Chap 3. Light-Matter Interaction ● Electromagnetic wave [SR 5.2] (outline) 1.  Maxwell Eq with vector (A) and scalar (φ) potentials 2.  Vacuum (no charge, no current) ρ = 0, i = 0, Coulomb gauge ∇A = 0 Wave Eq : ニュー 3.仲= ∇² A ⇒ Plane wave A = A 0 ε cos(k · r − ωt) 4. (Energy from fields amplitude) = (Photon density)×(Energy quantum) ⇒ Ao = 2c(2πћN/ωV)1/2 ● “Minimal” Electromagnetic Interaction (charged particles)
classical eq of motion with] Lorenz force F = q(E + v × B/c) Using : p → ih∇ , ∇ · A = 0 (Coulomb gauge)
(“weak-field approximation” neglects the 2nd term [SR 5.3.3]) ● Absorption and Emission Spectra (1st order processes)
(periodic interaction)
↑
(Absorption or Emission ⇐ sign of E k − Em) Dipole Approximation (long wavelength) Visible lights (electronic transitions) :
So far, we have treated one particle (with charge q ). By considering all nuclei and electrons in molecular systems,
We arrive at (Note : all calculations were linear)
µkm : transition dipole of the molecular system (Note : in [SR 5.4] only electrons are considered, but we should include nuclei to treat IR spectra etc.)
● Simpler derivation of Ukm [SR 5.5.2]
We get the same Ukm (more easily) starting from V = −μ · E with
(Coulomb gauge ∇φ = 0 ) Density of states of photon field Having specified the states ( |m) → |k) ) of the molecular system, we still need to sum over the photon field states Wabs/em(m → k) =    WkmPphoton(ñω)d(ñω) Insert wkm of page 2 :
● Calculation of Pphoton( hw ) 1.  Number of states N in wave vector k : dN = (ﾃπ )³dk Periodic boundary condition in length L : eikx(x+L) = eikxx ⇒ kxL = 2πnx (nx = 0, ±1, ±2, ··· ) 2.  Use polar coordinate dk = k²dkdΩ and k = ω/c :  dN =   V    ω²dwdΩ dN V  [(Differential) density of photon] 3. dpphoton(ñω) = d(ñω) = (2πc)³ ↑dΩ states having k directed to dΩ Thus, (differential) emission rate toward dΩ : dwem(m → k)   dΩ used |Ukm| = (A0ω/2c)|ε・μkm|,  A0 = 2c(2πћN/ωV)1/2,  ñω = Ek = Em ● |ε · μ km | = | μ km | sin θ          (θ ≡ angle between k and μ km ) [ ʃ dΩ sin² θ = ʃ 2 π  dφ ʃ π  sin θdθ sin² θ =  8 π ● Integration over the solid angle dΩ ↑ Total emission rate 4N | μ κ m | 2 c | E Wem (m → k) = 3h ●  However, it is not correct to have wem → 0 for N → 0, since in reality, we have “spontaneous emissions”. ● To remedy this, we just need to replace N by N + 1, which is indeed justified via quantization of radiation (photon) field.
● Quantization of radiation field : Photon state |n} = √ n! 〈n + 1〉b † 〈n〉 = √n + 1 〈n − 1〉b | n 〉 = √ n (emission) (absorption) (1) b T | 0 ⟩ = 1 ≠ 0 ⇒ spontaneous emission into the photon vacuum is possible. (On the other hand, we have no "spontaneous absorption" since b | 0 ) = 0 ) ● Natural lifetime of excited states of an excited state | m | ← summing over all | k ⟩ in lower energy E k < E m :
● Rate of photo-absorption (Similarly to the emission case)
[Wabs → 0 as N → 0 : no “spontaneous absorption” Miscellaneous ● Einstein's A and B factors [SR 5.4.6] Photon field density of states (or energy) ●  SR uses that for quantized plane-wave (temperature independent) ●  Many others use black-body Planck distribution (temperature dependent) or discuss without explicit form of the density of states. (Indeed, the A, B factors are defined from the photon-field independent part of the transition rate.) ● Oscillator strength
Defined as relative intensity to n = 0 → 1 transition of 3D harmonic oscillator (which gives f = 1 … (*)). Exercise : Derive (*) via the following two routes: 1.  Using creation-annihilation operator 2.  Carrying out Gaussian integration of the wavefunctions representation

● Correspondence with experiments
Beer-Lambert Law : I l = I oe-αCl α … absorption coefficient [ C ... molar concentration Integrated absorbance A ≡ | α(ν)dν Considering the decay of photon field energy, we get (skipping details) 4π² h¯¯¯¯ LB : | 2 | Lkm 3hc C L ... Avogadro number Vkm … central peak frequency of the |m) → |k) absorption band (or in the derivation, representative ν when approximating ʃֲנָּנָּאָּdν ~ｶʃ α(ν)dν ) ● 1st correction to long-wavelength approx. [SR 5.4.8] Dipole Approx : eik r ~ 1 (|k| = 2π/λ) To 1st-order : eik r = 1 + k · r + · ·
●  M =  zmc (r × p) ... magnetic dipole ●  Q ij = (er i )(er j ) … electric quadrupole tensor (r i = x, y, z ) Symmetry : ● Q ~ xy, x', etc. ● μ ～ x, y, z ,      ● M ～ Rx, Ry, Rz, (rotation) ⇒ different symmetry for each term Franck-Condon Factor
Transition dipole for (ε,ν) → (ε′,ν′) :
Then, με'ν',εν = µε'ιε(R0) · 〈Χε'ν'|Χεν〉+ (non-Condon terms) (Franck-Condon factor) In many cases, large Franck-Condon overlap at “vertical transitions” ⇔ Nuclei don’t move during fast electronic transitions (semi-classical interpretation)