Physics 123C Waves

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Physics 123C Waves
Lecture 23The Bohr Atom June 1, 2005

• John G. Cramer
• Professor of Physics
• B451 PAB
• cramer_at_phys.washington.edu

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Lecture 23 Announcements

• We will begin the class with a UW Teaching
  Evaluation survey. I will leave the room for 10
  minutes while this is in progress.
• You are required to complete the Tycho Course
  Survey by 1159 PM on June 5. Please complete it
  as soon as possible.
• On Friday, for the Final, we will have a
  comprehensive review of the material covered this
  quarter.
• The Final Exam will be in this room at 830 AM
  on Monday, June 6. (NOT at 1030 AM!) You may
  bring to the Final 3 pages of notes. Also bring
  a Scantron sheet and a calculator with good
  batteries.

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Lecture Schedule (Weeks 7-10)
We are here
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Classical Physics at the Limit
Rutherfords nuclear atom model matched
experimental evidence about the structure of
atoms, but it had a fundamental shortcoming it
was inconsistent with Newtons mechanics and
Maxwells electromagnetic theory. In
particular, on electron orbiting a nucleus would
represent an oscillating charge that should
radiate a broad spectrum of electromagnetic
radiation, not lines. Further, the loss of
energy by such radiation should cause the
orbiting electron to spiral into the nucleus.
As the 20th century dawned, physicists could
not explain the structure of atoms, the stability
of matter, discrete spectral lines, emission and
absorption spectra differences, or the origins of
X-rays and radioactivity.
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Niels Bohrs Atomic Model
In 1911, after receiving his PhD, Niels Bohr
went to Cambridge to work in Rutherfords
laboratory. This was about a year after
Rutherford had completed his nuclear model of the
atom. Bohr wanted to fix the model so that
the orbiting electrons would not radiate away
their energy. Starting from Einsteins idea of
light quanta, in 1913 he proposed a radically new
nuclear model of the atom that made the following
assumptions

• Atoms consist of negative electrons orbiting a
  small positive nucleus
• Atoms can exist only in certain stationary states
  with a particular setof electron orbits and
  characterized by the quantum number n 1, 2, 3,
  
• Each state has a discrete, well-defined energy
  En, with E1
• The lowest or ground state E1 of an atom is
  stable and can persist indefinitely. Other
  stationary states E2, E3, are called excited
  states.
• An atom can jump from one stationary state to
  another by emitting a photon of frequency f
  (Ef-Ei)/h, where Ei,f are the energies of the
  initial and final states.
• An atom can move from a lower to a higher energy
  state by absorbing energy in an inelastic
  collision with an electron or another atom, or by
  absorbing a photon.
• Atoms will seek the lowest energy state by a
  series of quantum jumps between states until the
  ground state is reached.

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The Bohr Model
The implications of the Bohr model are

• Matter is stable, because there are no states
  lower in energy than the ground state
• Atoms emit and absorb a discrete spectrum of
  light, only photons that match the interval
  between stationary states can be emitted or
  absorbed
• Emission spectra can be produced by collisions
• Absorption wavelengths are a subset of the
  emission wavelengths
• Each element in the periodic table has a
  different number of electrons in orbit, and
  therefore each has a unique signature of spectral
  lines.

