Introduction to Quantum Mechanics (2)

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Chapter 9

• Introduction to Quantum Mechanics (2)

(May. 25, 2005)
2
A brief summary to the last lecture

• Blackbody radiation and Plancks hypothesis
  (Stefan and Boltzmanns law, Weins displacement
  law, Rayleigh-Jeans formula, Weins formula,
  Plancks formula and his hypothesis)
• Photoelectric effect (shows that the
  absorption of light is as light quantum,
  photoelectric effect equation)

3
What does the Stefan-Boltzmanns Law explain?
M(T) is radiation energy, T is the absolute
temperature.
4
For the radiation power M?(T), it is a function
of ? for a constant T. What is Wiens
displacement law?
5
For the radiation power and the fixed
temperature, two formulas were obtained using
classical theories, which one is correct for
longer wavelength and which one is valid for
shorter wavelength?
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• Plancks Magic formula
• In 1900, after studying the above two formulas
  carefully, Planck proposed (??) an empirical
  formula.

How to get Stefan-Boltzmanns Law using above
formula?
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How to obtain Weins displacement law by Plancks
formula? How could you get Rayleigh and Jeans
formula, Weins formula by Plancks formula?
8

• Planck-Einstein Energy Quantization Law and
  photoelectric effect.

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9.3 Compton effect

• A phenomenon called Compton scattering
  (observed in 1924) provides additional direct
  confirmation of the quantum nature of
  electromagnetic radiation

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• X-rays impinges (??,??) on matter, some of the
  radiation is scattered (??), just as the visible
  light falling on a rough surface undergoes
  diffuse reflection (??,diffusion).
• Scattered radiation has smaller frequency and
  longer wavelength than the incident radiation.
• The wavelength of scattered radiation depends on
  the scattered angle f .

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• Classical explanation
• Based on the classical principles, the scattering
  mechanism is induced by motion of electrons in
  the material, caused by the incident radiation.
  This motion must have the same frequency as that
  of incident wave because of forced vibration, and
  so the scattered wave radiated by the oscillating
  charges should have the same frequency. There is
  no way that the frequency can be shifted by this
  mechanism.

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• Quantum explanation
• The quantum theory, by contrast, provides a
  beautifully simple explanation. We imagine the
  scattering process as a collision of two
  particles, the incident photon and an electron at
  rest as shown in Fig. 8.4. The photon gives up
  some of its energy and momentum to the electron,
  which recoils as a result of this impact and the
  final photon has less energy, smaller frequency
  and longer wavelength than the initial one.

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• Theoretical derivation
• Suppose that ?0 and ? are the wavelength of the
  incident and scattered radiation, respectively.
  The Compton effect could be described by the
  following procedures
• Using relativistic theory
• Energy conservation
• Momentum conservation (using cosine principle
  at the same time)

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• The key points of derivation procedure
• (1) Relativistic theory

When m0 0, the momentum is equal to
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(2) Energy conservation
(1)
For electron, we have
(2)
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(3) The momentum conservation
(3)
So the (mV)2 can be deleted from (2) and (3), so
we have
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(4)
Squaring (1) and minus (4), we have
The above equation can be rewritten as
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Fig. 8.4 Schematic diagram of Compton scattering.
where m0 is the electron mass.
called Compton wavelength.
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9.4 The duality of light
The concept that waves carrying energy may have a
corpuscular (particle) aspect and that particles
may have a wave aspect which of the two models
is more appropriate will depend on the
properties the model is seeking to explain. For
example, waves of electromagnetic radiation need
to be visualized as particles, called photons to
explain the photoelectric effect. Now you may
confuse the two properties of light and ask what
the light actually is?
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The fact is that the light shows the property of
waves in its interference and diffraction and
performances the particle property in blackbody
radiation, photoelectric effect and Compton
effect. Till now we say that the light has
duality property.
We can say that light is wave when it is involved
in its propagation only like interference and
diffraction.
This means that light interacts with itself. The
light shows photon property when it interact with
other materials.
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9.5 Line spectra and Energy quantization in atoms
The quantum hypothesis, used in the preceding
section for the analysis of the photoelectric
effect, also plays an important role in the
understanding of atomic spectra.
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9.5.1 Line spectra of Hydrogen atoms

• Hydrogen always gives a set of line spectra in
  the same position.

