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The resulting Huygens—Fresnel principle was extremely successful at reproducing light's behavior and was subsequently supported by Thomas Young 's discovery of wave interference of light by his double-slit experiment in James Clerk Maxwell discovered that he could apply his previously discovered Maxwell's equations , along with a slight modification to describe self-propagating waves of oscillating electric and magnetic fields. It quickly became apparent that visible light, ultraviolet light, and infrared light were all electromagnetic waves of differing frequency.

In , Max Planck published an analysis that succeeded in reproducing the observed spectrum of light emitted by a glowing object. To accomplish this, Planck had to make a mathematical assumption of quantized energy of the oscillators i. Einstein later proposed that electromagnetic radiation itself is quantized, not the energy of radiating atoms. Black-body radiation , the emission of electromagnetic energy due to an object's heat, could not be explained from classical arguments alone. The equipartition theorem of classical mechanics, the basis of all classical thermodynamic theories, stated that an object's energy is partitioned equally among the object's vibrational modes.

But applying the same reasoning to the electromagnetic emission of such a thermal object was not so successful.

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That thermal objects emit light had been long known. Since light was known to be waves of electromagnetism, physicists hoped to describe this emission via classical laws.

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This became known as the black body problem. Since the equipartition theorem worked so well in describing the vibrational modes of the thermal object itself, it was natural to assume that it would perform equally well in describing the radiative emission of such objects. But a problem quickly arose if each mode received an equal partition of energy, the short wavelength modes would consume all the energy.

This became clear when plotting the Rayleigh—Jeans law , which, while correctly predicting the intensity of long wavelength emissions, predicted infinite total energy as the intensity diverges to infinity for short wavelengths. This became known as the ultraviolet catastrophe. This was not an unsound proposal considering that macroscopic oscillators operate similarly when studying five simple harmonic oscillators of equal amplitude but different frequency, the oscillator with the highest frequency possesses the highest energy though this relationship is not linear like Planck's.

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  • By demanding that high-frequency light must be emitted by an oscillator of equal frequency, and further requiring that this oscillator occupy higher energy than one of a lesser frequency, Planck avoided any catastrophe, giving an equal partition to high-frequency oscillators produced successively fewer oscillators and less emitted light. And as in the Maxwell—Boltzmann distribution , the low-frequency, low-energy oscillators were suppressed by the onslaught of thermal jiggling from higher energy oscillators, which necessarily increased their energy and frequency.

    The most revolutionary aspect of Planck's treatment of the black body is that it inherently relies on an integer number of oscillators in thermal equilibrium with the electromagnetic field.

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    These oscillators give their entire energy to the electromagnetic field, creating a quantum of light, as often as they are excited by the electromagnetic field, absorbing a quantum of light and beginning to oscillate at the corresponding frequency. However, once realizing that he had quantized the electromagnetic field, he denounced particles of light as a limitation of his approximation, not a property of reality.

    While Planck had solved the ultraviolet catastrophe by using atoms and a quantized electromagnetic field, most contemporary physicists agreed that Planck's "light quanta" represented only flaws in his model. A more-complete derivation of black body radiation would yield a fully continuous and "wave-like" electromagnetic field with no quantization. However, in Albert Einstein took Planck's black body model to produce his solution to another outstanding problem of the day: the photoelectric effect , wherein electrons are emitted from atoms when they absorb energy from light.

    Since their existence was theorized eight years previously, phenomena had been studied with the electron model in mind in physics laboratories worldwide. In Philipp Lenard discovered that the energy of these ejected electrons did not depend on the intensity of the incoming light, but instead on its frequency.

    So if one shines a little low-frequency light upon a metal, a few low energy electrons are ejected.

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    If one now shines a very intense beam of low-frequency light upon the same metal, a whole slew of electrons are ejected; however they possess the same low energy, there are merely more of them. The more light there is, the more electrons are ejected. Whereas in order to get high energy electrons, one must illuminate the metal with high-frequency light.

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    Like blackbody radiation, this was at odds with a theory invoking continuous transfer of energy between radiation and matter. However, it can still be explained using a fully classical description of light, as long as matter is quantum mechanical in nature. Low-frequency light only ejects low-energy electrons because each electron is excited by the absorption of a single photon. Increasing the intensity of the low-frequency light increasing the number of photons only increases the number of excited electrons, not their energy, because the energy of each photon remains low.

    Only by increasing the frequency of the light, and thus increasing the energy of the photons, can one eject electrons with higher energy.

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    Thus, using Planck's constant h to determine the energy of the photons based upon their frequency, the energy of ejected electrons should also increase linearly with frequency, the gradient of the line being Planck's constant. These results were not confirmed until , when Robert Andrews Millikan produced experimental results in perfect accord with Einstein's predictions. While energy of ejected electrons reflected Planck's constant, the existence of photons was not explicitly proven until the discovery of the photon antibunching effect, of which a modern experiment can be performed in undergraduate-level labs.

    Einstein's "light quanta" would not be called photons until , but even in they represented the quintessential example of wave-particle duality. Electromagnetic radiation propagates following linear wave equations, but can only be emitted or absorbed as discrete elements, thus acting as a wave and a particle simultaneously.

    In , Albert Einstein provided an explanation of the photoelectric effect , an experiment that the wave theory of light failed to explain. He did so by postulating the existence of photons , quanta of light energy with particulate qualities. In the photoelectric effect , it was observed that shining a light on certain metals would lead to an electric current in a circuit.

