We consider the nonrelativistic regime so no pair production or annihilation.the hamiltonian for interaction of fields and sources is 1 (p

Transcription

1 .. RADIATIVE TRANSITIONS Marh 3, 5 Leture XXIV Quantization of the E-M field. Radiative transitions We onsider the nonrelativisti regime so no pair prodution or annihilation.the hamiltonian for interation of fields and soures is H γ + i (p m i e i A i) µ i B i + e i e j + everything else. (.) 8π r i r j i j H γ is the energy of the free field ω. µ is the magneti dipole moment µ e m i g is i and s i is the partile spin. Everything else ontains non-eletromagneti interations and fine and hyperfine struture. The interation term in our Hamiltonian has terms linear (H ) and quadrati (H ) in the E-B fields. H e m p A µ B ( e ) H A m (Coulomb gauge where A.) What are the relative strengths of the piees? We are thining about atomi transitions so non-relativisti eletrons and emssion or absorption of photons with energy harateristi of atomi biniding energy. e m p A µ B e m A Finally e m mv Eγ e m Eγ λ H λ eα (α m ) / αα 3 (m ) ( v α e E γ Eγ m () ) 3 m (.) α m (α m ) αα 4 m ( v ) 4 α m (.3) e m A e E ( γ v ) 4 m λ αα4 m α m.. Emission and absorption of photons Let s write the vetor potential field operator for future referene. A(x, t) ωv [a,λˆɛ λ e i( r ωt) + a,λˆɛλ e i( r ωt) ],λ (.4)

2 .. RADIATIVE TRANSITIONS The reation and annihilation operators onnet states that differ by preisely one photon of momentum and polarization λ. In partiular n + a λ n n + and n a λ n n Consider absorption of a photon by an atom transitioning from i to f. The annihilation operator omponent of the field operator is the only part that ontributes. f; n λ H i; n λ e f; n m ωv a,λe i( x ωt) p ɛ λ i; n i e n f e i( x) p ɛ λ i e iωt m ωv The annihilation gives zero for all but the photon and polarization in the initial state... Semi-lassial desription Let s revisit the semi-lassial desription. Then we said that a λ was a fourier oeffiient of the plane wave expansion of the vetor potential, rather than an annihilation operator. We ould write the absorption proess as nλ A ωv ɛ λe i( x ωt) The absorption probability is proportional to A. In the quantum theory the probablity sales linearly with n λ. Wors fine and the results are equivalent. What about the emission proess. Then we have something lie f; n λ + H i; n λ e f; n + m i e (n + ) m ωv Time dependent perturbation theory We have a hamiltonian H H + H I and state ωv a e i( x ωt),λ p ɛ λ i; n f e i( x) p ɛ λ i e iωt ψ(t) (t) u e ie t/ where H u E u

3 .. RADIATIVE TRANSITIONS Then substitution into Shrodinger s equation ċ m i H ψ i t ψ (H + H I ) ψ i ( ) t ie u u m (H + H I ) ψ iu m ( ) t ie u u m H I u e ie t/ m H I e i(em E )t/ i m t e iemt/ e ie t/ e iemt/ If at t the system is in the state l and if the perturbation is wea so that l () and all others are small, then ċ m i m H I l e i(em E l)t/ and m (t) t m H I (t ) l e i(em E l)/dt i If the time dependene of H I is sinusoidal then H I (t ) H I e ±iωt and Then m (t) i t m H I l e i((em E l)/ iω)dt m m H I l sin ((E m E l ω)/)t/ t[(e m E l ω)//] m H I l tπδ((e m E l ω)/) where we use Finally sin αx lim α αx δ(x) m m H I l πtδ(e m E l ω) and the transition rate is d dt m π m H I l δ(e m E l ω) To turn it into a real measureable rate we need to sum over all of the final states. For photons there are ρ V ddω (π) 3 d(ω) 3

4 .. RADIATIVE TRANSITIONS states per unit energy ω in the interval ω ω + d(ω). Then ρ V ω dω (π) 3 3. So the transition probability per unit time into dω is where ρ must satisfy the delta funtion...3 Density of States w π m H I l ρ For spontaneous emission, we multiply by the density of states, that is the number of states per unit energy available to the final state photon with energy ω. Then ρ(e) V d 3 (π) 3 d(ω) V dωdω (π) 3 dω V dω (π) 3 What about absorption? Suppose the atom is in a avity with modes with frequenies orresponding to the interesting transition. Now put a single photon into eah mode. There is one and only one mode (and photon) that an exite the transition and be absorbed. Now put a seond photon into every mode. The absorption rate will inrease by a fator of two. The number of photons per unit energy is ρ V dω (π) 3 n..4 Quantum mehanial one photon transitions The lowest order interation hamiltonian for single photon transitions is H e m ωv p ˆɛλ ()e i( r ωt) (.5) The transition rates are given by the Golden rule Γ i f π e m f p ˆɛ λ ()e i( r) i δ(e i E f ) (.6) ωv and Spontaneous emission dγ i f π e m ωv f p ˆɛ λ ()e i( r i V dω (π) 3 Here the initial state is i; i atom γ and the final state is f, f atom λ fa λ. The differential transition rate is dγ π f; λ H i; ρ f () 4

