De Broglie Hypothesis

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De Broglie Hypothesis

THE DE BROGLIE RELATIONS

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

:

\lambda = rac{h}{p} = rac {h}{mv} \sqrt{1 - rac{v^2}{c^2}}

where

\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 = rac{E}{h} = rac{mc^2}{h} \left(rac {1}{\sqrt{1 - rac{v^2}{c^2}}} - 1 
ight)

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

:

p = \hbar k

:

E = \hbar \omega

where

\hbar

is Dirac's Constant , ''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

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 .¹

WAVELENGTH OF LARGE OBJECTS

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)

\lambda = rac{h}{p}

where

p = mv

\lambda = rac {6.626 	imes 10^{-34} kg  m^2  /s}{0.15 kg 	imes 40 m/s}

\lambda = 1.10 x 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

Bohr Model

REFERENCES

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.