Download The Bohr Theory, Matter Waves, and Quantum Theory

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The Bohr Theory, Matter Waves, and Quantum Theory
At the beginning of the 20th 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 ε = kT. 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 nhν. He used the Maxwell-Boltzmann distribution to calculate the average value:
Let P(ε) = the distribution function
Then, for Planck’s model,
ε=
∑ (prob.of energy) × energy
energies
ε=
hν
∑ P(ε)× nhν =
energies
e
hν
kT
−1
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:
Electrons out
Light in
Metal
Here, light impinges on a surface and electrons may be emitted. Classically, the energy of the
light is controlled by the intensity of the light. Energy conservation products that there should be
no dependence on wavelength of light. If the intensity of the light is increased, the kinetic
energy of the electrons should increase.
This was not observed. Instead, the kinetic energies of the electrons depended on the frequency
(wavelength) of the light. If the frequency was below a threshold value, no electrons were
emitted. Increasing the intensity of the light just increased the number of electrons coming from
the surface (the current), but not their energies.
Einstein (1905) provided the explanation. He said that the energy from light is quantized, giving
light a particle-like character. E = hν. This energy could be used to overcome the binding
energy of the electron to the metal (work function), with the remainder going to the kinetic
energy of the electron. The significance of this result was the fact that the value of “h” that
Einstein used to fit data was the same as the Planck value. This was the basis for Einstein’s
Nobel Prize in 1921.
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A final class of experiment that could not be interpreted with classical theory was the
determination of the line spectra of atoms.
White light contains all possible
wavelengths, as shown by the
diagram:
Ordinary white light has all colors
and the spectrum is continuous
In contrast, the light emitted by an
atom occurs at discrete
wavelengths. There were many
examples of such spectra, and an
experimentalist named Ritz showed
that the frequencies of the emission
lines of hydrogen could be
cataloged with the expression
1 
1
ν = const  2 − 2 
m 
n
where n and m are integers.
Niels Bohr developed a theory for the hydrogen atom emission lines with the following
postulates:
•
•
•
The electron moves around the nucleus in a circular orbit with a well-defined radius
These states are called “stationary states”, and radiation is emitted or absorbed when the
electron “jumps” from one radius to another.
In these states, the angular momentum of the electron is quantized, and has values mvR =
h
n
2π
Here’s how the theory worked:
mv 2
Ze 2
=
R
4πε 0 R 2
•
•
mv2/R
Ze2/4πε0R2
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Therefore, kinetic energy =
Also, (mv)2 =
1
Ze 2
mv 2 =
2
8πε 0 R
mZe 2
4πε 0 R
The total energy E = kinetic energy + potential energy =
Ze 2
Ze 2
Ze 2
−
=−
8πε 0 R 4πε 0 R
8πε 0 R
We therefore need to determine exactly what the allowed values of R are.
To this end, Bohr hypothesized without any theoretical justification that angular momentum in
those orbits can only have discrete values.
mvR = n
h
≡ nh
2π
Aside from the existence of stationary states, this hypothesis was the most controversial of all.
n 2h 2
We get a second expression for (mv)2 =
R2
Equate these two expressions and solve for R:
mZe 2 n 2 h 2
= 2
4πε 0 R
R
so, R =
4πε 0 n 2 h 2
mZe 2
Note that R increases as n2. Increased nuclear charge Z corresponds to smaller values of R.
We now use this value of R in our expression for E:
Ze 2
Ze 2
E= −
=−
8πε 0 R
8πε 0
 mZe 2 
mZ 2 e 4  1 
=−
 

2 2 
32π 2 ε 02 h 2  n 2 
 4πε 0 n h 
So, we see that the energy levels go as 1/n2, in qualitative
agreement with the Ritz data-fitting.
In fact, this expression is in quantitative agreement with
experimental data.
Evaluating all the constants, we can calculate
1 
 1
∆E = −2.178 × 10 −18 J  2
− 2
 n final n initial 
By specifying the initial and final quantum numbers, we can
calculate the energy difference. So, the energy level
diagram shows the transitions that can occur when all
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transitions end on n = 2. This is the so-called Balmer series.
The transitions that terminate on n = 1 are called the Lyman series, and the lowest energy
transition in the Lyman series is the one from n = 2 to n = 1. This is called the Lyman-α
transition.
We can also calculate wavelengths for these transitions:
hc
hc
for a “quantum jump”
so, λ =
λ
∆E
−18
For the Lyman-α transition, ∆E = ¾( 2.178 × 10 J ) = 1.634 × 10-18 J
∆E =
λ = (6.63 × 10-34 J sec)(3.00 × 108 m/sec)/(1.634 × 10-18 J) = 1.216 × 10-7 m = 121.6 nm
So, this line is in the ultraviolet (UV) region of the spectrum.
The Bohr Theory, which represented an attempt to “patch up” classical mechanics with a few
additional postulates, (like angular momentum quantization and stationary states) was an
example of the “Old Quantum Theory”. However, there was no way to justify the assumptions
of the theory, and it could only be applied to the one-electron atom. The Old Quantum Theory
was an intellectual dead end. A new concept was needed.
deBroglie waves
The result of the analyses of blackbody radiation and the Photoelectric Effect showed that
light has properties of waves and particles.
A young prince, Louisa deBroglie, who was studying for the Ph.D. degree in France in the
1920’s suggested that matter should also have the properties of particles and waves.
For calculate the wavelength associated with the wavelike properties of matter, deBroglie
equated the energy of a relativistic particle moving at the speed of light with the energy of a
“particle” of light, the photon.
E = pc = hν =
hc
λ
From this idea, the deBroglie wavelength for a matter wave was assigned the value
λ=
h
h
=
=
p mv
h
2mE
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So, theory predicted the existence of matter waves. deBroglie’s professors rejected the
connections that he made, but Einstein endorsed them. The existence of matter waves was
quickly demonstrated to exist in electron diffraction experiments conducted by Davisson and
Germer.
DEMO: diffraction with a HeNe laser and a ruler
Applying Matter Waves
Once the concept of matter waves was advanced, it was quite
easy to rationalize the ad hoc quantization of angular momentum
that Bohr had introduced: stationary states occurred when
an integral number of deBroglie waves could fit exactly on the
circumference of the orbit:
2πR = n
h
mv
n = 1, 2, 3, 4
Panels (a) and (b) show cases where 4 or 5 deBroglie waves fit exactly. We say that standing
waves corresponding to complete constructive interference are formed. However, when we
attempt to fit a non-integral multiple of deBroglie wavelengths on the circle, as in panel (c),
complete destructive interference occurs quickly.
The conclusion: stationary states are a consequence of constructive interference of matter
waves in a fixed region of space. This was the conceptual foundation for the “New
Quantum Theory”, Schrödinger’s wave mechanics.