    Presumably, the light was knocking electrons out of the metal, causing current to flow. However, using the case of potassium as an example, it was also observed that while a dim blue light was enough to cause a current, even the strongest, brightest red light available with the technology of the time caused no current at all.

    According to the classical theory of light and matter, the strength or amplitude of a light wave was in proportion to its brightness: a bright light should have been easily strong enough to create a large current. Yet, oddly, this was not so. Einstein explained this enigma by postulating that the electrons can receive energy from electromagnetic field only in discrete units quanta or photons : an amount of energy E that was related to the frequency f of the light by. Only photons of a high enough frequency above a certain threshold value could knock an electron free.

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    For example, photons of blue light had sufficient energy to free an electron from the metal, but photons of red light did not. One photon of light above the threshold frequency could release only one electron; the higher the frequency of a photon, the higher the kinetic energy of the emitted electron, but no amount of light below the threshold frequency could release an electron.

    To violate this law would require extremely high-intensity lasers that had not yet been invented. Intensity-dependent phenomena have now been studied in detail with such lasers. Einstein was awarded the Nobel Prize in Physics in for his discovery of the law of the photoelectric effect.

    In , Louis-Victor de Broglie formulated the de Broglie hypothesis , claiming that all matter [15] [16] has a wave-like nature, he related wavelength and momentum :. De Broglie's formula was confirmed three years later for electrons with the observation of electron diffraction in two independent experiments. At the University of Aberdeen , George Paget Thomson passed a beam of electrons through a thin metal film and observed the predicted interference patterns.

    De Broglie was awarded the Nobel Prize for Physics in for his hypothesis. Thomson and Davisson shared the Nobel Prize for Physics in for their experimental work. In his work on formulating quantum mechanics, Werner Heisenberg postulated his uncertainty principle , which states:. Heisenberg originally explained this as a consequence of the process of measuring: Measuring position accurately would disturb momentum and vice versa, offering an example the "gamma-ray microscope" that depended crucially on the de Broglie hypothesis.

    The thought is now, however, that this only partly explains the phenomenon, but that the uncertainty also exists in the particle itself, even before the measurement is made. In fact, the modern explanation of the uncertainty principle, extending the Copenhagen interpretation first put forward by Bohr and Heisenberg , depends even more centrally on the wave nature of a particle. Just as it is nonsensical to discuss the precise location of a wave on a string, particles do not have perfectly precise positions; likewise, just as it is nonsensical to discuss the wavelength of a "pulse" wave traveling down a string, particles do not have perfectly precise momenta that corresponds to the inverse of wavelength.

    Moreover, when position is relatively well defined, the wave is pulse-like and has a very ill-defined wavelength, and thus momentum. And conversely, when momentum, and thus wavelength, is relatively well defined, the wave looks long and sinusoidal, and therefore it has a very ill-defined position. De Broglie himself had proposed a pilot wave construct to explain the observed wave-particle duality. The pilot wave theory was initially rejected because it generated non-local effects when applied to systems involving more than one particle.

    Non-locality, however, soon became established as an integral feature of quantum theory and David Bohm extended de Broglie's model to explicitly include it. In the resulting representation, also called the de Broglie—Bohm theory or Bohmian mechanics, [18] the wave-particle duality vanishes, and explains the wave behaviour as a scattering with wave appearance, because the particle's motion is subject to a guiding equation or quantum potential.

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    This idea seems to me so natural and simple, to resolve the wave—particle dilemma in such a clear and ordinary way, that it is a great mystery to me that it was so generally ignored. The best illustration of the pilot-wave model was given by Couder's "walking droplets" experiments, [20] demonstrating the pilot-wave behaviour in a macroscopic mechanical analog.

    Since the demonstrations of wave-like properties in photons and electrons , similar experiments have been conducted with neutrons and protons. Among the most famous experiments are those of Estermann and Otto Stern in A dramatic series of experiments emphasizing the action of gravity in relation to wave—particle duality was conducted in the s using the neutron interferometer.

    In the neutron interferometer, they act as quantum-mechanical waves directly subject to the force of gravity. While the results were not surprising since gravity was known to act on everything, including light see tests of general relativity and the Pound—Rebka falling photon experiment , the self-interference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before. In , the diffraction of C 60 fullerenes by researchers from the University of Vienna was reported.

    The de Broglie wavelength of the incident beam was about 2. In , these far-field diffraction experiments could be extended to phthalocyanine molecules and their heavier derivatives, which are composed of 58 and atoms respectively. In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity. For this demonstration they employed a near-field Talbot Lau interferometer. Whether objects heavier than the Planck mass about the weight of a large bacterium have a de Broglie wavelength is theoretically unclear and experimentally unreachable; above the Planck mass a particle's Compton wavelength would be smaller than the Planck length and its own Schwarzschild radius , a scale at which current theories of physics may break down or need to be replaced by more general ones.

    Recently Couder, Fort, et al. Wave—particle duality is deeply embedded into the foundations of quantum mechanics. In the formalism of the theory, all the information about a particle is encoded in its wave function , a complex-valued function roughly analogous to the amplitude of a wave at each point in space. For particles with mass this equation has solutions that follow the form of the wave equation. Propagation of such waves leads to wave-like phenomena such as interference and diffraction.

    The particle-like behaviour is most evident due to phenomena associated with measurement in quantum mechanics. Landau D. Lieb Elliott H. Montenbruck O. Mullan Dermott J.