# Matter Waves (Wave Particle Duality)

Special relativity said that

if m=0

Plank said he could fit the Black Body radiation data assuming that that

where = Plank's constant

Combining the two we have

photons have momentum like a particle (mv)

Do particles reciprocate and behave like photons?

## De Broglie's Hypothesis

If photons can behave like particles by having momentum

Then can a particle behave like a wave by having wavelength

or

de Broglie Hypothesis

## Davisson and Germer

We know that X-rays having a wavelength of make a diffraction pattern on an aluminum foil. The waves diffract around the obstacle and then constructively interfere according to

where is the spacing between crystal planes, is the reflected angle, and is the X-ray wavelength. The order of the diffraction maxima is given by .

Another way to calculate

What would be the energy of an electron with the same wavelength as the above X-ray?

relativistic total energy relation

= 511.3 keV

relativistic kinetic energy

Note classical physics may be used for electrons below 50 keV

Clinton Davisson and Lester Germer in 1927 published conclusive evidence for the diffraction of electron waves using 54 eV electrons impinging a crystal made of nickel.

One problem to overcome for the experiment was that such a low energy electron scatters in air. The had to do the experiment in a vacuum.

From hyperphysics: