Quantum Theory Chapter 5

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Quantum TheoryChapter 5
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Lecture Objectives

• Indicate what is meant by the duality of matter.
• Discuss the wavelike nature of matter as proposed
  by De Broglies Theory.
• List one way that matter acts in a manner that
  reveals its wavelike nature.
• State Heisenbergs Uncertainty Principle as it
  relates to atoms.
• Indicate what information Schroedingers Wave
  Equation gives us about the electron.

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Wave-Particle Duality

• Recall the Bohrs model only accurately predicted
  the emission spectrum of hydrogen.
• In the 1920s, De Broglie proposed 2 radical
  ideas
• The possible circular orbits of an electron are
  limited to whole numbers of complete wavelengths.
• If light, a wave, could also take on
  particle-like properties, couldnt the electron,
  a particle, also take on wave-like properties?

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The Standing Waves Caused by the Vibration of a
Guitar String Fastened at Both Ends
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The Hydrogen Electron Visualized as a Standing
Wave Around the Nucleus
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Wave-Particle Duality

• For these ideas to be true, the electron is
  therefore allowed only certain possible
  wavelengths, frequencies, and energies.
• De Broglies Equation
• l h
• mv
• This equation predicts the wavlength of a
  particle of mass (m) moving at velocity (v).

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Wave-Particle Duality

• i.e., all moving particles have wave
  characteristics.
• N.B. Normal objects dont have waves detectable
  with even very sensitive objects, but an electron
  has a wavelength of 2.9 x 10-5 m, which is easily
  measurable by experimentation.

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Electron Interference

• Richard Feynman liked to talk about wave-particle
  duality with the following analogy
• Imagine shooting a machine gun at an iron plate
  with two slots in it. If there were a concrete
  wall behind the iron plate, what kind of pattern
  do you think the bullets would make?
• Electron Gun Experiment
• http//www.colorado.edu/physics/2000/schroedinger/
  two-slit3.html

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G. P. Thomson

• Also in 1927, G. P. Thomson, the son of J. J.
  Thomson, reported his experiments, in which a
  beam of energetic electrons was diffracted by a
  thin foil.
• This is constructive-destructive interference,
  just like we saw with light!

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Getting the Prize

• Experiments by Davisson, Germer, and Thomson
  proved that de Broglie's waves are not simply
  mathematical conveniences, but have observable
  physical effects. They got the 1937 Nobel Prize
  in Physics for their pioneering work.

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The Heisenburg Uncertainty Principle

• It is fundamentally impossible to know precisely
  both the velocity and the position of a particle
  at the same time.

• x position
• mv momentum
• h Plancks constant
• The more accurately we know a particles
  position, the less accurately we can know its
  momentum.

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The Heisenburg Uncertainty Principle

• For large objects, the change in velocity
  produced by determining the position is so small
  that we can ignore it.
• This is why scientists, in the 1920s, found
  Heisenburgs idea so difficult to accept.

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The Heisenburg Uncertainty Principle

• For atomic particles, however, it is profound, as
  the uncertainty produced is on the order of 10-9
  m, an uncertainty one order of magnitude greater
  than the diameter of the entire atom.
• E.g. Using a photon to bump an electron in
  order to determine its location will cause an
  excitation of the electron that, according to
  quantum mechanics, will change its orbit and,
  therefore, both its wavelenth, velocity, and
  position.

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The Shroedinger Wave Equation

• In 1926, the Austrian physicist Erwin
  Schroedinger derived Bohrs equation for
  hydrogens electron by assuming it is a wave.
• It accurately predicted hydrogens energy levels!
• The Schrodinger equation is used to find the
  allowed energy levels of quantum mechanical
  systems (such as atoms, or transistors). The
  associated wavefunction gives the probability of
  finding the particle at a certain position.

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The Quantum Model

• This works because, while matter has duality,
  just like light, if you perform an experiment to
  see where a particle is, then you find something
  particle-like. But otherwise it's a wave that
  carries information about where the electron
  probably is.

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The Quantum Model

• Until you check where the electron is, it's
  really just a wave.
• Not only that, but Schrödinger showed that these
  electrons don't even move. The waves are
  stationary.
• Each time you check where an electron is you will
  find it in a different place, but that doesn't
  mean it's moving in between checks.
• For some energy levels, if you check position
  enough times you may see an "orbit-like" pattern,
  but the electron isnt actually moving in little
  circles.

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The Quantum Model

• An electron isn't in any particular place when
  you aren't looking because then it is a wave.
• Generally, for most physics we only care about
  how much energy it has, not where it is.
• Orbits, while misleading about where the electron
  is, do tell us how much energy it has.

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The Quantum Model

• We call this the Energy Level of the electron.
  Because the idea of orbits is so misleading,
  physicists started using a picture of the atom
  which just showed energy levels as relative
  heights.
• And we called this the "Schrödinger Model," of
  course.
• Born, a German-born Jewish physicist, coined the
  term quantum mechanics, which replaces
  Newtonian classical mechanics in explaining the
  behavior of matter at the atomic level.
• He was the person who discovered that the
  wavefunction in Schroedingers equation was a
  probability density function for the electron