Wave Function
In quantum mechanics the wave function is a term used to describe a quantum mechanical system. The function specifies conditions of space and time that satisfy Schrödinger's equation. In an alternative use, a wave function (i.e., a Dirac function) describes functions of space and time (using space and time coordinates) that can be used to describe particle spin properties.
In 1921, German American physicist Albert Einstein was awarded the Nobel Prize for physics. What is surprising, perhaps, is that Einstein was honored not for relativity (still considered in 1921 to be too controversial a subject) but rather for his explanation of the photoelectric effect.
Einstein had introduced the concept of light-quanta, or photons, and thus had ascribed particle-like behavior to light, which had been understood in terms of wave mechanics since English physicist Thomas Young's famous double-slit experiment in 1801.
A young French aristocrat had become captivated by Einstein's idea, and in his 1924 doctoral thesis in physics, Louis de Broglie took it to its logical conclusion: if waves behaved as particles, shouldn't particles behave as waves? Shouldn't nature admit only one type of entity, having both wave-like and particle-like properties? "After long reflection in solitude and meditation," de Broglie later explained, "I suddenly had the idea, during the year 1923, that the discovery made by Einstein in 1905 should be generalized by extending it to all material particles, and notably to electrons."
De Broglie associated a fictitious wave to electrons, whose wavelength was determined as a function of momentum. Combining Einstein's relativistic energy-momentum relation (E2 -p2 = m2 c4 , and his energy-frequency relation for photons (E = hv) de Broglie obtained the expression for what is now known as the de Broglie wavelength.
The new wave model of the electron gave a physical justification for Danish physicist Niels Bohrüs model of the hydrogen atom. The allowed atomic energy levels were those for which a standing wave pattern could be constructed: an integral number of wavelengths must fit into an allowed orbit's circumference. But what did the associated wave represent, and how did it react when acted upon by a force? In 1926, the latter question was answered by the Austrian physicist Erwin Schrödinger.
Schrödinger introduced a mathematical function, the wave function (x,t), which described the electron wave, and which obeyed a wave equation, today known as the Schrödinger equation. Consider a familiar wave phenomenon, ripples on the surface of a pond. The mathematical function representing the water level at a point on the surface obeys a wave equation; knowing the forces acting on the water allows the wave equation to be solved, and the water level at any point to be determined. In a like manner, knowing the forces acting on an electron allows the Schrödinger equation to be solved, and the associated wave at any point to be determined. The water level, however, is a measurable quantity, and its wave function, the height at any point, is real-valued; it is a real number. Schrödinger realized that the electron's wavefunction must be complex-valued; at any point it is a complex number, a number containing the imaginary. The wave function cannot represent something directly measurable.
The absolute square of a complex number is a real number. Thus, the absolute square of the electron's wave function can represent something physical and measurable. At first Schrödinger thought it might be the electron's density (no longer could electrons be considered point particles). Although it turned out that the absolute square of the electron's wave function could be non zero in disjoint regions of space, no one had ever seen an electron in pieces. Accordingly, density was excluded. The problem was solved by German physicist Max Born, also in 1926, who later explained: "Again an idea of Einstein's gave me the lead. He had tried to make the duality of particles (light quanta or photons) and waves comprehensible by interpreting the square of the [light-wave] amplitudes as probability density for the occurrence of photons. This concept could at once be carried over to the wave function: [In other words, the square of the electron's wave function] ought to represent the probability density for electrons (or other particles)."
Born's probability interpretation is today the standard. The absolute square of the wave function represents the probability that the particle will be found at that point. Quantum mechanics does not predict precisely where a particle will be found, as does Newtonian mechanics, but rather specifies a probability region wherein a particle is likely to be found.
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