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Behaviors mimicked via this hydrodynamic pilot-wave system include quantum single particle diffraction,  tunneling, quantized orbits, orbital level splitting, spin, and multimodal statistics.
It is also possible to infer uncertainty relations and exclusion principles. Videos are available illustrating various features of this system.
See the External links. However, more complicated systems that involve two or more particles in superposition are not amenable to such a simple, classically intuitive explanation.
Much of the behaviour of light can be modelled using classical wave theory. The Huygens—Fresnel principle is one such model; it states that each point on a wavefront generates a secondary wavelet, and that the disturbance at any subsequent point can be found by summing the contributions of the individual wavelets at that point.
This summation needs to take into account the phase as well as the amplitude of the individual wavelets. Only the intensity of a light field can be measured—this is proportional to the square of the amplitude.
In the double-slit experiment, the two slits are illuminated by a single laser beam. If the width of the slits is small enough less than the wavelength of the laser light , the slits diffract the light into cylindrical waves.
These two cylindrical wavefronts are superimposed, and the amplitude, and therefore the intensity, at any point in the combined wavefronts depends on both the magnitude and the phase of the two wavefronts.
The difference in phase between the two waves is determined by the difference in the distance travelled by the two waves.
If the viewing distance is large compared with the separation of the slits the far field , the phase difference can be found using the geometry shown in the figure below right.
Where d is the distance between the two slits. When the two waves are in phase, i. This effect is known as interference.
The interference fringe maxima occur at angles. The spacing of the fringes at a distance z from the slits is given by. For example, if two slits are separated by 0.
If the width of the slits b is greater than the wavelength, the Fraunhofer diffraction equation gives the intensity of the diffracted light as: .
This is illustrated in the figure above, where the first pattern is the diffraction pattern of a single slit, given by the sinc function in this equation, and the second figure shows the combined intensity of the light diffracted from the two slits, where the cos function represent the fine structure, and the coarser structure represents diffraction by the individual slits as described by the sinc function.
Similar calculations for the near field can be done using the Fresnel diffraction equation. As the plane of observation gets closer to the plane in which the slits are located, the diffraction patterns associated with each slit decrease in size, so that the area in which interference occurs is reduced, and may vanish altogether when there is no overlap in the two diffracted patterns.
Like the Schrödinger's cat thought experiment , the double-slit experiment is often used to highlight the differences and similarities between the various interpretations of quantum mechanics.
The Copenhagen interpretation , put forth by some of the pioneers in the field of quantum mechanics, asserts that it is undesirable to posit anything that goes beyond the mathematical formulae and the kinds of physical apparatus and reactions that enable us to gain some knowledge of what goes on at the atomic scale.
One of the mathematical constructs that enables experimenters to predict very accurately certain experimental results is sometimes called a probability wave.
In its mathematical form it is analogous to the description of a physical wave, but its "crests" and "troughs" indicate levels of probability for the occurrence of certain phenomena e.
The probability "wave" can be said to "pass through space" because the probability values that one can compute from its mathematical representation are dependent on time.
One cannot speak of the location of any particle such as a photon between the time it is emitted and the time it is detected simply because in order to say that something is located somewhere at a certain time one has to detect it.
The requirement for the eventual appearance of an interference pattern is that particles be emitted, and that there be a screen with at least two distinct paths for the particle to take from the emitter to the detection screen.
Experiments observe nothing whatsoever between the time of emission of the particle and its arrival at the detection screen.
If a ray tracing is next made as if a light wave as understood in classical physics is wide enough to take both paths, then that ray tracing will accurately predict the appearance of maxima and minima on the detector screen when many particles pass through the apparatus and gradually "paint" the expected interference pattern.
The Copenhagen interpretation is similar to the path integral formulation of quantum mechanics provided by Feynman. The path integral formulation replaces the classical notion of a single, unique trajectory for a system, with a sum over all possible trajectories.
The trajectories are added together by using functional integration. Each path is considered equally likely, and thus contributes the same amount.
However, the phase of this contribution at any given point along the path is determined by the action along the path:.
All these contributions are then added together, and the magnitude of the final result is squared , to get the probability distribution for the position of a particle:.
As is always the case when calculating probability , the results must then be normalized by imposing:. To summarize, the probability distribution of the outcome is the normalized square of the norm of the superposition , over all paths from the point of origin to the final point, of waves propagating proportionally to the action along each path.
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