# De-Broglie or Matter Wave

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when physicists Lester Germerand Clinton Davisson fired electrons at a crystalline nickeltarget and the resulting diffraction pattern was found to match the predicted values.

[2]In de

Broglie's equation an electron's wavelength is a function ofPlanck's constant (6.6261034

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 anobject'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, thewavelength of a person would be so small (on the order of 1035 nanometer 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 better spatial resolution.

[edit] The de Broglie relations

The de Broglie equations relate the wavelength and frequency to the momentum andkinetic energy , respectively, as

and

where is Planck's constant. The two equations are also written as

where is the reduced Planck's constant (also known as Dirac's constant,

pronounced "h-bar"), is the angular wavenumber, and is the angular frequency.

Using results from special relativity, the equations can be written as

and

where is the particle's rest mass, is the particle's velocity, is the Lorentz factor, and

is the speed of light in a vacuum.

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See t e arti le on group velocit for detail on t e argument and derivation oft e de Broglierelations. Group velocit (equalto t e particle's speed) should not be confused withphase

velocit (equalto the product ofthe particle's frequency and its wavelength).

[edi E perimental nfirmati n

[edit] Elementary parti les

In 1927 atBell Labs, Clinton Davisson and Lester Germerfired slow-moving electrons at a

crystallinenickeltarget. The angular dependence ofthe reflected electron intensity was

measured, and was determined to have the same diffraction pattern as those predicted by

Bragg forx-rays. Before the acceptance ofthe de Broglie hypothesis, diffraction was aproperty that was thoughtto be only exhibited by waves. Therefore, the presence of any

diffraction effects by matter demonstrated the wave-like nature of matter. When the deBroglie 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 resultin the development ofquantum mechanics. Just as ArthurComptondemonstrated the particle nature oflight, the Davisson-Germer experiment showed the wave-

nature of matter, and completed the theory ofwave-particle duality. Forphysiciststhis ideawas important because it means that not only can any particle exhibit wave characteristics,

butthat one can use wave equationsto describe phenomena in matterif one uses the de

Broglie wavelength.

Since the original Davisson-Germer experiment for electrons, the de Broglie hypothesis has

been confirmed for otherelementary particles.

The wavelength of a thermali ed electron in a non-metal at room temperature is about

8 nm.[cit ti

]

[edit] Neutral atoms

Experiments with Fresnel diffraction[3] and specular reflection

[4][5] of neutral atoms confirm

the application ofthe de Broglie hypothesis to atoms, i.e. the existence of atomic waves

which undergo diffraction, interference and allow quantum reflection by the tails ofthe

attractive potential.[6]

Advances in laser cooling have allowed cooling of neutral atoms down

to nanokelvin temperatures. Atthese temperatures, the thermal de Broglie wavelengths comeinto the micrometre range. Using Bragg diffraction of atoms and a Ramsey interferometry

technique, the de Broglie wavelength of cold sodium atoms was explicitly measured andfound to be consistent with the temperature measured by a different method.[7]

This effect has been used to demonstrate atomic holography, and it may allow the

construction of an atom probe imaging system with nanometer resolution.[8][9]

The description

ofthese phenomena is based on the wave properties of neutral atoms, confirming the de

Broglie hypothesis.

[edit] Waves ofmolecules

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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.[10]

Theresearchers calculated a De Broglie wavelength ofthe most probable C60 velocity as 2.5pm.

More recent experiments proofthe quantum nature of molecules with a mass up to 6910

amu.[11]

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

[edit] Spatial Zeno effect

The matter wave leads to the spatial version ofthe Zeno effect. If an object (particle) is

observed with frequency in a half-space (say,y < 0), then this observation prevents

the particle, which stays in the half-spacey > 0 from entry into this half-spacey < 0. Suchan "observation" can be reali ed 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 aspecular reflection of a particle from a ridged mirror, assuming the grazing incidence (small

values ofthe grazing angle). Such a ridged mirroris universal; while we considertheidealised "absorption" ofthe de Broglie wave atthe ridges, the reflectivity is determined by

wavenumberkand does not depend on other properties of a particle.[5]

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