Lecture

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Lecture 17Review - Quantum Mechanics The
Hydrogen Atom

• March, 29th

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• The bundles of energy later came to be called
  photons.
• They are particle-like. The energy cannot be
  subdivided to a value less than hf.
• But Interference experiments indicate that light
  is a wave. Example The Young 2-slit experiment.

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  Duality Duality is the modern view of light
(and also of all radiation). Light acts as both
a wave and a particle. The experiment performed
on light dictates which aspect is
revealed. Example The photoelectric effect
reveals the particle like (photon) aspect. The
diffraction experiment reveals the wave-like
aspect.  

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•  
•  

  Matter Waves The Wave Nature of Matter Louis
de Broglie (1923) proposed that particles also
have duality. They can behave like a wave, or
like a particle. He proposed All particles also
have a wavelength ?. This wavelength displays
the wave-aspects. He postulated that the value of
the wavelength is ? h/mv. Here, v the
particle speed. Can also re-write this relation
as ? h/p. Here, p is defined by p mv,
and is called the momentum of the particle.
     
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Experimental Confirmation Electron
diffraction was observed in 1925 by Davisson and
Germer. A crystal served to diffract the
electrons. No explanation, but this Matter waves
exist! Do electron diffraction demo. Later,
precise experiments confirmed the de Broglie
relation for ?. Conclusion Duality
holds for both matter and radiation.  
 
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• Light from Gases
• What happens in the tube containing the gas?
• There is a large electric potential difference
  across the tube.
• Electrons rush across tube and collide with gas
  atoms.
• Energy is transferred to the gas atoms.
• They then de-excite.
• In this de-excitation process, the atomic energy
  is turned into light.

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• Explanation of the discrete frequencies
• We know, from the Einstein Planck relation, E
  hf, that photon energies are discrete.
• The energy in a beam of light can only take on
  the values hf, 2hf, 3hf, ..etc.
• The energy cannot be in-between these values.
• But energy is conserved!
• Conclude The atom energies must also be
  discrete.
• The word quantized is often used, when a
  physically measurable quantity can only take on
  discrete values.

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• Summary So Far
• Values of energy for both radiation and matter
  are quantized.
• Quantum mechanics is born as a replacement for
  classical mechanics.
• Need a model for atoms that can explain how its
  energy levels are discrete.

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• The First Step
• The Bohr model of hydrogen atom (1913) was the
  first step, in understanding the unique
  properties of the energy values for atoms.
• Assumptions
• Electron circles the proton. But, only certain
  discrete values of the radius r are allowed.
• Each orbit forms a state.
• Bohrs Key Postulate
• These states satisfy
• L nh/2p, where n is an integer.
• n 1, 2, 3, .etc.
• Here, L angular momentum mvr for a circle.
  v speed of electron, r orbit radius.

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Results of Bohr    The only allowed radii r are r
rn.   Here, rn n2r1.   r1 is called the
Bohr radius.   The value of r1 is  
h2/4p2(mke2) 0.529 x 10-10 meters.   k is
Coulombs Constant. e is the electron charge.  
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How do we make Polarized Light?
II. Reflection -
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The Photoelectric Effect

• The phenomenon that when light shines on a metal
  surface, electrons are emitted

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Photoelectric Effect
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Photoelectric Effect
Most commonly observed phenomena with light can
be explained by waves. But the photoelectric
effect suggested a particle nature for light.
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• Results of Bohr
• Apply Newtons 2nd Law and Coulombs Law to
  show
• The only allowed radii r are r rn.
• Here, rn n2r1.
• r1 is called the Bohr radius.
• The value of r1 is
• h2/4p2(mke2) 0.529 x 10-10 meters.
• k is Coulombs Constant.
• e is the electron charge.

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• Allowed values for the energy of the hydrogen
  atom
• The allowed energy values En are found to be
• En E1/n2
• Here, E1 is the lowest energy allowed.
• Its value is
• E1 -e2/(2kr1) -13.6 eV.
• This lowest allowed energy is known as the
  ground-state energy.
• Note The electron-volt is a convenient unit for
  the energy, when discussed atoms or molecules.
  One definition that can be used is the
  conversion-factor
• 1eV 1.6 x 10-19 Joules.

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• Picture of the energy values
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• Recall The general result is En E1 /n2 .
• The lowest energy (ground-state, n 1)
  corresponds to the smallest orbit radius
• r1 0.529 Angstroms.
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• The Angstrom is a convenient length-unit for
  atoms. Its definition is
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• 1 Angstrom 10-10 meters.
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The second-smallest energy is obtained by setting
n 2 in En E1/n2 (first excited
state)   E2 E1/22 E1/4. Its numerical
value is   E2 -3.40 eV.   The orbit-radius
is r2 22r1 4 r1 2.16 Angstroms.
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• Energy Level Diagram
• Region of Unbound States
• _______ 0eV (n infinity).
• ________ -3.40 eV (n 2).
• ________ -13.6 eV (n1).
• In this diagram, energy values are plotted on an
  upward scale.
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•  

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• Ionization
• At n infinity, the energy is zero, and the
  electron is freed from the proton. 
• The ionization energy (also called the binding
  energy) is defined as the energy to liberate the
  electron, if the system is initially in its
  ground state.
• So, for hydrogen, the ionization energy is  
• E (at infinity) - E1 0 (-13.6) eV 13.6
  eV.
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  Quiz 3  1. What is the energy of photons,
if their wavelength is 515 nm?2. What is the
color of light in Question 1?3. The work
function of potassium is 2 eV. 1 eV1.6 10-19
J. Blue light with a wavelength of 400 nm is
shined on a potassium plate. A. What is the
energy of incident photons?B. Are the electrons
ejected off?C. If so, what is their kinetic
energy in Joules?