The wave–particle duality relation, often loosely referred to as the Englert–Greenberger–Yasin duality relation, or the Englert–Greenberger relation, relates the visibility, , of interference fringes with the definiteness, or distinguishability, , of the photons' paths in quantum optics. As an inequality:
Although it is treated as a single relation, it actually involves two separate relations, which mathematically look very similar. The first relation, derived by Greenberger and Yasin in 1988, is expressed as . It was later extended to, providing an equality for the case of pure quantum states by Jaeger, Shimony, and Vaidman in 1995. This relation involves correctly guessing which of the two paths the particle would have taken, based on the initial preparation. Here can be called the predictability. A year later Englert, in 1996, derived a related relation dealing with experimentally acquiring knowledge of the two paths using an apparatus, as opposed to predicting the path based on initial preparation. This relation is . Here is called the distinguishability.
The significance of the relations is that they express quantitatively the complementarity of wave and particle viewpoints in double-slit experiments. The complementarity principle in quantum mechanics, formulated by Niels Bohr, says that the wave and particle aspects of quantum objects cannot be observed at the same time. The wave–particle duality relations makes Bohr's statement more quantitative – an experiment can yield partial information about the wave and particle aspects of a photon simultaneously, but the more information a particular experiment gives about one, the less it will give about the other. The predictability which expresses the degree of probability with which path of the particle can be correctly guessed, and the distinguishability which is the degree to which one can experimentally acquire information about the path of the particle, are measures of the particle information, while the visibility of the fringes is a measure of the wave information. The relations shows that they are inversely related, as one goes up, the other goes down. Fringes are visible over a wide range of distinguishability.
The mathematics of two-slit diffraction
This section reviews the mathematical formulation of the double-slit experiment. The formulation is in terms of the diffraction and interference of waves. The culmination of the development is a presentation of two numbers that characterizes the visibility of the interference fringes in the experiment, linked together as the Englert–Greenberger duality relation. The next section will discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave–particle duality. Of this experiment, Richard Feynman once said that it "has in it the heart of quantum mechanics. In reality it contains the only mystery."
The wave function in the Young double-aperture experiment can be written as
The function
is the wave function associated with the pinhole at A centered on ; a similar relation holds for pinhole B. The variable is a position in space downstream of the slits. The constants and are proportionality factors for the corresponding wave amplitudes, and is the single hole wave function for an aperture centered on the origin. The single-hole wave-function is taken to be that of Fraunhofer diffraction; the pinhole shape is irrelevant, and the pinholes are considered to be idealized. The wave is taken to have a fixed incident momentum :
where is the radial distance from the pinhole.
To distinguish which pinhole a photon passed through, one needs some measure of the distinguishability between pinholes. Such a measure is given by
where and are the probabilities of finding that the particle passed through aperture A and aperture B respectively.
Since the Born probability measure is given by
and
then we get:
We have in particular for two symmetric holes and for a single aperture (perfect distinguishability). In the far-field of the two pinholes the two waves interfere and produce fringes. The intensity of the interference pattern at a point y in the focal plane is given by
where is the momentum of the particle along the y direction, is a fixed phase shift, and is the separation between the two pinholes. The angle α from the horizontal is given by where is the distance between the aperture screen and the far field analysis plane. If a lens is used to observe the fringes in the rear focal plane, the angle is given by where is the focal length of the lens.