View source for Materials in Electronics/Wave-Particle Duality/Electrons as Waves

We have seen that electrons can be observed to act like waves by diffracting through narrow slits. This page will deal with the derivation of the electron's properties from the value of the wavelength. We will consider a situation where electrons of a known energy from an electron gun are diffracted through a grating, and the wavelength deduced as [[Materials_in_Electronics/Wave-Particle_Duality/The_Two-Slit_Experiment/Electrons|before]].

Although this page deals only with results obtained through experimental methods, the same results can be obtained from relativity, but the derivation is beyond the scope of this book. A diagram of the set-up is shown below. Note that the scale is very much exaggerated - the crystal spacing is minute compared to the distance to the screen.

[[Image:Electron diffraction setup.svg|center]]

By varying the voltage, ''V'', of the electron gun, we find, empirically, that:

::<math>\lambda \propto \frac{1}{\sqrt{V}}</math>

We also know the following relation about the velocity, ''v'', of electrons produced by electron guns:

::<math>v=\sqrt{\frac{2eV}{m}}</math>

where
:*''e'' is the charge on an electron
:*''m'' is the mass of an electron

Therefore, velocity is proportial to the square root of the voltage:

::<math>v \propto \sqrt{V}</math>

This means that

::<math>\lambda \propto \frac{1}{v}</math>

Since the mass is constant (neglecting relativity), we can bring it out of the constant of proportionality and say that

::<math>\lambda \propto \frac{1}{mv}</math>

We call the remaining constant of proportionality '''Planck's Constant''', or ''h'':
{{eqn|de Broglie's Relationship}}
::<math>\lambda = \frac{h}{mv}=\frac h p</math>

where ''p'' is momentum. Plank's Constant is very useful, and it turns up in all aspects of quantum physics and therefore also in electronics.

Sometimes it is convenient to use the wavenumber, rather than the wavelength:

::<math>p = \frac{h}{2 \pi}k</math>

We also sometime use ''ħ'', which is called the '''Reduced Planck's Constant''' or '''Dirac's Constant''' and is given by:

::<math>\hbar = \frac{h}{2 \pi}</math>

We therefore get:

::<math>p = \hbar k</math>

The values for ''h'' and ''ħ'' are:
:*<math>h     = 6.63 \times 10^{-34} \mbox{ J s}</math>
:*<math>\hbar = 1.05 \times 10^{-34} \mbox{ J s}</math>

{| class=wikitable style="text-align:center" align=center
|+'''Wave and Particle Properties'''
!width=100px|Property!!width=100px|Particle!!width=100px|Wave
|-
!Momentum
|<math>mv \,</math>||<math>\hbar k</math>
|-
!Kinetic Energy
|<math>\frac{mv^2}{2}=\frac{p^2}{2m}</math>||<math>\frac{\hbar^2 k^2}{2m}</math>
|}


==Uncertainty==

In fact, if we were to slow the electron beam down until only one electron was fired at a time, the diffraction fringes still build up, one dot at a time, implying that the solitary electron diffracts through ''both'' slits and interferes with itself.

{{BookCat}}