Demonstration for Class 23  Matter waves and Tunneling
After DeBroglie said that all matter has a wavelength, physicists knew that a
theory to describe matter must itself have all the properties of a wave.
An
equation was constructed to meet those requirements:
d
2
Ψ/dx
2
+
8 π
2
m/h [E  U(x)]Ψ = 0
This is the time independent or "stationary state"
form of Schrodinger's
Equation for a particle in one dimension.
The solution can have the simple
form
Ψ(x)
= (2/L)
1/2
sin (nπ/L) x
which is for a particle of matter in an infinitely strong
potential confined to
a width "L".
The interesting thing is its similarity to our equations for ordinary waves.
We said that the solution for any sort of undamped classical or "real world"
wave is a sinusoid, and the equation describing the motion must have the
function plus its second derivative somehow equal to zero.
"Waves is
waves", be they electromagnetic, vibrations or matter waves.
Three things to remember about matter waves:
 if the potential confining the particle is zero, the particle is "free"
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 Spring '06
 Stoler
 Energy, Total internal reflection, air gap, matter waves

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