A Brief History of Quantum Mechanics

The problems with light (1670 - 1900).

The nature of light has been contested throughout much of the history of modern science. Robert Hooke and Christiaan Huygens begun working on a wave theory of light in the 1670s, this was reliant on the idea that light waves must propagate through a medium and so the void between the Sun and the Earth must be filled with an æther. Around the same time Newton suggested a corpuscular theory of light and gave experimental results which he claimed proved this. This was controversial, Newton needed to appeal to wave like qualities to explain diffraction for example. One way to experimentally distinguish the two theories was to measure the speed of light in different mediums, with Newton's corpuscular theory suggesting that light would travel faster in a denser medium. It was not until 1850 that it was possible to do these experiments and Foucault's results proved Newton wrong in this respect.

Fifty years earlier than this, at the turn of the 19th century Thomas Young and Augustin-Jean Fresnel provided experimental evidence in favour of the wave theory of light. In the famous double-slit experiments it was shown that light produces interference patterns which are expected from waves and Young went on to explain Newton's results in terms of his wave theory.

Towards the end of the 1800s Maxwell proposed that light could be understood as the propagation of electromagnetic waves; at any point of a beam of light there is an electric and a magnetic force moving perpendicularly to each other in the direction of the beams propagation. These force fields oscillate periodically and are therefore detected as waves. Maxwell showed this by verifying a set of four equations which describe the interrelationship between electric field, magnetic field, electric charge, and electric current.

Quantum Theory (1900 - 1926).

In 1900 Max Plank first introduced the concept of quanta. Plank was interested in explaining the relationship between the amount of radiation a blackbody emits and its temperature. He found that the experimental data only made sense if it was assumed that energy radiated in discrete 'quanta', or photons, each containing a quanta of energy proportional to the frequency with which they radiate. The proportionality constant has come to be known as Plank's constant.

Einstein applied the idea of Plank's constant to the problem of the photoelectric effect in 1905. The photoelectric effect shows that electrons can be released from certain metals by interacting with light. This could not be explained by Maxwell's theory since the rate that the electrons were released depended not on the intensity of light, but the frequency. Einstein showed that this could be explained with a quantum theory of light whereby electrons are released only when particular frequencies are reached corresponding to multiples of Plank's constant.

In 1913 Bohr used the idea of quantised energy to explain how electrons orbit a nucleus. Bohr essentially showed that it was not classical but quantum laws that apply to electrons orbiting a nucleus. Just over a decade later de Broglie proposed that all matter has a wave-like nature. Bohr related the angular momentum of electrons to Plank's constant and deduced that electrons orbit with momenta, speeds and energies quantised to multiples proportional to Plank's constant. All other values are not accessible to the electron, including certain spatial regions, when travelling between orbits the electrons seem to disappear from one quantised distance to another simultaneously. Electrons do not loose energy as they orbit the nucleus and only do so when 'jumping' between orbits.

The idea of action at a distance in quantum mechanics could be seen as analogous to Newtonian action at a distance, but differs in two respects. Firstly, quantum action at a distance does not have the symmetry that gravitational force has because in quantum mechanics it is the first measurement which always determines the outcome of the other, they are not of mutual influence. Secondly in quantum mechanics the effects are irrespective of distance, whereas in the Newtonian model the force decreases proportionally to the square of the distance between objects.

In 1926, Schrödinger showed that quantum states can be represented by a complex function which evolves according to a second-order differential wave equation. Schrödinger's wave equation shows that a quantum state has a unitary evolution, with quanta existing in all physically possible states at once, this is known as a superposition. Around the same time as Schrödinger produced the wave equation Niels Bohr and his assistant at Copenhagen, Werner Heisenberg, used a matrix theory to interpret quantum mechanics, leading to the formulation of Heisenberg's uncertainty principle. Bohr and Heisenberg showed that for an object in a superpositional state, properties corresponding to more than one physical possibility cannot be measured simultaneously, and are said to be non-commuting.

Later that year Max Born proposed a statistical interpretation of Schrödinger's wave function, with the square of the wavefunction interpreted as a probability amplitude. This is known as the Born rule and was interpreted by Bohr and Heisenberg to give the probability for the outcome of a quantum experiment given that a measurement is made, it says nothing of the object when it is in a superpositional state.

The mathematical interpretation of quantum mechanics was completed when the matrix mechanics used in Heisenberg's theories and the wave mechanics used in the Schrödinger equation were made compatable. This problem was tackled most notably by John von Neumann and Paul Dirac. In the standard von Neumann-Dirac theory a quantum system is thought of as a point in Hilbert space. Hilbert space is analogous to the dimensional phase space of classical mechanics, but includes an infinite amount of dimensions representing the infinite amount of linear combinations of vectors corresponding to the possible states. Observables, measurable properties, are represented as linear Hermitian operators on Hilbert spaces. The uncertainty principle can be explained by the fact that the two operators are non-commutating. A different approach came in 1926 when Jordan, in Göttingen, and Dirac, in Cambridge, provided an independent unification of matrix and wave mechanics known as transformation theory.