## Time-dependent approximations

### Time-dependent perturbation theory

Let H = H0 + W(t).
Let {|Φp>} be an orthonormal eigenbasis of H0,  H0p> = Epp>.
Let  ωfi = (Ef - Ei)/ħ and Wfi(t) = <Φf|W(t)|Φi>.
Assume that at t = 0 the system is in the state |Φi>.
The probability of finding the system in the state |Φf> (f ≠ i) at time t is
Pif(t) = (1/ħ2)|∫0texp(iωfit')Wfi(t')dt'|2,
in first order time-dependent perturbation theory.  (Derivation)

### Fermi's golden rule

Assume there exists a group of states nearly equal in energy  E = Ei + ħω.
Let ρ(β,E) be the density of final states, i.e. ρ(β,E)dE is the number of final states in the interval dE characterized by some discrete index β.
Let W(t) = Wexp(±iωt).  Then the transition probability per unit time is given by
w(i,βE) = (2π/ħ)ρ(β,E)|WEi|2δE-Ei,ħω, where WEi = <ΦE|W|Φi>.
This is Fermi's golden rule.
The transition probability increases linearly with time.

Details:
Assume there is a group of states n, nearly equal in energy E, and that Wni = <Φn|W(t)|Φi> is nearly independent of n for these states.  Take for example continuum states.  We may label continuum states by |α>, where α is continuous and <α|α'> = δ(α - α').
The probability of making a transition to one of these states in a small range Δα is ∫Δα|<α|Ψ(t)>|2dα.
If |α> = |β,E> then dα = ρ(β,E)dE, where ρ(β,E) is the density of states
We assume β to be some discrete index.  We then have
δP(i,βE) = ∫ΔE ρ(β,E)|<βE|Ψ(t)>|2dE.
If W is a constant perturbation, then δP(i,βE) = ∫ΔE dE ρ(β,E)[|WEi|22] sin2Eit/2)/(ωEi/2)2.
The function sin2Eit/2)/(ωEi/2)2 peaks at ωEi = 0 and has an appreciable amplitude only in a small interval ΔωEi or ΔE about ωEi = 0.  We assume that ρ(β,E) and |WEi|2 are nearly constant in that small interval and therefore may be taken out of the integral.  Then
δP(i,βE) = ρ(β,E)[|WEi|22]∫ dħωEi sin2Eit/2)/(ωEi/2)2
= ρ(β,E)[|WEi|22]2ħ ∫dx sin2(xt)/x2.
-∞dy sin2(y)/y2 = π.
We have (πt)-1∫dx sin2(xt)/x2 = 1.  Therefore
δP(i,βE) = (2π/ħ) ρ(β,E) |WEi|2 δ(E - Ei)t.
(It is understood that this expression is integrated with respect to dE.)
The transition probability per unit time is the given by Fermi's golden rule,
w(i,βE) = (2π/ħ) ρ(β,E) |WEi|2 δ(E - Ei).
Similarly, for a sinusoidal perturbation W(t) = Wsinωt or W(t) = Wcosωt we obtain
w(i,βE) = (π/2ħ) ρ(β,E) |WEi|2 δ(E - Ei - ħω).
and for W(t) = Wexp(±iωt) we obtain w(i,βE) = (2π/ħ) ρ(β,E) |WEi|2 δ(E - Ei - ħω).

### Dipole transition selection rules

When an atom interacts with an electromagnetic wave, the electromagnetic field is most likely to induce a transition between an initial and a final atomic state if these selection rules are satisfied.  If these selection rules are not satisfied a transition is less likely and is said to be forbidden.
The selection rules are:
If H0 = p2/2m + U(r), i.e. if we are neglecting the spin-orbit coupling,
Δl = ±1, Δm = 0, ±1.
If H0 contains a spin orbit coupling term f(r)L∙S, then the dipole selection rules then become
Δj = 0, ±1, (except ji = jf = 0), Δl = ±1, Δmj = 0, ±1.