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Example Wavelength ofan Emitted Photons
An atom has stationary states with energies
Ej 4.0 eV and Ek 6.0 eV. What is the
wavelength of a photon emitted in a quantum jump
from state k to state j?
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Energy Level Diagrams
It is convenient to represent the energy
states of an atom using an energy level diagram.
Each energy level is represented by a
horizontal line at at appropriate height scaled
by relative energy and labeled with the state
energy and quantum numbers. De-excitation photon
emissions are indicated by downward arrows.
Absorption excitations are indicated by upward
arrows.
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ExampleEmission and Absorption
An atom has only three stationary states E1
0.0 eV, E2 3.0 eV, and E3 5.0 eV. What
wavelengths are observed in the absorption
spectrum and in the emission spectrum of this
atom?
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Clicker Question 1
A photon with a wavelength of 414 nm has an
energy of 3.0 eV. For an atom with the
energy levels shown, would you expect to see a
line with this wavelength in the absorption
spectrum? In the emission spectrum?
(a) yes, yes (b) yes, no (c) no,
yes (d) no, no (e) cannot tell.
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The Bohr Hydrogen Atom
This is a version of the mathematical
formulation of the Bohr Model of the atom
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Hydrogen Atom Energy Levels
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Example Stationary States of the Hydrogen Atom
Can a electron in a hydrogen atom have a
speed of 3.60x105 m/s? If so what are its energy
and the radius of its orbit?
X
What about a speed of 3.65x105 m/s?
OK
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Binding Energy andIonization Energy
The binding energy of an electron in
stationary state n is defined as the energy that
would be required to remove the electron an
infinite distance from the nucleus. Therefore,
the binding energy of the n1 stare of hydrogen
is EB 13.60 eV. It would be necessary to
supply 13.60 eV of energy to free the electron
from the proton, and one would say that the
electron in the ground state of hydrogen is
bound by 13.60 eV.
The ionization energy is the energy required
to remove the least bound electron from an atom.
For hydrogen, this energy is 13.60 eV. For other
atoms it will typically be less.
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Quantization ofAngular Momentum
Actually, in constructing his atomic model,
Niels Bohr did not assume that an integer number
of de Broglie wavelengths fitted into the
circumference of the orbit. (Bohr did not know
about de Broglie waves when he formulated his
theory.) Instead, he assumed that the angular
momentum L of the orbit was quantized in units of
h (h/2p).
Thus, Bohrs assumption that angular
momentum is quantized in units of h is equivalent
to assuming that an integer number of de Broglie
wave lengths of the electron fits into the
circumference of a Bohr orbit.
Where does Lnh come from? It comes from
the symmetry that if you rotate an object by
360O, it should return to the same state. This
symmetry produces an angular momentum condition
on the electrons wave function.
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Clicker Question 2
What is the quantum number n of the electron
orbit in this hydrogen atom?
(a) 2 (b) 3 (c) 4 (d) 6 (e)
12.
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The Hydrogen Spectrum
The figure shows the energy-level diagram
for hydrogen. The top rung is the ionization
limit, which corresponds to n?8 and to completely
removing the electron from the atom. The higher
energy levels of hydrogen are crowded together
just below the ionization limit. The arrows
show a photon absorption 1?4 transition and a
photon emission 4?2 transition.
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Transitions
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Example The Lyman a Forest
When astronomers look at distant quasars,
they find that the light has been strongly
absorbed at the local wavelength of the Lyman a
1?2 transition of the Lyman series of hydrogen.
This absorption tells us that interstellar space
is filled with vast clouds of hydrogen left over
from the Big Bang. What is the wavelength of
the Lyman a 1?2 absorption line in hydrogen?
This wavelength is in the ultraviolet.
However, the cosmic recession velocity of quasars
Doppler shifts the Lyman a line into the visible,
and because the absorption occurs at many
distances, a forest of absorption line is
observed.
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Hydrogen-like Atoms
The Bohr model also works well for
hydrogen-like atoms, i.e., atoms with Z protons
in the nucleus and only one orbital electron.
For such atoms, the orbits are shifted by the
increased Coulomb force
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Success and Failure
Bohrs analysis of the hydrogen atom was a
resounding success. By introducing stationary
states, together with Einsteins ideas about
light quanta, Bohr was able to explain the
stability of atoms, provide the first solid
understanding of discrete spectra, and justify
the Balmer formula. However, there were
problems. The model failed when two or more
electrons were in orbit. The Bohr model could
not predict even the spectrum of helium, an atom
with two electrons orbiting a charge-2 mass-4
nucleus. Something in Bohrs assumptions worked
correctly for a single electron but failed when
two or more electrons were involved. It is
important to distinguish between the Bohr Model,
which assumes that stationary states exist, and
the Bohr hydrogen atom, which gives concrete
expressions for such stationary states. The
problem for multi-electron atoms was not Bohrs
model, but his method for finding the stationary
states. The missing technique, which was
developed in the mid-1920s, was quantum mechanics.
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Five Things You Should Have Learned from This
Lecture

• Niels Bohr proposed a model for explaining the
  stability of atoms in terms of stationary states
  and quantum jumps between states.
• In the Bohr model for hydrogen, orbits are
  allowed only if an integer number of de Broglie
  wavelengths of the orbiting electron fit into the
  circumference of the orbit.
• The Bohr model predicts the ionization potential
  and spectrum of Hydrogen, and gives justification
  to the Balmer formula.
• The Bohr model also predicts the observable
  properties of hydrogen-like atoms, i.e., heavier
  atoms with a single electron.
• The Bohr model fails to predict the spectra and
  properties of atoms with more than one electron.

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Chapter 38 - Summary (1)
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Chapter 38 - Summary (2)
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Chapter 38 - Summary (3)
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End of Lecture 23

• The Final Exam will be in this room at 830 AM
  on Monday, June 6. (NOT at 1030 AM!) You may
  bring to the Final 3 pages of notes. Also bring
  a Scantron sheet and a calculator with good
  batteries.
• You are required to complete the Tycho course
  survey by 1159 PM on June 5. Please complete it
  as soon as possible
• On Friday we will have a comprehensive Review of
  the material covered this quarter.
• Look up your assigned seat on Tycho before
  coming to the Final.