like both particles and waves, only one of their properties can be measured at a time. For instance, if one measures the position of a particle at a given instant, one cannot know its velocity to any accuracy. This is referred to as the uncertainty principle. The uncertainty principle sets the smallest meaningful lengths and energies that can be measured &mdash the Plank scales.
Second, the properties of interacting particles &mdash their energy, positions, and angular momentum &mdash are predicted to have discrete values that are integer multiples of some minimum value. The energies are therefore referred to as being quantized.
Finally, quantum mechanics predicts that identical particles are indistinguishable. This means that if two particles are exchanged, their waveform has to stay the same. This turns out to divide particles into two types: bosons, in which the waveform of a pair of particles stays the same when they are exchanged, and fermions, in which the waveform of a pair of particles changes sign when they are exchanged.
Bosons and fermions behave very differently. Two fermions cannot occupy the same state (a combination of energy and spin), and so obey the Pauli exclusion principle. The structure of electrons in atoms is determined in part by the exclusion principle for fermions. In contrast multiple Bosons can have the same state. As a result, when a collection of bosons is cool, they can behave as if they were one particle. This produces superfluidity and superconductivity.
Quantum mechanics was formulated in the early 20th century to explain a number of puzzling observations. First, physicists were puzzled by the spectrum of radiation (light) emerging from a box held at a fix temperature (blackbody radiation). Starting at long wavelengths (low energies) the spectrum of the light increased in intensity as one looked at shorter and shorter wavelengths. Eventually, the spectrum turned over. In 1901, Max Planck derived a formula that explained the entire spectrum, in which the wavelength of light was quantized, forming modes within the box. The temperature of the box determined the number of modes that were filled, which in turn explained the spectrum. The quanta became known as photons. However, the origin of the quantization was unexplained.
Second, it was noted that light had the properties of both a particle and a wave. The wave theory of light had been developed in the 17th century to explain optics and diffraction. However, experiments in the 19th century showed that when bombarding a metal object with light, electrons would be produced, but only if the light had a frequency above some limit. In 1905, Einstein explained this effect by following Planck, and suggesting that the minimum frequency needed to eject an electron was determined by the energy carried by the photon of light. Einstein's hypothesis was confirmed by Arthur Compton (1922) in experiments scattering X-rays off of electrons. In 1924, Satyendra Bose brought photons full-circle, by applying a statistical description of them to explain the spectrum of light.
Third, by the beginning of the 20th century, physicists were very confused about atoms. They knew they were made of protons, neutrons, and electrons, but electromagnetic theory predicted that the electron should continually radiate energy (light) as it orbited the proton, which would cause it to eventually spiral into the nucleus of the atom. This prediction was clearly wrong. Instead, experiments showed that light radiated by a hydrogen atom came out in discrete lines. Johannes Rydberg developed a formula in which the wavelengths at which the light emerged were related by integer fractions. Niels Bohr suggested that electrons had a discrete set of fixed orbits, which Louis de Broglie later explained could be thought of standing as waves with an integer number of wavelengths in their orbit. When the electrons changed orbits, the energy they would lose could by predicted by the fractions in Rydberg's formula.
These observations were eventually explained by mathematical theories that were developed by Werner Heisenberg, Erwin Schr\"{o}dinger, and Max Born, and later unified. The theory came to be known as quantum mechanics.
Quantum mechanics is at the same time one of the strangest and the most successful physical theories ever developed. Two discoveries about electrons were predicted by early quantum mechanical theories. In 1922, Stern and Gerlach showed that electrons had a quantum mechanical parameter referred to as spin, which was related to angular momentum. It is this spin that gives electrons their status as Fermions, and forces them to obey the Pauli exclusion principle. Moreover, Clinton Davisson and Lester Germer in 1927 showed that showed that electrons behaved like waves in some situations. This confirmed de Broglie's interpretation of Bohr's model for the atom.
Special relativity and electromagnetic fields were incorporated into quantum mechanics, starting with Paul Dirac, which allowed the prediction of the hyperfine structure of atoms, and even the existence of the positron (anti-matter electron).
Since then, experiments have revealed a wide range of quantum effects. Quantum tunneling, in which a particle has a finite probability of finding itself on the other side of a seemingly-impenetrable barrier, has been observed in many situations. The predicted interactions between atomic nuclei and magnetic fields were verified in experiments on nuclear magnetic resonance. Free electrons in a magnetic field are also observed to have quantized energy levels (Landau levels).
Recently, physicists have manipulated matter to explore a range of quantum phenomena. They have produced Bose-Einstein condensates &mdash collections of atoms that behave as if they have a single wave function &mdash in the lab. They have also created quantum dots, which are clusters of a few atoms that have unique electrical properties.
Quantum mechanical principles underly the laser, semiconductors (most notably, the transistor, nuclear magnetic resonance devices used in medical imaging, precise atomic clocks, X-ray and neutron scattering beams that are used to study the structures of crystals, scanning electron microscopes, and SQUID detectors that are designed to measure weak magnetic fields.
Looking to the future, quantum entangled states may eventually be used as the bits in quantum computers, quantum dots might be used to make properties with unique photoelectric properties (LEDs or photovoltaics), and quantum mechanical principles might be applied to make nanomachines.
In physics, quantum mechanics can be used in statistical calculations in order to derive the thermodynamical properties of matter, and in detailed calculations of the structure of atoms. Quantum mechanics is applied in chemistry and biology to compute the structures of molecules. It is applied in astronomy to compute the rate of nuclear reactions in the cores of stars, and the structures of white dwarfs and neutron stars, the way in which radiation (light) interacts with matter in the space between stars, and the wavelengths of light expected from atoms and molecules.
Quantum mechanics poses some serious philosophical difficulties, because in its common interpretation (the Copenhagen school), it implies a couple strange things. First, it implies that the atomic world is governed by chance, which Einstein famously objected to by saying, "God does not play dice." Second, if two particles have entangled waveforms, measuring the state of one will allow one to determine the state of the other, even if they are separated by a large distance. There is debate as to whether this represents a signal propagating faster than the speed of light, which would violate causality. As a result, other interpretations of quantum mechanics been developed, including the suggestion that there are hidden variables that we have not yet identified (the Bohm interpretation, which was largely rejected after the work of John Bell in the 1960s), and the notion that the universe we live in is one of many concurrently-existing universes (the latter idea inspired the movie Donnie Darko, and Neal Stephenson's recent book Anathema).
Quantum mechanics also has yet to be incorporated with our current understanding of gravity, General Relativity. In the mathematical framework of General Relativity, space-time is inherently smooth. In quantum mechanics, there is a smallest meaningful length (and time) scale, named after Max Planck. Quantum mechanics can incorporate the presence of a gravitational potential in an experiment, but it cannot yet describe how particles might interact gravitationally.
Like science itself, these pages are under construction. OK, so they are in a lot worse shape than science. I welcome your comments.