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Maxwell's famous equations, the triumph of late 19th Century physics, unify the description of all electrical and magnetic phenomena in terms of the electric field E due to charge, and the magnetic field B due to current. Maxwell found that his equations also describe Electromagnetic Radiation. When a charge is oscillated at frequency ν it generates transverse kinks in the electric field, which propagate at speed c, while the corresponding oscillating current generates propagating transverse kinks in the magnetic field. The fact that this radiation propagates to infinity at the speed of light suggested that light itself is an electromagnetic wave. But there was a huge conceptual problem! The electric and magnetic fields were just invented by Michael Faraday, to allow him to visualize forces exerted by charges and currents! To describe light as propagating disturbances in something that is not physically real was clearly unsatisfactory. Maxwell and other physicists were driven to imagine that perhaps “empty” space is filled with a material of unimaginably low density, yet somehow incredibly stiff enough to sustain wave disturbances in it that propagated at the incredible speed of light! By 1900 it had become clear that such a material did not exist, since the speed of light did not depend on either the motion of the source or the observer through the material, the “luminiferous ether.” Importantly, Maxwell had shown that electromagnetic radiation transports not just energy but also momentum, with the momentum of a wave of total energy U being given by p = U/c. Of course the radiation also has a wavelength given by λ = c/ν.
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In 1905 Einstein published a famous paper which won him the Nobel Prize. He showed that the phenomenon in which light, shining on a metal surface, ejects electrons from the surface, cannot be understood in any way if light consists of electromagnetic waves! To describe all the details of the process correctly, light must be assumed to consist of particles, photons, each of which has a kinetic energy K = hν, with of course h being Planck's constant. Thus a charge oscillated at frequency ν emits a spray of individual pointlike particles, massless and chargeless photons, with that kinetic energy. Since p = K/c, each massless photon would carry a momentum of p = hν/c = h/λ. [Planck's constant h is a fundamental physical constant of nature, equal to about 4 × 10-15 eV-sec. More commonly we encounter ℏ, which is h divided by 2π.] |
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In 1924, Louis de Broglie wrote a Ph. D. thesis in which he suggested that all particles in nature somehow must have a wave associated with them... exactly how, and what the wave might be physically, he did not speculate upon in detail. The important point was that physicists should look for all quantum particles to display wave diffraction and interference under appropriate physical circumstances where the photons of light displayed those phenomena. The wavelength associated with a particle of momentum p should be λ = h/p, just as for photons! His speculation was quickly confirmed by a series of experiments. But what were these “matter waves”? What were they waves in? Maxwell's problem was generalized to a universal degree, but still unsolved! Importantly, de Broglie's idea motivated Erwin Schroedinger to search for a wave equation for the supposed matter waves. What he ultimately came up with really was not in any shape or form anything resembling a classical wave equation. Its solution was a field Ψ which assigned a complex number to every point in space-time. How could this field, with its imaginary part, be related to any physical phenomenon? |
 Classical wave equation. vp is the phase speed of the wave. |
 Schroedinger equation: Note the explicit presence of i, the square root of minus 1. |
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Quantum physics really began in 1926 when Max Born provided the interpretation which has been the basis of all applications of quantum ideas, ever since. This is what he proposed: (1) To every process and system in nature there corresponds an amplitude Ψ(r,t). (2) The absolute value squared of this amplitude is the probability of seeing the process happen at, or finding the system at, the specific point (r,t). In other words, the probability distribution is P(r,t) = |Ψ(r,t)|2. Now comes a key point. (3) Suppose in an experiment a process can occur in several different ways, and the experiment can distinguish between the various possibilities. Then the total probability is just the sum of the probabilities for each way the process can occur. However, if the experiment cannot distinguish between the various possibilities, first you sum the amplitudes and then square, to get the total probability. That is, P = |Ψ1 + Ψ2 + ...|2. In other words, the basic individual possibilities interfere with one another! |
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How about a single slit? Here a particle with a definite momentum (and thus a probability distribution extending over all space) is drawn like a classical plane wave. We know there is diffraction from a single slit... because we have no information about where the particle is, if it passed through the slit, we don't know where... in the image there are indicated six possible points... of course the actual number is infinite if the particle is pointlike... and those possibilities interfere to form the characteristic single slit pattern.
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Quantum physics suffers from the fact that physicists casually often still use the nomenclature established in the era 1900 - 1925, before anyone had any idea what was in fact going on. Thus, the probability amplitude Ψ(r,t) is correctly called the state function of a system or process, and incorrectly but often (alas) called “the wave function.” In general, it is not at all wave-like, it is a scalar field that assigns a complex number to every point in space-time. The measurable quantity is P(r,t) = |Ψ(r,t)|2. By the way, if you know what the complex conjugate is, the quick way to get P is by computing the complex conjugate of the state function, times the state function, P = Ψ* × Ψ.
