# De Broglie Waves

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## De Broglie Waves

Waves That Are Associated With Any Moving Microscopic Particle And That Reflect The Quantum Nature Of Particles.

Quantum properties were first observed in electromagnetic fields. After M. Planck’s research on the laws of the thermal radiation of bodies (1900), the concept of “light portions”—quanta of an electromagnetic field—began to be used in science. These quanta, photons, are in many respects similar to particles (corpuscles): they have a specific energy and momentum and interact with matter as a whole unit. The wave properties of electromagnetic radiation were already known—for example, they are apparent in the phenomena of light diffraction and interference. Thus, it is possible to speak of the dual nature of the photon—the corpuscular-wave dualism.

In 1924, L. de Broglie introduced the daring hypothesis that the corpuscular-wave dualism is intrinsic to all forms of matter (electrons, protons, atoms, and so on); also, quantitative relations between the wave and corpuscular properties of particles are the same as those established earlier for photons. Namely, if a particle has energy *δ* and momentum *p*, then a wave whose frequency is *v* = δ/*h* and whose wavelength is λ = *h/p* is associated with it; here *h* ≈ 6 x 10^{-27} erg · sec is Planck’s constant. These waves came to be called de Broglie waves.

For particles whose energy is not very high « = *h/mv*, where *m* is the particle’s mass and *v* is its velocity. Thus, the length of a de Broglie wave decreases as the particle’s mass and velocity increase. For example, a de Broglie wave with λ ≈ 10^{-18} angstrom (Å), which lies outside the observable region, will correspond to a particle with a mass of 1 g, moving with a velocity of 1 m/sec. It is therefore clear that wave properties are insignificant in the mechanics of macroscopic objects. For electrons with energies from 1 to 10,000 electron volts (eV; 1 eV = 1.6 x 10^{-19} joule), the length of the de Broglie wave lies in the range from 10 to 0.1 Å—that is, in the X-ray range. Consequently, the wave properties of electrons must appear, for example, in their scattering on the same crystals for which X-ray diffraction is observed.

The first experimental confirmation of de Broglie’s hypothesis was obtained in 1927 by C. Davisson and L. Germer. An electron beam was accelerated in an electrical field with a potential difference of 100-150 volts (the energies of such electrons are 100-150 eV, which corresponds to λ ≈ 1 Å) and fell onto a nickel crystal, which played the role of a spatial diffraction grating. It was established that the electrons were diffracted on the crystal in exactly the same way as waves whose length is defined by the de Broglie relation. The wave properties of electrons, neutrons, and other particles, as well as atoms and molecules, are now not only reliably demonstrated by direct experiments but are also utilized in devices with high resolving power, so that it is possible to speak of the engineering use of de Broglie waves.

De Broglie’s idea of the dual nature of microscopic particles, which was confirmed experimentally, changed in principle the concept of the microscopic world. Whereas particles—for example, electrons—were previously absolutely contrasted to waves, in particular electromagnetic waves, the hypothesis of the universality of the particle-wave dualism significantly changed the situation. Since both particle and wave properties are intrinsic to all microscopic objects (traditionally referred to as “particles”), then obviously none of these particles can be considered either a particle or a wave in the classical sense. The need arose for a theory in which the wave and particle properties of matter were complementary to each other rather than mutually exclusive. The basis of such a theory—the theory of wave or quantum mechanics—was de Broglie’s concept, the refinement of which led to the probabilistic interpretation of de Broglie waves.

However, even before the construction of quantum mechanics, several attempts were made to tie together particle and wave properties. The most interesting is the attempt to consider a particle as a wave packet. The superposition of a number (generally an infinite number) of monochromatic waves of close frequencies that are propagating in approximately the same direction can yield a resultant wave in the form of a “burst” flying in space—that is, in some region the amplitude of such an assemblage of waves is significant, and outside this region it is negligible. Such a burst, or packet, of waves was also considered as a particle composed of de Broglie waves. A strong argument in favor of this idea was the fact that the velocity of propagation of the center of the packet (group velocity) proved to be equal to the mechanical velocity of the particle. However, the velocity of the wave depends on its frequency; therefore, the velocities of the de Broglie waves that compose the packet are different, and with time the packet must spread out. (Under certain conditions, the packet may even separate into several packets.) Consequently, the representation of particles as wave packets is erroneous.

The generally accepted interpretation of de Broglie waves was given by M. Born (1926), who put forward the idea that the quantity that describes the state of the particle—that is, its wave function Ψ, whose square defines the probability of finding a particle at various points and at various moments of time—is subject to wave laws. The wave function of a free particle with an accurately defined momentum is also a de Broglie wave. In this case, ǀΨǀ^{2} = const—that is, the probability of finding a particle at all points is the same. Thus, de Broglie waves are probability waves rather than physical, material waves.

V. I. GRIGOR’EV