Quantum Mechanics 102

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Quantum Mechanics 102

• Tunneling and its Applications

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Interference of Waves and the Double Slit
Experiment

• Waves spreading out from two points, such as
  waves passing through two slits, will interfere

l
Wave crest Wave trough
Spot of constructive interference Spot of
destructive interference
d
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Interpretation

• The probability of finding a particle in a
  particular region within a particular time
  interval is found by integrating the square of
  the wave function
• P (x,t) ? Y(x,t)2 dx ? c(x)2 dx
• c(x)2 dx is called the probability density
  the area under a curve of probability density
  yields the probability the particle is in that
  region
• When a measurement is made, we say the wave
  function collapses to a point, and a particle
  is detected at some particular location

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Particle in a box

• c(x) B sin (npx/a)

n3
c(x)
c(x)2
n2

• Only certain wavelengths l 2a/n are allowed
• Only certain momenta p h/l hn/2a are allowed
• Only certain energies E p2/2m h2n2/8ma2 are
  allowed - energy is QUANTIZED
• Allowed energies depend on well width

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What about the real world?

• Solution has non-trivial form, but only certain
  states (integer n) are solutions
• Each state has one allowed energy, so energy is
  again quantized
• Energy depends on well width a
• Can pick energies for electron by adjusting a

c(x)2
n2
n1
x
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Putting Several Wells Together

• How does the number of energy bands compare with
  the number of energy levels in a single well?
• As atom spacing decreases, what happens to energy
  bands?
• What happens when impurities are added?

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Quantum wells

• An electron is trapped since no empty energy
  states exist on either side of the well

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Escaping quantum wells

• Classically, an electron could gain thermal
  energy and escape
• For a deep well, this is not very probable

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Escaping quantum wells

• Thanks to quantum mechanics, an electron has a
  non-zero probability of appearing outside of the
  well
• This happens more often than thermal escape

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What if free electron encounters barrier?

• Do Todays Activity

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What Have You Seen?

• What happens when electron energy is less than
  barrier height?
• What happens when electron energy is greater than
  barrier height?
• What affects tunneling probability?
• T ? e2kL
• k 8p2m(Epot E)½/h

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A classical diode

• According to classical physics, to get to the
  holes on the other side of the junction, the
  conduction electrons must first gain enough
  energy to get to the conduction band on the
  p-side
• This does not happen often once the energy
• barrier gets large
• Applying a bias increases
• the current by decreasing
• the barrier

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A tunnel diode

• According to quantum physics, electrons could
  tunnel through to holes on the other side of the
  junction with comparable energy to the electron
• This happens fairly often
• Applying a bias moves the
• electrons out of the p-side
• so more can tunnel in

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Negative resistance

• As the bias is increased, however, the energy of
  the empty states in the p-side decreases
• A tunneling electron would then end up in the
  band gap - no allowed energy

• So as the potential difference is increased, the
  current actually decreases negative R

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No more negative resistance

• As bias continues to increase, it becomes easier
  for conduction electrons on the n-side to
  surmount the energy barrier with thermal energy
• So resistance becomes positive again

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The tunneling transistor

• Only electrons with energies equal to the energy
  state in the well will get through

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The tunneling transistor

• As the potential difference increases, the energy
  levels on the positive side are lowered toward
  the electrons energy
• Once the energy state in the well equals the
  electrons energy, the electron can go through,
  and the current increases.

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The tunneling transistor

• The current through the transistor increases as
  each successive energy level reaches the
  electrons energy, then decreases as the energy
  level sinks below the electrons energy

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Randomness

• Consider photons going through beam splitters
• NO way to predict whether photon will be
  reflected or transmitted!

(Color of line is NOT related to actual color of
laser all beams have same wavelength!)
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Randomness Revisited

• If particle/probabilistic theory correct, half
  the intensity always arrives in top detector,
  half in bottom
• BUT, can move mirror so no light in bottom!

(Color of line is NOT related to actual color of
laser all beams have same wavelength!)
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Interference effects

• Laser light taking different paths interferes,
  causing zero intensity at bottom detector
• EVEN IF INTENSITY SO LOW THAT ONE PHOTON TRAVELS
  THROUGH AT A TIME
• What happens if I detect path with bomb?

No interference, even if bomb does not detonate!
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Interpretation

• Wave theory does not explain why bomb detonates
  half the time
• Particle probability theory does not explain why
  changing position of mirrors affects detection
• Neither explains why presence of bomb destroys
  interference
• Quantum theory explains both!
• Amplitudes, not probabilities add - interference
• Measurement yields probability, not amplitude -
  bomb detonates half the time
• Once path determined, wavefunction reflects only
  that possibility - presence of bomb destroys
  interference

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Quantum Theory meets Bomb

• Four possible paths RR and TT hit upper
  detector, TR and RT hit lower detector
  (Rreflected, Ttransmitted)
• Classically, 4 equally-likely paths, so prob of
  each is 1/4, so prob at each detector is 1/4
  1/4 1/2
• Quantum mechanically, square of amplitudes must
  each be 1/4 (prob for particular path), but
  amplitudes can be imaginary or complex!
• e.g.,

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Adding amplitudes

• Lower detector
• Upper detector

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What wave function would give 50 at each
detector?

• Must have a b c d 1/4
• Need a b2 cd2 1/2