Fig. 9.5 the Balmer series of atomic hydrogen.
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• It is impossible to explain such a line
  spectrum phenomenon without using quantum theory.
  
• For many years, unsuccessful attempts were
  made to correlate the observed frequencies with
  those of a fundamental and its overtones (??)
  (denoting other lines here). Finally, in 1885,
  Balmer found a simple formula which gave the
  frequencies of a group lines emitted by atomic
  hydrogen. Since the spectrum of this element is
  relatively simple, and fairly typical of a number
  of others, we shall consider it in more detail.

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Under the proper conditions of excitation, atomic
hydrogen may be made to emit the sequence of
lines illustrated in Fig. 9.5. This sequence is
called series.
There is evidently a certain order in this
spectrum, the lines becoming crowded more and
more closely together as the limit of the series
is approached. The line of
Fig. 9.5 the Balmer series of atomic hydrogen.
longest wavelength or lowest frequency, in the
red, is known as H?, the next, in the blue-green,
as H?, the third as H?, and so on.
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Balmer found that the wavelength of these lines
were given accurately by the simple formula
(9.5.1)
where ? is the wavelength, R is a constant called
the Rydberg constant, and n may have the integral
values 3, 4, 5, etc. if ? is in meters,
(9.5.2)
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Substituting R and n 3 into the above formula,
one obtains the wavelength of the H?-line
For n 4, one obtains the wavelength of the
H?-line, etc. for n ?, one obtains the limit of
the series, at ? 364.6nm shortest wavelength
in the series.
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Other series spectra for hydrogen have since been
discovered. These are known, after their
discoveries, as Lymann, Paschen, Brackett and
Pfund series. The formulas for these are
Lymann series
Paschen series
(9.5.3)
Brackett series
Pfund series
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The Lymann series is in the ultraviolet, and the
Paschen, Brackett, and Pfund series are in the
infrared. All these formulas can be generalized
into one formula which is called the general
Balmer series.
(9.5.4)
All the spectra of atomic hydrogen can be
described by this simple formula. As no one can
explain this formula, it was ever called Balmer
formula puzzle.
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9.5.2 Bohrs atomic theory
Bohrs theory was not by any means the first
attempt to understand the internal structure of
atoms. Starting in 1906, Rutherford and his
co-workers had performed experiments on the
scattering of alpha Particles by thin metallic.
These experiments showed that each atom contains
a massive nucleus whose size is much smaller than
overall size of the atom.
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• The atomic model of Rutherford
• The nucleus is surrounded by a swarm (???) of
  electrons. To account for the fact, Rutherford
  postulated that the electrons revolve (??) about
  the nucleus in orbits, more or less as the
  planets in the solar system revolve around the
  sun, but with electrical attraction providing the
  necessary centripetal force (???). This
  assumption, however, has an unfortunate
  consequence.

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(1) Accelerated electron will emit
electromagnetic waves Its energy will be used up
sometimes later. Dead atoms
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A body moving in a circle is continuously
accelerated toward the center of the circle and,
according to classical electromagnetic theory, an
accelerated electron radiates energy. The total
energy of the electrons would therefore gradually
decrease, their orbits would become smaller and
smaller, and eventually they would spiral (??)
into the nucleus and come to rest. Go to next
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(2) The emitted frequency should be that of
revolution and they should emit continuous
frequency.
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Furthermore, according to classical theory, the
frequency of the electromagnetic waves emitted by
a revolving electron is equal to the frequency of
revolution. Their angular velocities would
change continuously and they would emit a
continuous spectrum (a mixture of frequencies),
in contradiction to the line spectrum actually
observed.
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• In order to solve the above contradictions, Bohr
  made his hypotheses
• Static (or stable-orbit) postulate
• Faced with the dilemma, Bohr concluded that , in
  spite of the success of electromagnetic theory in
  explaining large scale phenomenon, it could not
  be applied to the processes on an atomic scale.
  He therefore postulated that an electron in an
  atom can revolve in certain stable orbits, each
  having a definite associated energy, without
  emitting radiation.

