Quantum Mechanics, part 3 Trapped electrons

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Quantum Mechanics, part 3Trapped electrons
Confinement leads to quantization

• Infinite Potential Well
• Finite Potential Well
• Quantum Traps
• Nanocrystallites
• Quantum Dots
• Quantum corrals
• 2-D and 3-D Traps
• Hydrogen Atom
• Bohr Theory
• Solution to Schrödinger Equation

A quantum corral of iron atoms
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Electron Trap
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Energy Level Diagrams---DISCRETE LEVELS NOT
CONTINUOUS!!!!!!!!!!!11
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Particle in a Box by analogy(Infinite Potential
Well)
Standing waves in a string

• Classically - any energy and momentum just like a
  free particle

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Particle in a Box

• QM - Boundary conditions for the matter wave

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Particle in a Box
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Introduction to Wave Mechanics (review)

• The wave function
• Interpretation - Probability function and density
  
• Normalization
• Probability of locating a particle
• Expectation value

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Infinite Potential Well Boundary conditions are
everything!!!!!!!!!!!!!!Solution using
Schrödinger Wave Equation
The general solution is that of SHM i.e.,
U 0 inside the well and everywhere else, so
y 0 if x lt 0 or x gt L.
Apply the boundary Conditions at x 0
Also
requires
where
or
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Determining the constant A in the Infinite
Potential Well Solution using Schrödinger Wave
Equation example prob 39-2
Normalize the probability
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Infinite Potential Well Solution using
Schrödinger Wave Equation
or
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Infinite Potential Well Solution using
Schrödinger Wave Equation
Verify that the above is a solution to the
differential equation.
Why isnt n 0 a valid quantum number?
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Infinite Potential Well Solution using
Schrödinger Wave Equation
Energy level transitions
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Particle Finite Potential Well
Regions of the potential well
Matter wave leaks into the walls. For any
quantum state the wavelength is longer so
the corresponding energy is less for the finite
well than the infinite trap/well.
Wave function and probability functions
Energy level diagram for L 100 pm and Uo 450
eV
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Finite Well Cont.

• Given U0450 eV, L100 pm
• Remove the portion of the energy diagram of the
  infinite well above E450 eV and shift the
  remaining levels (three in this case) down.

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Examples of quantum electron traps
Nanocrystallites
A quantum corral of iron atoms
Quantum Dot
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2 D and 3 D rectangular corrals
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Simple Harmonic Oscillator
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The Nature of the Nuclear Atom

• Rutherford 1911 (w/grad students Geiger and
  Marsden) Scattering
• Scattering alpha particles from gold foil
• Some alphas bounced back as if a cannonball
  bouncing off tissue paper
• Established the nuclear atom
• Electron outside a very small positive nucleus
• Classical theory leads to contradiction

An electron would spiral into the nucleus in a
time
AAargh.
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Electrons are trapped by the Nucleus

• Could the energy states be discrete?
• Stability of the atom is due to quantization of
  energy much like the trapped electron in the
  finite well!!!!
• Bohr postulates that angular momentum and thus
  energy is quantized in units of Planck's constant
  
• There is a hint from the signature of atomic
  spectrathis weeks lab.

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Hydrogen Line Spectra
Johannes Balmer 1897 Balmer Series Spectrum
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Atomic Line Spectra

• Rydberg Formula

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Bohr Model of the Atom (1913)

• Semiclassical nuclear model
• Assumes Electrostatic Forces
• Stationary Orbits hypothesized

Note
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Bohr Model of the Atom
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Bohr Model of the Atom
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The Bohr Model and Standing Electron
Waves(Arthur Sommerfeld)
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Results

• Consistent with basic Hydrogen spectrum
• Explains origin of photons
• Fails to explain more complex spectra and fine
  points of Hydrogen spectrum

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The Solution to the Schrödinger Equation for
Hydrogen
Solution is the product of 3 functions

• Predicts 3 quantum numbers n,l,ml
• Successfully describes atomic spectra

Note
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The Solution to the Schrödinger Equation for
Hydrogen
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The Solution to the Schrödinger Equation for
Hydrogen
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Correspondence Principle

• For large quantum numbers, the results of quantum
  mechanical calculations approach those of
  classical mechanics