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.
A more-complete derivation of black-body radiation would yield a fully continuous and “wave-like” electromagnetic field with no quantization.
Resonant interaction between the droplet and its own wave field exhibits behaviour analogous to quantum particles: interference in double-slit experiment, unpredictable
tunneling (depending in complicated way on practically hidden state of field), orbit quantization (that particle has to ‘find a resonance’ with field perturbations it creates—after one orbit, its internal phase has to return to the
initial state) and Zeeman effect.
However, in 1905 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.
Most physicists accept wave–particle duality as the best explanation for a broad range of observed phenomena; however, it is not without controversy.
Planck had intentionally created an atomic theory of the black body, but had unintentionally generated an atomic theory of light, where the black body never generates quanta
of light at a given frequency with an energy less than hf.
 If one used Planck’s energy quanta, and demanded that electromagnetic radiation at a given frequency could only transfer energy to matter in integer multiples of an energy
quantum hf, then the photoelectric effect could be explained very simply.
Albert Einstein, who, in his search for a Unified Field Theory, did not accept wave–particle duality, wrote: This double nature of radiation (and of material corpuscles)
… has been interpreted by quantum-mechanics in an ingenious and amazingly successful fashion.
In these experiments the build-up of such interference patterns could be recorded in real time and with single molecule sensitivity.
Black-body radiation, the emission of electromagnetic energy due to an object’s heat, could not be explained from classical arguments alone.
Einstein’s “light quanta” would not be called photons until 1925, but even in 1905 they represented the quintessential example of wave–particle duality.
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.
He saw it in what is called second quantization, which generates an entirely new concept of fields that exist in ordinary space-time, causality still being visualizable.
Beginning in 1670 and progressing over three decades, Isaac Newton developed and championed his corpuscular theory, arguing that the perfectly straight lines of reflection
demonstrated light’s particle nature, only particles could travel in such straight lines.
However, it can still be explained using a fully classical description of light, as long as matter is quantum mechanical in nature.
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.
Mead cites as the gross evidence of the exclusively wave nature of both light and matter the discovery between 1933 and 1996 of ten examples of pure wave phenomena, including
the ubiquitous laser of CD players, the self-propagating electrical currents of superconductors, and the Bose–Einstein condensate of atoms.
Since light was known to be waves of electromagnetism, physicists hoped to describe this emission via classical laws.
Hegerfeldt’s theorem, which appears to demonstrate the incompatibility of the existence of localized discrete particles with the combination of the principles of quantum mechanics
and special relativity, has also been used to support the conclusion that reality must be described solely in terms of field-based formulations.
Within the limits of the wave–particle duality the quantum field theory gives the same results.
 Although the use of the wave–particle duality has worked well in physics, the meaning or interpretation has not been satisfactorily resolved; see interpretations of quantum
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.
Upon measuring the location of the particle, the particle will be forced into a more localized state as given by the uncertainty principle.
The pilot wave theory was initially rejected because it generated non-local effects when applied to systems involving more than one particle.
 The wave view did not immediately displace the ray and particle view, but began to dominate scientific thinking about light in the mid 19th century, since it could explain
polarization phenomena that the alternatives could not.
 Authors of similar recent experiments with atoms and molecules, described below, claim that these larger particles also act like waves.
 Particle-only view Still in the days of the old quantum theory, a pre-quantum-mechanical version of wave–particle duality was pioneered by William Duane, and
developed by others including Alfred Landé.
The more localized the position-space wavefunction, the more likely the particle is to be found with the position coordinates in that region, and correspondingly the momentum-space
wavefunction is less localized so the possible momentum components the particle could have are more widespread.
Moreover, when position is relatively well-defined, the wave is pulse-like and has a very ill-defined wavelength, and thus momentum.
 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.
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.
The probability (shown as the colour opacity) of finding the particle at a given point x is spread out like a waveform; there is no definite position of the particle.
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.
In this view, each particle has a well-defined position and momentum, but is guided by a wave function derived from Schrödinger’s equation.
In the resulting representation, also called the de Broglie–Bohm theory or Bohmian mechanics, 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.
Conversely, the more localized the momentum-space wavefunction, the more likely the particle is to be found with those values of momentum components in that region, and correspondingly
the less localized the position-space wavefunction, so the position coordinates the particle could occupy are more widespread.
And conversely, when momentum, and thus wavelength, is relatively well-defined, the wave looks long and (complex-)sinusoidal, and therefore it has a very ill-defined position.
Around the same time, Newton’s contemporaries Robert Hooke and Christiaan Huygens, and later Augustin-Jean Fresnel, mathematically refined the wave viewpoint, showing that
if light traveled at different speeds in different media, refraction could be easily explained as the medium-dependent propagation of light waves.
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.
Whereas in order to get high energy electrons, one must illuminate the metal with high-frequency light.
In 1630, René Descartes popularized the opposing wave description in his treatise on light, The World, showing that the behaviour of light could be re-created by modeling
wave-like disturbances in a universal medium i.e.
 Bohr regarded renunciation of the cause-effect relation, or complementarity, of the space-time picture, as essential to the quantum mechanical account.
Below is an illustration of wave–particle duality as it relates to de Broglie’s hypothesis and Heisenberg’s Uncertainty principle, in terms of the position and momentum space
wavefunctions for one spinless particle with mass in one dimension.
 The Afshar experiment (2007) may suggest that it is possible to simultaneously observe both wave and particle properties of photons.
the electric and magnetic field strengths of Maxwell) are replaced by an entirely new kind of field value, as considered in quantum field theory.
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.
 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.
The resulting Huygens–Fresnel principle was extremely successful at reproducing light’s behaviour and was consistent with Thomas Young’s discovery of wave interference of
light by his double-slit experiment in 1801.
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 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
But a problem quickly arose if each mode received an equal partition of energy, the short wavelength modes would consume all the energy.
 Other single and double slit experiments  have shown that wall-droplet interactions rather than diffraction or interference of the pilot wave may be responsible
for the observed hydrodynamic patterns, which are different from slit-induced interference patterns exhibited by quantum particles.
The field permits solutions that follow the wave equation, which are referred to as the wave functions.
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.
He saw the duality as present for all quantic entities, but not quite in the usual quantum mechanical account considered by Bohr.
A quantum object will sometimes exhibit wave, sometimes particle character in different physical settings.
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).
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.
 Experimental confirmation of wave–particle duality Single-particle interferometry has become a classic for its clarity in expressing the central puzzles of quantum mechanics.
Because it demonstrates the fundamental limitation of the ability of the observer to predict experimental results, Richard Feynman called it “a phenomenon which is impossible
[…] to explain in any classical way, and which has in it the heart 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.
In the photoelectric effect, it was observed that shining a light on certain metals would lead to an electric current in a circuit.
An interaction as in a Feynman diagram is accepted as a calculationally convenient approximation where the outgoing legs are known to be simplifications of the propagation
and the internal lines are for some order in an expansion of the field interaction.
Since the field is non-local and quantized, the phenomena that previously were thought of as paradoxes are explained.
Following the development of quantum field theory the ambiguity disappeared.
 He related wavelength and momentum: This is a generalization of Einstein’s equation above, since the momentum of a photon is given by and the wavelength (in a vacuum)
by , where c is the speed of light in vacuum.
 The particle-like behaviour is most evident due to phenomena associated with measurement in quantum mechanics.
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