Physics II

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Physics II

• DeBroglie waves

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Reference

• Max Born, Atomic Physics, Dover, 1969

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A prince of a fellow

• Prince Debroglies (1929) Nobel lecture

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An extraordinary thesis
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Motivations

• By 1925, the year of DeBroglies work
• The wave-particle duality of light was well known
  (but still inexplicable)
• Einsteins Special Theory of Relativity was
  well-known

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DeBroglie leaned heavily on Einsteins work

• Energy of photon E hf
• Invariance of 4-vectors, esp. energy-momentum 4
  vector
• (p, E)
• p2c2 E2 invariant for a given system
• Assign same dualism to particles that had been
  attached to light

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Reasoning by analogy

• For photon, E hf, so assign a frequency to a
  particle with energy E in the same fashion.
• To be consistent with special relativity we argue
  that we can assign momentum by phk
• Treat space and time on equal footing
• Since f is number of vibrations in unit time,
  then k must be the number of waves per unit
  length
• k 1/?
• Then p h / ?
• The famed DeBroglie wavelength

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One-dimension DeBroglie waves

• Plane wave of momentum p and energy E moving
  along the x axis
• U(x,t) Aexp2pi(Et px)/h
• U(x,t) A exp2pi(ft kx)

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Phase velocity

• The locus of waves of constant phase moves at
  the phase velocity
• Constant phase ft kx constant
• d (ft kx) / dt d (constant) / dt
• f kdx/dt 0
• dx / dt f/k ? / T f ?
• Since frequency often depends on wavelength these
  waves show dispersion (ala white light passing
  through a prism)

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Phase velocity is artificial

• Phase velocity is a purely artificial concept
  since we cannot measure it (a positivist view)
• We cannot mark an infinite plane wave of fixed
  frequency w/o superimposing another wave train on
  the plane wave to make a bump on it which we
  can follow
• The bump moves at the group velocity

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Group velocity

• To the original wave we literally add a similar
  wave of slightly different frequency and
  wavelength (f and ?)
• We will end up with a wave which has an amplitude
  that varies ala beat notes used in tuning
  guitars, etc.
• We follow the maximum of this varying amplitude
  as our marker

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Results

• The resulting wave of superpositions moves at a
  phase velocity of (f f)/(k k)
• If f and k are very near f and k, this is close
  to the original phase velocity f / k
• The group (beat maximum) moves at a speed (f
  f) / (k k)
• As superposed wave approaches identicality with
  original wave, this becomes df / dk
• df /dk df / d(1/) - (1 / ? )2 df / d ?

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Hokey justification to Bohrs quantum condition

• Fit integral number of deBroglie waves into
  circumference of allowed circular orbits
• To assure constructive interference
• n ? 2pr
• Plug in DeBroglie
• nh/p 2pr
• nh / (mv) 2pr
• L mvr n (h / 2p )

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Davison-Germer and GP Thomson

• These gentlemen performed experiments
  demonstrating interference of electrons, as close
  as one can get to nailing down a wave nature of
  something.
• GP got Nobel prize for showing electron is a wave
• His daddy JJ won prize years earlier for showing
  electron is a particle
• Sic transit gloria mundi

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Group velocity
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