5 .. RADIATIVE TRANSITIONS where the density of fnal photon states is ρ f V d 3 (π) 3 dω V (π) 3 dω Only the part of the interation hamiltonian proportional to the reation operator ontributes. And that ontribution is a single photon...5 Eletri Dipole transition We will need to evaluate f p ˆɛ λ ()e i( r) i. The energy of a photon emitted in an atomi transition is of order the binding energy α m. The wavelength λ α m. The size of the atom is of order a me mα. Therefore λ/a α, that is, the wavelength is muh bigger than the atom so that we an expand e i( r) + i r +... and eep only the lowest order term, namely. This is the dipole approximation. Then f p ˆɛ λ ()e i( r) i f p ˆɛ λ () i We use the fat that to rewrite [H, r] m [p, r] i p m f p i ˆɛ λ () m f [H, r] i ˆɛ λ () and sine initial and final states are eigenets of H with E i E f ω Spherial tensor f [H, r i ˆɛ λ () imω f r i ˆɛ λ () Let s wor in the spherial basis with unit vetors ˆɛ ± (ˆx ± iŷ), ˆɛ ẑ. and write r as a spherial tensor V ± (x ± iy) r 3 Y ±, V z r 3 Y Let s onsider λ + and in the θ diretion with respet to the z-axis of the atom, and the matrix element nf, l f, m f V ˆɛ + ˆɛ λ+ () The polarization vetor ˆɛ + () (ˆx +iŷ ) where ˆx and ŷ are the unit vetors in the oordinate system with ẑ. Then ˆɛ + () (ˆx os θ + ẑ sin θ + iŷ). (We assume that the angle θ is about the ŷ axis.) Now nf, l f, m f V n i, l i, m i ˆɛ + ˆɛ λ+ () n f, l f, m f V n i, l i, m i ( + os θ) 5

6 .. RADIATIVE TRANSITIONS Similarly nf, l f, m f V ˆɛ ˆɛ λ+ () n f, l f, m f V sin θ nf, l f, m f V n i, l i, m i ˆɛ ˆɛ λ+ () n f, l f, m f V n i, l i, m i ( os θ) The total rate from n i, l)i, m i to final state n f, l f, m f with a photon into angle θ with polarization + is proportional to dγ dω n f, l f, m f V ( + os θ) + + n f, l f, m f V sin θ + n f, l f, m f V ( os θ) (.7) Remembering the Wigner Ehart theorem nf, j f, m f V q n i, j i, m i jf, m f q,, j i, m i n f, j f V n i, j i where j f, m f q,, j i, m i is a Clebsh-Gordon oeffiient. The C-G oeffiient is zero unless m f + m i and j i + q j f j i q. Evidently only one term in Equation.7 is non-zero for any given hoie of initial and final state. Also, the dipole operator has odd parity so the expetation value is nonzero only if initial and final states have opposite parity. We onlude that l ±. Let s suppose m f m i +. Then only the first term in Equation.7 ontributes. Sum over photon polarizatons, dγ dω n f, l i ±, m i + V n i, l i, m i ( ( + os θ) + ( os θ) ) sin θdθ 4 and integrate over photon diretions. Γ n f, l i ±, m i + V 8π 3 (.8) Next we might sum over all m f + possible final states. That will involve the other terms in Equ..7. But they are all related (aording to WignerEhart) by Clebsh-Gordon oeffiients so it won t be too painful. And if the inital state is P and the final state S, the only final state is m f. We might also average over initial states. But if the final state has l, then by symmetry the rate from l, m i, ± must all be the same. Matrix element Next we want to evaluate nf, l f, m f V q n i, l i, m i Initial and final atomi states are x n i R ni (r)y mi l i (θ, φ), and x n f R nf (r)y m f l f (θ, φ) and Vq r 3 Y q (θ, φ). Then nf, l f, m f V q n i, l i, m i r 3 R ni (r)rn f 6 Y m f l f (θ, φ)y q (θ, φ)y mi l i (θ, φ)dω (.9)

7 .. RADIATIVE TRANSITIONS r 3 R ni (r)r n f (q + )(l i + ) l i q; l i q : l f l i q; m i l i q; l f m f (l f + ) (.) Note that the triple produt of spherial harmonis enfores the Wigner-Ehart seletion rules. Again if the initial state is l i, m i and the final state l f, m f then and nf,, V n i,, 3 r 3 R ni (r)rn (3)(3) f ; ; 3 () r 3 R ni (r)rn (3)(3) f 3 () r 3 R ni (r)r n f where ; ; Note that if the atom were in a magneti field, then it would be polarized and the levels might be split there would be a orrelation between diretion, energy, and polarization. 7