Table of contents |

2 Einstein and photons 3 De Broglie |

In the early 1800s, diffraction experiments by Young and Fresnel provided evidence for Huygens' theories: these experiments showed that when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves; the wavelength of light can be computed from such patterns. Maxwell, during the late-1800s, explained light as the propagation of electromagnetic waves with the Maxwell equations. These equations were verified by experiment, and Huygens' view became widely accepted.

In 1905, Einstein reconciled Huygens' view with that of Newton; he explained the photoelectric effect (an effect in which light did not seem to act as a wave) by postulating the existence of photons, quanta of energy with particulate qualities. Einstein postulated that light's frequency, *ν*, is related to the energy, *E*, of its photons:

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In 1924, De Broglie claimed that *all* matter has a wave-like nature; he related wavelength, λ, and momentum, *p*:

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De Broglie's formula was confirmed three years later by guiding a beam of electrons (which have mass) through a crystalline grid and observing the predicted interference patterns. Similar experiments have since been conducted with protons and even with whole molecules, and the formula has been confirmed in every case.

The Planck constant *h* is extremely small and that explains why we don't perceive a wave-like quality of everyday objects:
their wavelengths are exceedingly small. The fact that matter can have very short wavelengths is exploited in electron microscopy.

In quantum mechanics, the wave-particle duality is explained as follows: every system and particle is described by wave functions which encode the probability distributions of all measurable variables. The position of the particle is one such variable. Before an observation is made the position of the particle is described in terms of probability waves which can interfere with each other.

An intriguingly simple experiment, the double-slit experiment, summarizes the duality: Shoot electrons (or anything else for that matter) at a screen with two slits and record their position of impact at a detector behind the screen. You will observe an interference pattern just like the one produced by diffraction of a light or water wave at two slits. This pattern will even appear if you slow down the electron source so that only one electron per second comes through. "Classically speaking", every electron either travels through the first or through the second slit. So we should be able to produce the same interference pattern if we ran the experiment twice as long, closing slit number one for the first half, then closing slit number two for the second half. But no: the pattern won't emerge. Furthermore, if we build little detectors around the slits in order to determine which path a particular electron takes, then this very measurement will destroy the interference pattern as well.

The pattern is a result of the electron's wavefunction being diffracted by *both* slits and interfering with itself. The wavefunction is a complex valued function of space and time. The square of the magnitude of this function describes the probability of finding the electron at a given location at a given time. Interference is due to the fact that the square of the magnitude of the sum of two complex number may be different from the sum of the squares of their magnitudes.

The experiment also illustrates an interesting feature of quantum mechanics. Until an observation is made the position of a particle is described in terms of probability waves, but after the particle is observed, it is described as a fixed value. How to conceptualize the process of measurement is one of the great unresolved questions of quantum mechanics. The standard interpretation is the Copenhagen interpretation which leads to interesting thought experiments such as Schrödinger's cat. Another interpretation is the many-worlds interpretation.