In physics, the de Broglie hypothesis is the statement that all matter 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 1923 in his PhD thesisL. de Broglie, PhD thesis, reprinted in Ann. Found. Louis de Broglie 17 (1992) p. 22.; 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.

The de Broglie relations

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

$\backslash lambda\; =\; \backslash frac\{h\}\{p\}\; =\; \backslash frac\; \{h\}\{mv\}\; \backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}$

where $\backslash lambda$ is the particle's wavelength, h is Planck's constant, p is the particle's momentum, m is the particle's rest mass, v is the particle's velocity, 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 a particle to the kinetic energy such that

$f\; =\; \backslash frac\{E\_k\}\{h\}\; =\; \backslash frac\{mc^2\}\{h\}\; \backslash left(\backslash frac\; \{1\}\{\backslash sqrt\{1\; -\; \backslash frac\{v^2\}\{c^2\}\}\}\; -\; 1\; \backslash right)$

where f is the frequency and $E\_k$ is the kinetic energy. The two equations are often written as

$p\; =\; \backslash hbar\; k$

$E\_k\; =\; \backslash hbar\; \backslash omega$

where $\backslash hbar$ is Dirac's constant, k is the wavenumber, and $\backslash omega$ is the angular frequency.

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

Experimental confirmation

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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 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. Recent experiments even confirm the relations for 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.

Wavelength of large objects

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Theoretically, all objects, not just sub-atomic particles, exhibit wave properties according to the De Broglie Hypothesis.

Consider the following example:

A baseball has a mass of 0.15 kg and is thrown by a professional baseball player at 40 m/s. The de Broglie wavelength of the baseball is given by:

$m\; =\; 0.15\; kg$

$v\; =\; 40\; m/s$ (about 90 mph)

$\backslash lambda\; =\; \backslash frac\{h\}\{p\}$ where $p\; =\; mv$

$\backslash lambda\; =\; \backslash frac\; \{6.626\; \backslash times\; 10^\{-34\}\; kg\; m^2\; /s\}\{0.15\; kg\; \backslash times\; 40\; m/s\}$

$\backslash lambda\; =\; 1.10\; \backslash times\; 10^\{-34\}\; m$

This wavelength is considerably smaller than the diameter of a proton (about $10^\{-15\}\; m$) and is approaching the Planck Length. As such, the wave-like properties of this baseball are so small as to be unobservable.

See also

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• Bohr model

References

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• Steven S. Zumdahl, Chemical Principles 5th Edition, (2005) Houghton Mifflin Company.
• Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. pp. 203-4, 222-3, 236.

Foundational quantum physics | Hypotheses

Hypothèse de De Broglie | השערת דה ברויי | ド・ブロイ波 | Hipoteza de Broglie'a | De Broglie-våglängd