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• The angular momentum mvr of the electron on the
  stable orbits is supposed to be equal to the
  integer multiple of h/2p. This condition may be
  stated as

(9.5.5)
where n is quantum number, this is the hypothesis
of stable state and it is called the quantization
condition of orbital angular momentum.
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• Transition hypothesis,
• Bohr postulated that the radiation happens only
  at the transition of electron from one stable
  state to another stable state. The radiation
  frequency or the energy of the photon is equal to
  the difference of the energies corresponding to
  the two stable states.

(9.5.6)
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• Corresponding principle
• The new theory should come to the old theory
  under the limited conditions.

• Important conclusions
• Another equation can be obtained by the
  electrostatic force of attraction between two
  charges and Newtons law

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(9.5.7)
Solving the simultaneous equation of (9.5.5) and
(9.5.7), we have
So the total energy of the electron on the nth
orbit is
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(9.5.8)
It is easy to see that all the energy in atoms
should be discrete. When the electron transits
from the nth orbit to kth orbit, the frequency
and wavelength can be calculated as
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(9.5.9)
where
is Rydberg constant.
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It is found that the value of R is matched with
experimental data very well. Till then, the
30-years puzzle of line spectra of atoms had been
solved by Bohr since equation (9.5.9) is exactly
the general Balmer formula.
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When Bohrs theory met problems in explaining a
little bit more complex atoms (He) or molecules
(H2), Bohr realized that his theory is full of
contradictions as he used both quantum and
classical theories. The problem was solved
completely after De Broglie proposed that
electron also should have the wave-particle
duality. Since then, the proper theory describing
the motion of the micro-particles, quantum
mechanics, has been gradually established by
many scientists.
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9.6 De Broglie Wave
In the previous sections we traced the
development of the quantum character of
electromagnetic waves. Now we will turn to the
consequences of the discovery that particles of
classical physics also possess a wave nature. The
first person to propose this idea was the French
scientist Louis De Broglie.
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9.6.1 De Broglie wave
De Broglies result came from the study of
relativity. He noted that the formula for the
photon momentum can also be written in terms of
wavelength
(9.6.1)
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If the relationship is true for massive particles
as well as for photons, the view of matter and
light would be much more unified. De Broglies
point was the assumption that momentum-wavelength
relation is true for both photons and massive
particles. So De Broglie wave equations are
(9.6.2)
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Where p is the momentum of particles, ? is the
wavelength of particles. At first sight, to claim
that a particle such as an electron has a
wavelength seems somewhat absurd. The classical
concept of an electron is a point particle of
definite mass and charge, but De Broglie argued
that the wavelength of the wave associated with
an electron might be so small that it had not
been previously noticed. If we wish to prove that
an electron has a wave nature, we must perform an
experiment in which electrons behave as waves.
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9.6.2 Electron diffraction
In order to show the wave nature of electrons, we
must demonstrate interference and diffraction for
beams of electrons. At this point, recall that
interference and diffraction of light become
noticeable when light travels through slits whose
width and separation are comparable with the
wavelength of the light. So let us first look at
an example to determine the magnitude of the
expected wavelength for some representative
objects.
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For example, de Broglie wavelength for an
electron whose kinetic energy is 600 eV is
0.0501nm. The de Broglie wavelength for a golf
ball of mass 45g traveling at 40m/s is 3.68
10-34 m. Such a short wave is hardly
observed. We must now consider whether we could
observe diffraction of electrons whose wavelength
is a small fraction of a nanometer. For a grating
to show observable diffraction, the slit
separation should be comparable to the
wavelength, but we cannot rule a series of lines
that are only a small fraction of a nanometer
apart, as such a length is less than the
separation of the atoms in solid materials.
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When electrons pass through a thin gold or other
metal foils (?), we can get diffraction patterns.
So it indicates the wave nature of electrons.
Look at the pictures on page 265 and 231 in your
Chinese. Introduce Electron single and double
slits experiments electronic microscope, nuclear
reactor etc.