Quantum Theory

• Dipole moment (μ)


  Gives interaction between atom and electric field (F)
• Dipole moment

  • Produced due to linear displacement or separation of electric charge
  • μ = -er
  • μ = r (vector) -e
  • E = -μ.F
  • (-) sign denotes that μ is parallel, therefore most favorable arrangement
• Oscillation of electric field (F) and magnetic field (B)

  1. F (Fo = amplitude) oscillates in xz-plane

  2. B (Bo = amplitude) oscillates in yz-plane

  3. Perpendicular to each other
• The polarization of a photon is based on the electric (E) field
• Magnetic field moment (m)

  m = 2μe
• Angular momentum (l) = √l(l + 1)ħ
• Allowed values of l are -l, -l+1, ..., 0, ..., l-1, l
• Transition is not observed due to symmetrically spherical shape when m = 0 but l = 0
• Meaning: even without linear separation (r) but rotation may still occur
• Inversion

  • Symmetry operation: rotation through 180 degrees followed by a reflection in a plane perpendicular to the rotation axis
  • Symmetry element: inversion center, symbol: i
• Parity

  • Represents center of inversion
  • Gerade = symmetric with inversion
  • Ungerade = antisymmetric with inversion
• Parity transformation is the flip in the sign of one spatial coordinate
• Transition dipole moment (d)


  • d = ∫ψ₂*μψ₁dv
  • μ is the dipole moment operator
• In terms of parity, ψ₄₂*x ψ₁s = odd x odd x even = even
• Overall contribution of (-) and (+) regions cancels out
• Laporte rule

  • Electric dipole transitions occur only between orbitals of opposite parity
  • Allowed transitions: s→p, p→d
  • Forbidden transitions: s→s, d→d, p→p
• Laporte rule correlates with angular momentum change (Δl = ±1)
• Laporte rule is not applicable for complex transitions due to changes in both photons and atoms
• Selection rule for m

  • Δm = 0, ±1
  • Important only if atom is placed in an external electric and magnetic field
• Spin magnetic moments cannot interact with an electric field
• Spin-orbit coupling breaks all selection rules for heavy atoms
• Larmor frequency is the rate of precession of the magnetic moment (μ) of a photon around an external magnetic field
• In classical physics, l, lx, and ly can be determined with precision, but lz cannot
• In quantum mechanics, lz can be determined with precision, but lx and ly cannot
• Orbital momenta


  L = l₁ + l₂ + l₃ + ... + ln
• Spin momenta

  S = S₁ + S₂ + S₃ + ... + Sn
• Total Angular momenta


  J = L + S
• J

  Quantized according to the rule that J = √j(j+1)ħ
• Russel-Saunders Coupling

  • Coupling of spin & orbital angular momenta
• Russel-Saunders Coupling


  1. L = l₁ + l₂

  2. S = S₁ + S₂

  3. J = L + S
• Spin-Orbit Coupling

  • j-j coupling is strong for heavy metal (n>70)
• Spin-Orbit Coupling

  Es.o = ζ₄₂[j(j + 1) - l(l + 1) - s(s + 1)]
• Spin-orbit coupling constant ζ depends on the 4th power of n, important in heavy metals
• Heavy atom

  • Electrons strongly attract the heavy charged nucleus
  • If confined in small space volume, curvature of ψ increases, therefore speed increases
• Russel-Saunders coupling

  Not a property of nature of e, true for lighter atoms
• Spin-orbit coupling


  Property of nature of e, for heavy atoms (n>70)
• Term Symbols

  Help define the state in which an atom finds itself, convey the information about the state and its multiplicity
• Term Symbols

  • 2S+1L
  • 2S+1LJ
• Multiplicity of state

  Defined as 2s+1, s=total spin NOT state symbol
• Electric Dipole Selection Rules

  • ΔL = 0 or +1 but 0 ↔ 0
  • ΔS = 0
  • Δg,u,gg, u ↔ u
  • ΔJ = 0 or +1 but 0 ↔ 0