2. Wave Nature of Matter
Just as light sometimes behaves as a particle, matter sometimes behaves like a wave.
The wavelength of a particle of matter is:
This wavelength is extraordinarily small.
The wave nature of matter becomes more important for very light particles such as the electron.
In 1923 Louis de Broglie postulated that perhaps matter exhibits the same “duality” that light exhibits
The above equations suggest that the wavelength of light is related to its momentum

3. Energy, Mass, and Momentum of a Photon
Clearly, a photon must travel at the speed of light. Looking at the relativistic equation for momentum, it is clear that this can only happen if its rest mass is zero.
We already know that the energy is hf; we can put this in the relativistic energy-momentum relation and find the momentum:

4. Compton Scattering Recoiling electron Stationary electron f q Incident
If light is like a particle does it have momentum?
In Compton scattering x-rays impart momentum to matter, scattering electrons like billiard balls
Thus photons also have momentum. The momentum of a photon is given by
Recoiling electron
Stationary
electron f q
Incident
Photon, l0
Scattered
Photon, l’

5. The wavelength of matter waves is very small
The wavelength of matter waves is very small. This is why we do not see them in our every day experience
To measure λ, a diffraction grating with a very small slit width is required (eg the space between two atoms in a crystal)
Davisson and Germer were the first to show this… they used the close spacing between atoms in a crystal lattice to diffract electron waves thus proving that matter can also exhibit diffraction and interference

6. [1] What is the wavelength of a neutron ( m = 1.67 x 10-27 kg )
traveling at 6.5 x 104 m/s.
De Broglie
wavelength
[2] What is the wavelength of an electron of energy 100 eV?
De Broglie wavelength

7. From Chapter 29…
The solution to the blackbody spectrum leads to the concept of photons, and to a solution for the photoelectric effect
The maximum excess energy of a photoelectron is
The particle nature of light is also shown by Compton scattering of electrons by photons
Scattering shows that photons have momentum given by
This implies that matter also has wavelike properties given by the de Broglie formula
The de Broglie wavelength leads to phenomena such as electron diffraction. A common tool in modern crystallography