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Interaction with electromagnetic radiation Electric Dipole approximation: dipole r r E (r , t ) r r d = er r r interacts with field E (r , t ) a0 = 1 Å r k r r B (r , t ) Interaction hamiltonian: r r r H = er ⋅ E (r , t ) r r r −iωt ikr ⋅rr E (r , t ) = E0e e r r 1 r r2 r − iωt ⎡ n ⎤ ( ) ( ) 1 ... = E0e + i k ⋅ r + k ⋅ r + + O r ⎢⎣ ⎥⎦ 2 r − iωt ≈ E0e ( ) Note: λ = 5000 Å ⎛ 2πa ⎞ ⎟ ⎝ λ ⎠ Higher order terms in O⎜ r r 2πa << 1 k ⋅ r ≈ ka = λ E-field does not vary over size of the atom, hence: H EM Electric quadrupole, octupole r r r H = μ ⋅ B (r , t ) Magnetic interactions: M Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs r r = er ⋅ E (t ) Quantum analog of electromagnetic radiation Classical electric dipole radiation Classical oscillator I rad Transition dipole moment Quantum jump r2 ∝ er 2 * r ψ 1 er ψ 2 2 I rad ∝ μ fi = The atom does not radiate when it is in a stationary state ! The atom has no dipole moment μii = ∫ *r ψ 1 r ψ 1dτ =0 Intensity of spectral lines linked to Einstein coefficient for absorption: Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Bif = μ fi 2 6ε 0h 2 Selection rules depend on angular behavior of the wave functions Parity operator r r θ φ r −r r r Pr = −r ( x , y , z ) → ( − x, − y , − z ) (r ,θ , φ ) → (r , π − θ , φ + π ) Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs All quantum mechanical wave functions have a definite parity r r Ψ (− r ) = ± Ψ (r ) r Ψ f r Ψi ≠ 0 If Ψf and Rule about the Ψi Ylm have opposite parity functions PYlm (θ , φ ) = (− )l Ylm (θ , φ ) Selection rules Mathematical background: function of odd parity gives 0 when integrated over space In one dimension: ∞ Ψ f x Ψi = ∞ ∫ Ψ *f xΨi dx −∞ = with ∫ f ( x)dx f ( x) = Ψ *f xΨi −∞ ∞ 0 ∞ 0 ∞ ∞ ∞ −∞ −∞ 0 ∞ 0 0 0 ∫ f ( x)dx = ∫ f ( x)dx + ∫ f ( x)dx = ∫ f (− x)d (− x ) + ∫ f ( x)dx = ∫ f (− x)dx + ∫ f ( x)dx ∞ = 2 ∫ f ( x)dx ≠ 0 if Ψi f ( − x) = f ( x) and Ψ f opposite parity 0 =0 if f (− x) = − f ( x) Ψi and Ψ f same parity Electric dipole radiation connects states of opposite parity ! Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Selection rule r Ψ f r Ψi Δl Transition dipole matrix element Parity consideration r r P Ψ f r Ψi = P RnlYlm r Rn'l 'Yl 'm' = (− )l (− )(− )l ' = even if l + 1 + l' = even Δl = ±1,±3,... Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Selection rule Δm cos φ x Ψ f y Ψi = r Ψ f sin φ Ψi = r Ψ f 1 z [x]: 2π = ∫e −im 'φ 0 2π [y]: [z]: = ∫e −im 'φ 0 2π = (e φ + e φ )/ 2 (e φ − e φ )/ 2i Ψ i −i i −i i 1 ei (−m'±1+ m )φ dφ ≠ 0 2 if m − m'±1 = 0 2π i ( − m ' m1+ m )φ eiφ − e − iφ imφ e e dφ = ∫ dφ ≠ 0 2i 2i 0 if m − m'±1 = 0 eiφ + e −iφ imφ e dφ = 2 ∫e −im 'φ imφ e dφ ≠ 0 2π ∫ 0 if m − m'= 0 0 Combinations of x, y, z related to the polarization of light Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Polarization vectors ( ) 1 rˆ = xê x + yê y + zê z r = sin θ cos φê x + sin θ sin φê y + cos θê z rˆ ∝ Y1,−1 ê x +iê y 2 ê rad = A σ− + Y10ê z + Y11 ê x −iê y 2 2π Y1,−1 − Y11 3 2π sin θ sin φ = i Y1,−1 + Y11 3 ) ( -ê x +iê y cos θ = 2 ⎛ ê x +iê y ⎞ ⎟ ⎜− σ+⎜ 2 ⎟⎠ ⎝ ( sin θ cos φ = ) 4π Y10 3 Definition of Yij + Aπ ê z + A definition of polarized light vectors r̂ ⋅ ê rad = A Y + Aπ Y10 σ − 1,−1 +A Y σ + 11 Verify selection rules for x+iy and x-iy Circular polarization Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs m=0 m=-1 σ- σz m=1 σ+ l=0,m=0 Properties of Write r-vector in terms of spherical harmonics Ylm Y11 = − Y1,−1 = − Y10 = * Y ∫ l f m f Y1,mYl imi sin θdθdφ ≠ 0 if l f = li ±1 m f = mi ± m Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs z r − 1 ( x + iy ) 3 3 sin θeiφ = 4π 8π 2 r 1 ( x − iy ) 3 3 sin θe −iφ = 4π 8π 2 r 3 4π Triangular property of Spherical harmonics Verify with “Mathematica” Photon picture of selection rules r J r J' absorption r J f =1 photon r r r J '= J + 1 so J ' = J +1 J'= J J ' = J −1 Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs ΔJ = 0,±1 For the total angular momentum Non-electric dipole selection rules Higher order transitions Scale like xy, r2 * Y ∫ l f m f Y2,mYl imi sin θdθdφ ≠ 0 l f − l i = 0,±2 Magnetic dipole interactions have different selection rules Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Selection rules in Hydrogen atom Intensity of spectral lines given by r r μ fi = ∫ Ψ*f μΨi = Ψ f − er Ψi n 1) Quantum number no restrictions Balmer series 2) Parity rule for l Δl = odd 3) Laporte rule for l Angular momentum rule: r r r l f = li + 1 From 2. and 3. so Δl ≤ 1 Δl = ±1 Lecture Notes Structure of Matter: Atoms and Molecules; W. Ubachs Lyman series