Ain Shams U' Faculty of Engineering Mathematics and Engineering Physics Department Quantum Mechanics

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Ain Shams U.Faculty of EngineeringMathematics
and Engineering Physics DepartmentQuantum
Mechanics 2

• ?. ???? ???????
• ?????? 2004

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Contents

• Applications of Schrödinger equation
• Hydrogen atom
• Summary of Bohr model.
• Quantum mechanical solution.
• Particle in infinite potential well
• Introduction to band theory
• What is a quantum well.
• Solution of Schrödinger equation for the particle
  in infinite quantum well.

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Websites

• http//www.colorado.edu/physics/2000/quantumzone/b
  ohr.html
• http//www.launc.tased.edu.au/online/sciences/phys
  ics/brhydrogen.html
• http//www.falstad.com/mathphysics.html
• http//www.physics.nwu.edu/ugrad/vpl/atomic/hydrog
  en.html
• http//www.falstad.com/qmatom/

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Atomic spectroscopy

• What is the problem ?
• If an atom is exited through heat or bombardment
  with high speed electrons, electromagnetic
  radiation is generated from this atom.
• When this electromagnetic radiation is analyzed
  (for example through a prism), one finds that it
  is composed of discrete wavelengths.
• Each element in the periodic table has a unique
  line spectrum.
• The Hydrogen atom is the simplest case to model
  since it is composed of a single proton and a
  single electron. (applet)

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Hydrogen atom (Rutherford)

• Rutherford (1911) proved that the atomic system
  is a miniature solar system, but didnt know why
  the electron orbits should take certain radii.

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Hydrogen atom (Bohrs assumptions)

• In addition to the fixed orbits proposed by
  Rutherford, Bohr proposed that
• The orbits are defined such that the angular
  momentum (mvr) is quantized

• Electromagnetic radiation occur when an electron
  goes from one allowed orbit to another of lower
  energy

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Hydrogen atom (Bohrs model results)

• With a complete classical model (using Coulomb's
  and Newtons laws), Bohr succeeded
• in calculating the electron energy rotating in
  each orbit. (applet)

• in calculating the orbits radii

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Hydrogen atom (Bohrs model results)

• In calculating all spectrum lines of the Hydrogen
  atom as measured experimentally.

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Hydrogen atom (Bohrs model failure)

• This model visualizes the Hydrogen atom as a
  planar system, not a 3-D one.
• While this model is successful in describing
  other single electron systems such as He and
  Li, it failed to explain systems with multiple
  electrons.

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Hydrogen atom (QM)
Time independent SE in spherical coordinates
(note the 3-D nature of the problem)
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Hydrogen atom (QM)

• Using separation of variable

• Substituting in SE, one can find an expression
  for R(r),? (?), and ? (?) where each function is
  associated with a unique quantum number,
  resulting in 3 quantum numbers1 quantum number
  for spin.

• All s states have no angular (?) or azimuthal
  (?) dependence.

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Hydrogen atom (QM)
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Hydrogen atom (QM)

• 2s state

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• (applet)

1s state
2s state
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Hydrogen atom (QM)

• The probability of finding the electron in a
  spherical shell dV at a distance r from the
  nucleus

• We define the radial probability density P(r) as

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Hydrogen atom (QM)
ro
5ro
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Hydrogen atom (QM)
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Hydrogen atom (QM)

• Electron cloud instead of specific orbits.
• 3-D model for the atom.
• Applicable for all atomic systems, molecules.

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Band theory
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Band theory

• If N sodium atoms are brought together, each
  atomic level splits into N new level, each level
  accommodates 2 electrons of opposite spin.
• If one atomic state are fully occupied in all N
  atoms, the corresponding band is full.
• Since the 3s state in sodium atom has just one
  electron, the corresponding 3s band is half full

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Band theory

• The conduction band is partially filled

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Quantum well
Conduction band
Electrons confined in a finite quantum well
Energy gap
Valence band
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Finite QW
Energy

• Electron in a finite quantum well.

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Infinite QW
?
?
Energy
Distance
Electron in an infinite quantum well.
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Quantum well Potential well
Particle in a quantum well Particle in a box
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Infinite potential well

• This is a problem in 1D space.

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Infinite potential well

• In the region 0ltxlta

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Infinite potential well
B0
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Infinite potential well

• Applying the 2nd boundary condition

kan? n1,2,3,
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Infinite potential well

• The energy is quantized

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Infinite potential well

• Normalization

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Infinite potential well

• Normalization

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Infinite potential well
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Infinite potential well