Semi-Classical Theory of Radiative Transitions Massimo Ricotti ricotti@astro.umd.edu University of Maryland Semi-Classical Theory of Radiative Transitions p.1/13
Atomic Structure (recap) Time-dependent Schroedinger equation: i ψ t = Hψ Stationary solution: ψ(r, t) = ϕe iet/, where Hϕ = Eϕ is time-independent Schroedinger equation. For hydrogen atom, neglecting spin, relativistic effects, nuclear effects, the Hamiltonian is H = p 2 2m e eφ, where the momentum p = i is an operator and φ(r) = e/r. Semi-Classical Theory of Radiative Transitions p.2/13
Solution for hydrogen atom in terms of eigenfunctions (complete orthonormal base): ϕ(r, θ, φ) = R(n, l) r Y l,m (θ, φ) Where spherical harmonics obey the eigenvalue problem, L 2 Y l,m = l(l + 1) 2 Y l,m, (1) L z Y l,m = m Y l,m, (2) and the radial function obeys the differential equation d 2 R n,l dr 2 + { 2me [E 2 n e ] r l(l + 1) r 2 where E n = e 2 /2n 2, with n = l + 1, l + 2, l + 3,... } R n,l = 0. Semi-Classical Theory of Radiative Transitions p.3/13
Non-relativistic limit of EM Hamiltonian For hydrogen atom: m e v 2 /2 e 2 /2a 0 where a 0 = 2 /m e e 2. Thus, velocity v/c e 2 / c α = 1/137 is NR. The NR Hamiltonian of single particle in EM fied is: H = 1 2m e p + e c A 2 eφ, where m e ẋ = p + (e/c)a is the particle momentum and A and φ are the EM vector and scalar potentials. In Coulomb gauge ( A = φ EM = 0) can be shown that A represent EM in vacuum (i.e., A = 0) and φ represent the static potential of atom. Semi-Classical Theory of Radiative Transitions p.4/13
Thus, Hamiltonian can be separated in H = H st + H int, with H int = H 1 + H 2 where H 1 e (p A + A p) = e 2m e c m e c A p, (note that in Coulomb gauge [A,p] = 0), and H 2 e2 2m e c2a A (two photon processes). Can show that H 2 H 1 H st, with H 2 /H 1 H 1 /H st (n ph a 3 0) 1/2 1 Semi-Classical Theory of Radiative Transitions p.5/13
Because A = 0 we can write: A = [ ] e α (ˆk)a α (k)e i(k x ωt) + c.c. k,α From Parsival theorem we have H rad = 1 dx 3 ( E 2 + B 2 ) = 2V 8π 8π V ( E α (k) 2 + B α (k) 2 ) k,α In Coulomb gauge we have E = ( A/ t)/c, B = A. Thus, E α = ika α e α, B α = ika α (ˆk e α ) and H rad = V 2π k 2 a α (k) 2 k,α In terms of photon occupation number: H rad = [ ] hnα (k) ωn α (k), a α (k) = c V ω k,α Semi-Classical Theory of Radiative Transitions p.6/13
Thus, H 1 = [Hα abs e iωt + Hα em e iωt ] where Hα abs = e [ ] 1/2 h m e V ω N α(k) e ik x e α (ˆk) p, (3) Hα am = e { } 1/2 h m e V ω [1 + N α(k)] e ik x e α (ˆk) p, (4) Note, we added 1 to 2nd eq. to account for spontaneous emission processes. Our semi-classic treatment in which the EM field is not quantized. a and a should be operators (creation/annihilation operators) that do not commute: [a, a ] = hc/ωv. This gives rise to spontaneous emission term. (5) Semi-Classical Theory of Radiative Transitions p.7/13
Perturbation theory We may expand the perturbed wave function ψ as follows: ψ(x, t) = j c j (t)ϕ j (x)e ie jt/ because H 0 is Hermitian operator and ϕ j satisfying H 0 ϕ j = E j ϕ j forms a complete orthonormal basis for representing any wave function for the atomic system. Thus, eliminating the zero-th order terms we have H 1 ψ = j c j H 1 ϕ j e ie jt/ = i j ċ j ϕ j e ie jt/ = Now we can multiply by ϕ f = ϕ f eie ft/ e iωfjt c j (t) ϕ f H 1 ϕ j = i ċ f (t) where ω fi (E f E j )/. j Semi-Classical Theory of Radiative Transitions p.8/13
