Dilemma

1
Dilemma

• Existence of quanta could no longer be questioned
• e/m radiation exhibits diffraction gt wave-like
• photoelectric Compton effect gt localized
  packets of energy gt particle-like
• wave-particle duality

2
Wave properties
Photon detector clicks when a photon is absorbed
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Probability Waves

• Photon detector might be a photoelectric device
• at any point, the clicks will be randomly spaced
  in time
• cannot predict when a photon will be detected at
  any point on the screen
• if we move the detector, the click rate increases
  near an intensity maximum
• the relative probability that a single photon is
  detected at a particular point in a specified
  time ? Intensity at that point
• intensity ? Em2 Probability of detection
  ? Em2

4
Probability Wave

• Probability (per unit time) that a photon is
  detected in some small volume is proportional to
  the square of the amplitude of the waves
  electric field in that region
• Postulate that light travels not as a stream of
  photons but as a probability wave
• photons only manifest themselves when light
  interacts with matter
• photons originate in the source that produces the
  light wave (interaction)
• photons vanish on the screen (interaction)

5
Single-photon version (1909)

• Light source is so weak that it emits only one
  photon at a time at random intervals
• interference fringes still build up
• raises the question if the photons move through
  the apparatus one at a time, through which slit
  does the photon pass?
• How does a given photon know that there is
  another slit?
• Can a single photon pass through both slits and
  interfere with itself?

6
Electrons and Matter Waves

• Light is a wave but can transfer energy and
  momentum to matter in photon sized lumps
• can a particle have the same properties?
• can it behave as a wave?
• a matter wave
• de Broglie (1924) suggested ?h/p
• de Broglie
  wavelength
• a beam of electrons has a wavelength and should
  diffract if slit width comparable to ?h/p
• 1927 Davisson and Germer observed diffraction of
  electrons from crystals

7
Diffraction

• Recall for a single slit, the diffraction minima
  are at a sin? m?
• if ? ltlta, then all ?m?/a are small and the
  light essentially travels as a ray and does not
  spread out
• need ? ? a for strong diffraction effects
• eg. 1 kg billiard ball moving at 5.0 m/s
• ?h/p h/mv (6.63x10-34 J.s)/(1kg)(5.0m/s)
  1.3 x 10-34 m ltlt size of atoms
• gt no diffraction

8
Diffraction

• eg. Electrons with kinetic energy of 54.0 eV
• if Kp2/2me , then p(2meK)1/2
  (2 x 9.11x10-31 kg x 54eV x 1.6x10-19
  J/eV)1/2 3.97 x 10-24 kg. m/s
• ?h/p (6.63x10-34 J.s)/(3.97x10-24 kg.m/s)
  1.67 x 10-10 m .167 nm
• smaller mass gt larger ?
• typical atom has diameter of 10-10 m

9
Diffraction

• regular arrays of atoms (crystals) should
  diffract electrons!
• Neutrons can also diffract gt study structures of
  solids and liquids

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Double Slit Experiment with electrons (1989)
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(No Transcript)
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X-ray beam (light wave)
Electron beam (matter wave)
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Problem

• Singly charged Na ions are accelerated through a
  potential difference of 300 V. What is (a) the
  momentum acquired? (b) what is the de Broglie
  wavelength?
• Solution K qV (1.6x10-19 C)(300V)
  4.80 x10-17 J 300 eV
• m(22.9898 g/mole)/(6.02x1023 atom/mole)
  3.819 x10-23 g 3.819 x 10-26 kg

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Solution

• (a) p (2mK)1/2 2(3.819x10-26)(4.8x10-17)1/2
  1.91 x10-21 kg.m/s
• (b) ? h/p (6.63x10-34 J.s)/(1.91x10-21
  kg.m/s) 3.46 x10 -13 m

15
What is Waving?

• If particles behave as waves, what is waving?
• Wave on a string gt particles in string execute
  SHM
• sound wave in air gt air molecules oscillate in
  SHM
• light wave gt electric and magnetic fields
  oscillate
• e.g. E(x,y,z,t) electric field varies from
  place to place and with time
• intensity ? E2
• what varies from place to place for a matter
  wave?
• Wave function ?(x,y,z,t) psi

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Schrödinger Equation

• Schrödinger Equation 1926 H?E?
• ?(x,t) is a solution of this equation
• the wave equation for matter waves
• probability waves
• probability density is P(x,t) ?(x,t)
  ?(x,t) ?(x,t)2

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A harmonic wave has a definite value of k but
extends to infinity
A wave packet has a spread of k- values but is
localized
?k.?x ? 1
??.?t ? 1
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Example

• Radar transmitter emits pulses of electromagnetic
  radiation which last 0.15 ?s at a wavelength of ?
  1.2 cm
• (a) to what central frequency should the radar
  receiver be set?
• (b) what is the length of the wave packet?
• (c ) how much bandwidth should the receiver have?
• (a) f0 c/?0 3 x 108 m/s /1.2 x 10-2 m 26
  GHz
• (b) ?x c?t (3 x 108 m/s)(.15 x 10-6 s) 45 m
• (c ) ?f ??/2? 1/(2??t) 1/(6.28 x .15 x 10-6
  s) 1.1 MHz