Definitions

de Broglie equation

De Broglie hypothesis

In physics, the de Broglie hypothesis (pronounced /brœj/, as French breuil, close to "broy") is the statement that all matter (any object) has a wave-like nature (wave-particle duality). The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle's kinetic energy. The hypothesis was advanced by Louis de Broglie in 1924 in his PhD thesis; he was awarded the Nobel Prize for Physics in 1929 for this work, which made him the first person to receive a Nobel Prize on a PhD thesis.

Historical context

After strides made by Max Planck (1858-1947) and Albert Einstein (1879-1955) in understanding the behavior of electrons and what would be known as quantum physics, Niels Bohr (1885-1962) began (among other things) trying to explain how electrons behave. He came up with new fundamental ideas about electrons and mathematically derived the Rydberg equation, an equation that was discovered only through trial and error. This equation explains the energies of the light emitted when hydrogen gas is compressed and electrified (similar to neon signs, but with hydrogen in this case). Unfortunately, his model only worked for the hydrogen-atom-configuration, but his ideas were so revolutionary that they broke up the classical view of electrons' behavior and paved the way for fresh new ideas in what would become quantum physics and quantum mechanics.

Louis de Broglie (1892-1987) tried to expand on Bohr's ideas, and he pushed for their application beyond hydrogen. In fact he looked for an equation which could explain the wavelength characteristics of all matter. His equation was experimentally confirmed in 1927 when physicists Lester Germer and Clinton Davisson fired electrons at a crystalline nickel target and the resulting diffraction pattern was found to match the predicted values. . Nevertheless, his hypothesis would hold true for both electrons and for everyday objects. In de Broglie's equation an electron's wavelength will be a function of Planck's constant (6.626 times 10^{-34} joule-seconds) divided by the object's momentum (nonrelativistically, its mass multiplied by its velocity). When this momentum is very large (relative to Planck's constant), then an object's wavelength is very small. This is the case with every-day objects, such as a person. Given the enormous momentum of a person compared with the very tiny Planck constant, the wavelength of a person would be so small (on the order of 10^{-35} meters or smaller) as to be undetectable by any current measurement tools. On the other hand, many small particles (such as typical electrons in everyday materials) have a very low momentum compared to macroscopic objects. In this case, the de Broglie wavelength may be large enough that the particle's wave-like nature gives observable effects.

The wave-like behavior of small-momentum particles is analogous to that of light. As an example, electron microscopes use electrons, instead of light, to see very small objects. Since electrons typically have more momentum than photons, their de Broglie wavelength will be smaller, resulting in a greater spatial resolution.

The de Broglie relations

The first de Broglie equation relates the wavelength lambda to the particle momentum ~p~ as

lambda = frac{h}{p} = frac {h}{gamma mv} = frac {h}{mv} sqrt{1 - frac{v^2}{c^2}}

where ~h~ is Planck's constant, ~m~ is the particle's rest mass, ~v~ is the particle's velocity, ~gamma~ is the Lorentz factor, and ~c~ is the speed of light in a vacuum.

The greater the energy, the larger the frequency and the shorter (smaller) the wavelength. Given the relationship between wavelength and frequency, it follows that short wavelengths are more energetic than long wavelengths. The second de Broglie equation relates the frequency of the wave associated to a particle to the total energy of the particle such that

f = frac{E}{h} = frac{gamma,mc^2}{h} = frac {1}{sqrt{1 - frac{v^2}{c^2}}} cdot frac{mc^2}{h}

where ~f~ is the frequency and ~E~ is the total energy. The two equations are often written as

p = hbar k
E = hbar omega

where ~p~ is momentum, ~hbar=h/(2pi)~ is the reduced Planck's constant (also known as Dirac's constant, pronounced "h-bar"), ~k~ is the wavenumber, and ~omega~ is the angular frequency.

See the article on group velocity for detail on the argument and derivation of the de Broglie relations.

Experimental confirmation

Elementary particles

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for X-Rays. Before the acceptance of the de Broglie hypothesis, diffraction was a property that was thought to be only exhibited by waves. Therefore, the presence of any diffraction effects by matter demonstrated the wave-like nature of matter. When the de Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the de Broglie hypothesis for electrons.

This was a pivotal result in the development of quantum mechanics. Just as Arthur Compton demonstrated the particle nature of light, the Davisson-Germer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength.

Since the original Davisson-Germer experiment for electrons, the de Broglie hypothesis has been confirmed for other elementary particles.

Neutral atoms

Experiments with Fresnel diffraction and specular reflection of neutral atoms confirm the application of the De Broglie hypothesis to atoms, i.e. the existence of atomic waves which undergo diffraction, interference and allow quantum reflection by the tails of the attractive potential. This effect has been used to demonstrate atomic holography, and it may allow the construction of an atom probe imaging system with nanometer resolution. The description of these phenomena is based on the wave properties of neutral atoms, confirming the de Broglie hypothesis.

Waves of molecules

Recent experiments even confirm the relations for molecules and even macromolecules, which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes.

In general, the De Broglie hypothesis is expected to apply to any well isolated object.

Spatial Zeno effect

The De Broglie hypothesis leads to the spatial version of the Zeno effect. If an object (particle) is observed with frequency ~Omegaggomega~ in a half-space (say, ~y<0~), then this observation prevents the particle, which stays in the half-space ~y>0~ from entry into this half-space ~y<0~. Such an "observation" can be realized with a set of rapidly moving absorbing ridges, filling one half-space. In the system of coordinates related to the ridges, this phenomenon appears as a specular reflection of a particle from a ridged mirror, assuming the grazing incidence (small values of the grazing angle). Such a ridged mirror is universal; while we consider the idealised "absorption" of the de Broglie wave at the ridges, the reflectivity is determined by wavenumber ~k~ and does not depend on other properties of a particle.

See also

References

  • Steven S. Zumdahl, Chemical Principles 5th Edition, (2005) Houghton Mifflin Company.
  • Broglie, Louis de, The wave nature of the electron, Nobel Lecture, 12, 1929
  • Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Co. ISBN 0-7167-4345-0. pp. 203-4, 222-3, 236.
  • Web version of Thesis, translated (English): http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.htm

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