Absorption transition probability Because at t = 0 we have c j = δ ji to zero-th order we can drop all terms j i in the summation: c f (t) = i 1 t 0 < ϕ f H 1 ϕ i > e iω fit dt = 1 < ϕ f H abs α ϕ i > [ e i(ω fi ω)t ] 1 (ω fi ω) Thus, going to the continuous limit and using dk 3 = c 3 ω 2 dωdω, the transition probability P if = k,α c f 2 is P if = V (2π) 2 d 3 k 2 ϕ f Hα abs ϕ i 2 sin2 [(ω ω fi )t/2] [(ω ω fi )/2] 2 tn α ϕ f e ik x e α p ϕ i Thus, the transition rate probability is dp if /dt const(t). Semi-Classical Theory of Radiative Transitions p.9/13
Dipole Approximation Approximate e ik x = 1 + k x +... 1 thus < ϕ f e ik x e α p ϕ i > e α < ϕ f p ϕ i > It is useful to express the momentum operator as the commutator Thus, [H 0,x] H 0 x xh 0 = 2 2m e ( 2 x x 2 ) = 2 m e = i m e p < ϕ f p ϕ i >= im e ω fi X fi, where X fi < ϕ f x ϕ i > Semi-Classical Theory of Radiative Transitions p.10/13
Bound-bound absorption cross section Finally, from the transition probability rate we derive the cross section σ ν for a flux of photons cn integrated over phase-space elements: dp if dt = 4πe2 ω 3 fi 3hc 3 N(ω fi ) X fi 2 = 1 (2π) 3 0 σ ν cn(ω)d 3 k. Thus, σ ν = 4π2 3 α X fi 2 ωδ(ω ω fi ), where α is the fine structure constant. In terms of the classical cross section for bound-bound transitions we have: σ ν = πe2 m e c f 12φ 12 (ν), where the oscillator strength in terms of the matrix elements is: f 12 = 2m e(ω 21 X 21 ) 2 3 ω 21 1. (ratio of kinetic energy of electron to the emitted photon Semi-Classical energy). Theory of Radiative Transitions p.11/13
Relativistic Electromagnetic Hamiltonian Relativistic Hamiltonian with EM field: H = [(cp ea) 2 + m 2 c 4 ] 1/2 + eφ Relativistic hard to separate due to square root. Two approaches: 1) Klein-Gordon (without electromagnetic potentials for simplicity) square operators in Schroedinger eq before applying to ψ: H 2 ψ = 2 ψ/ t 2 [( 2 1 c 2 2 t 2 ) ( mc ) ] 2 ψ = 0 Operator is d Alambertian where the second term is the Compton wavenumber of particle of mass m. The solution represent the equation for scalar field ψ in QFT. Scalar field represent a gauge boson of mass m and spin s = 0. Photon is a massless boson of spin s = 1. Semi-Classical Theory of Radiative Transitions p.12/13
Dirac approach Rewrite relativistic equation as linear in p: H = a Pc + bmc 2 + eφ [(c 2 P 2 + m 2 c 4 ] 1/2 + eφ where P = p e/ca is the relativistic particle momentum. The coefficient a and b need to be 4 4 matrices to satisfy the equation. The solution gives rise to concepts of spin and anti-matter. For particle at rest in vacuum there are 2 possible eigenvalues of energy: E = ±m e c 2. What represent a negative rest mass energy? Dirac interpretation of anti-particle: a hole in sea of negative rest-mass energy particles (not quite rigorous). Feynman interpretation of anti-particle: positive rest-mass energy but Semi-Classical Theory of Radiative Transitions p.13/13 moving backward in time (more rigorous interpretation).