97
The Bohr Theory, Matter Waves, and Quantum Theory
At the beginning of the 20
th
century, classical physics was thought to be in “good shape”.
There
were only a few problems that could not be explained by Newton’s Laws.
Matter was described
by Newton’s Laws, and light was described as a wave, in accordance with Maxwell’s equations.
Light waves are characterized by a speed, a wavelength, and a frequency:
This relation is given by
ν
= c/
λ
.
As the frequency
increases, the wavelength decreases.
The electromagnetic
spectrum shows that visible radiation is in the wavelength
range from 400 to 700 nm.
Some of the problems that could not be understood with the classical theory included:
•
Blackbody radiation – the distribution of wavelengths of light emitted by an object at
a given temperature
•
The photoelectric effect
•
The line spectrum of hydrogen
“
Ultraviolet Catastrophe”
Blackbody radiation can be understood as follows:
put a fireplace poker in the fireplace.
First, it glows
dull red, then orange, then yellow.
Analyze the light
with a prism.
The results are shown in the figure.
Note that as T increases, the maximum in the
wavelength shifts to shorter values.
Theory said that
oscillators associated with the atoms in the lattice
emitted light.
Classically, the average value of the energy of one of
these oscillators is
.
kT
=
ε
The result is the
Ultraviolet Catastrophe, indicated by the
red
line.
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Planck
said that the energy of an oscillator was proportional to a fundamental “quantum”, with
energy nh
ν
.
He used the MaxwellBoltzmann distribution to calculate the average value:
Let P(
ε
) = the distribution function
Then, for Planck’s model,
()
∑
×
=
ε
energies
energy
energy
of
.
prob
1
e
h
nh
)
(
P
kT
h
energies
−
ν
=
ν
×
ε
=
ε
ν
∑
This last expression has the right behavior (goes to zero) at short wavelengths (high frequencies),
and also at long wavelengths.
By data fitting, Planck was able to fit the data in the plot.
He
found that a value of h = 6.6
×
10
34
J sec fit the data.
Planck won the Nobel Prize in 1918 for
this work.
Photoelectric effect:
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 Spring '08
 farrar
 pH, Angular Momentum, Energy, matter waves, stationary states

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