Quantum Mechanics 101.5

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Quantum Mechanics 101.5

• Wave-Particle Duality and the Wavefunction

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Review of Last Time

• Quantum mechanics is AWESOME, but it challenges
  our physical intuition
• Light and particles behave like waves when
  traveling and like particles when interacting or
  being observed
• Since they propagate like waves, both light and
  particles can produce interference patterns
• We can describe this duality through the use of a
  wave function Y(x,t) which describes the
  (unobserved) propagation through space and time

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Application particle in a box

• If a particle is confined to a region by
  infinitely-high walls, the probability of finding
  it outside that region is zero.
• Since nature is generally continuous (no
  instantaneous changes), the probability of
  finding it at the edges of the region is zero.
• The position-dependent solution to the
  Schrödinger equation for this case has the form
  of a sine function
• c(x) B sin (npx/a) (awell width)

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More 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?

• So confinement yields quantized energies
• In the real world, infinitely high wells dont
  exist
• Finite wells, however, are quite common
• Schrödinger equation is slightly more
  complicated, since Ep is finite outside well
• Solution has non-trivial form (trust me)

c(x)2
n2
n1
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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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What have we learned today?

• Integrating the square of the wave function over
  a region gives us the probability of finding the
  object in that region
• A particle confined to an infinitely-high box
  is described by a wave function of a sine.
• particles in finite wells or in atoms are
  described by more complicated wave functions
• All three situations result in quantized (only
  certain values allowed) energies