Atomic Quantum Mechanics - Hydrogen Atom (15.1-15.3)

1
Atomic Quantum Mechanics - Hydrogen Atom
(15.1-15.3)

• Assuming an atom doesnt move in space
  (translate), the SE is reduced to solving for the
  electrons only
• For single electron atoms and ions (e.g.,
  hydrogen), only the attraction of the electron to
  the nucleus is needed in the potential energy
  term
• For many electron atoms and ions, repulsion
  between the electrons is needed in the PE
• For the hydrogen atom, the solution to the SE is
  a set of atomic orbitals
• The wavefunction involves three quantum numbers
  (n, l, ml) since three sets of boundary
  conditions are needed (1 for r, ?, and f)
• The number of nodes in the wavefunction increases
  with increasing values of n and l
• The energy of the hydrogen atom only depends on
  the principle quantum number n

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Atomic QM Many Electron Atoms (15.6-15.7)

• When more than one electron is present, Vee must
  be included in SE
• Electrons are in constant motion, so potential
  energy is a constantly changing variable
  (electron correlation)
• SE is no longer exactly solvable!
• Solutions to many electron atom SE are very
  similar to hydrogen orbitals
• The wavefunction for each electron is a
  hydrogen-like orbital (1s, 2s, 2p, etc.)
• The energy associated with each electron now
  depends on n and l
  (orbital filling diagram)
• Two electrons can have the same principle,
  angular, and magnetic quantum numbers
• Electron configurations are used to show what the
  atomic wavefunction looks like
• Electrons posses another quantum number
  associated with their spin
• Electrons have another quantum number called the
  spin quantum number ( 1/2)
• Pauli exclusion principle states no two electrons
  can have the same quantum numbers, so one
  electron in an orbital must be spin-up and the
  other spin-down

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Hydrogen Wavefunctions
4
Radial Nodes in Hydrogen Orbitals
5
Angular Nodes in Hydrogen Orbitals
6
Probability Distribution Functions for Hydrogen
Orbitals
7
Orbital Filling Diagram
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Atomic Orbital Overlap