Final Exam

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Final Exam

• Comprehensive
• Date Tuesday May 4, 2004
• Location MSC 220
• Time 6-8 pm

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27.6 Photons and Electromagnetic Waves

• Light has a dual nature. It exhibits both wave
  and particle characteristics
• Applies to all electromagnetic radiations
• The photoelectric effect and Compton scattering
  offer evidence for the particle nature of light
• When light and matter interact, light behaves as
  if it were composed of particles
• Interference and diffraction offer evidence of
  the wave nature of light

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Possible Application of the Particle Feature in
Space

• Sunlight striking the sail creates a force that
  pushes the spaceship away from the sun, much as
  the wind propels a sail-boat.

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27.7 Wave Properties of Particles

• In 1924, Louis de Broglie postulated that because
  photons have wave and particle characteristics,
  perhaps all forms of matter have both properties
• Furthermore, the frequency and wavelength of
  matter waves can be determined

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Matter waves The de Broglie Wavelength and
Frequency

• Since a photon travels with speed of light, we
  treat it as a massless particle with the
  momentum
• pE/chf/ch/l (Photons only m0!)
• Wavelength of the matter wave 
• lh/ph/(mv) (Classical particles m?0!) 
• A particle has a wave with frequency fE/h
  associated with it!

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The Davisson-Germer Experiment

• In 1927, Davisson and Germer, working at Bell
  Labs, were studying the nature of the surface of
  a nickel crystal by directing a beam of electrons
  at the surface and observing the electrons
  reflected at various angles. They found that the
  electrons were reflected in almost the same way
  that x-ray would be reflected. The results gave
  strong support to de Broglies hypothesis.

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The Transmission Electron Microscope

• The electron microscope depends on the wave
  characteristics of electrons
• Microscopes can only resolve details that are
  slightly smaller than the wavelength of the
  radiation used to illuminate the object
• The electrons can be accelerated to high energies
  and have small wavelengths

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27.8 The Wave Function

• In 1926 Schrödinger proposed a wave equation that
  describes the manner in which matter waves change
  in space and time
• Schrödingers wave equation is a key element in
  quantum mechanics
• Schrödingers wave equation is generally solved
  for the wave function, ?

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Wavefunction, cont.

• In quantum mechanics ???2 is proportional to the
  probability of finding the particle at a given
  location.

The probability of finding the ground state
hydrogen electron (n1) as a function of the
radial distance from the proton.
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The Wave Function, final

• The wave function depends on the particles
  position and the time
• The value of ???2 at some location at a given
  time is proportional to the probability of
  finding the particle at that location at that time

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27.9 The Uncertainty Principle
Moving car
Past
Future
-x
x
Now
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The Uncertainty Principle, cont.

• What means exact? ? Deterministic view of nature
• ? This is ok in our world! But not in the world
  of the electron

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The Uncertainty Principle, cont.

• (a) A photon hits an electron and (b) transfers
  momentum to the electron.

The observation "destroys" the phenomen
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The Uncertainty Principle, cont.

• It is impossible to know (or to measure)
  simultaneously an electrons EXACT position and
  momentum!
• Erwin Schroedinger (1887-1961) and Werner
  Heisenberg (1901-1976) independently developed a
  new branch of physics called quantum mechanics.
  Quantum theory (its a small world!) predicts
  limits on the accuracy of measurements. This idea
  was introduced in 1927 by Heisenberg and
  complemented Schroedingers wave theory.

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The Uncertainty Principle, cont.
To see more clearly into the nature of
uncertainty, we consider electrons passing
through a slit
Momentum uncertainty in the y component
We apply the formula for single slit diffraction,
sin?l/W, and postulate that l is the de Broglie
wavelength.
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The Uncertainty Principle, cont.

• sinq?tanq (for small q)
• tanqDpy/px ? Dpy/px?l/W

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The Uncertainty Principle, cont.

• Since the electron can pass the slit through
  anywhere over the width W, the uncertainty in the
  y position of the electron is DyW.

• DpyDy?h

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The Uncertainty Principle, cont.

• Heisenberg found that at the very best
• DpyDy?h/(4p)

Momentum
Position
Uncertainty
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The Uncertainty Principle, cont.

• Dy/c?l/c ? Dt?l/c (This is the time that the
  particle needs to traverse its own uncertainty,
  which is on the order of its wavelength)  
• DE?hc/l
• (DE)(Dt)?(hcl)/(lc)
• (DE)(Dt)?h

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The Uncertainty Principle, final
(DE)(Dt)?h/(4p)
Energy uncertainty of a particle when the
particle is in a certain state
Time interval during which the particle is in the
state
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• Example Assume the position of an object is
  known so precisely that the uncertainty in the
  position is only Dy1.5?10-11 m. Determine the
  minimum uncertainty in the momentum of the object
  and find the corresponding minimum uncertainty in
  the speed if the object in an electron.
• Dpyh/(4pDy)(6.63?10-34 Js)/(4p1.5?10-11 m)
• Dpy3.5?10-24 kgm/s ? small
• DvyDpy/m(3.5?10-24 kgm/s)/(9.1?10-31 kg)
• Dvy3.9?106 m/s ? large

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Scanning Tunneling Microscope (STM)

• Allows highly detailed images with resolution
  comparable to the size of a single atom
• A conducting probe with a sharp tip is brought
  near the surface
• The electrons can tunnel across the barrier of
  empty space

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Scanning Tunneling Microscope, cont.

• By applying a voltage between the surface and the
  tip, the electrons can be made to tunnel
  preferentially from surface to tip
• The tip samples the distribution of electrons
  just above the surface
• The STM is very sensitive to the distance between
  the surface and the tip
• Allows measurements of the height of surface
  features within 0.001 nm

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Limitation of the STM

• There is a serious limitation to the STM since it
  depends on the conductivity of the surface and
  the tip
• Most materials are not conductive at their
  surface
• An atomic force microscope (AFM) has been
  developed that overcomes this limitation
• It measures the force between the tip and the
  sample surface
• Has comparable sensitivity