Let H = H_{0 }+ W(t).

Let {|Φ_{p}>}
be an orthonormal eigenbasis of H_{0}, H_{0}|Φ_{p}>
= E_{p}|Φ_{p}>.

Let ω_{fi}
= (E_{f} - E_{i})/ħ
and W_{fi}(t) = <Φ_{f}|W(t)|Φ_{i}>.

Assume that at t = 0 the system is in the state |Φ_{i}>.

The probability of finding the system in the state |Φ_{f}>
(f ≠ i) at time t is

P_{if}(t)
= (1/ħ^{2})|∫_{0}^{t}exp(iω_{fi}t')W_{fi}(t')dt'|^{2},

in **first order time-dependent perturbation theory**.

Assume there exists a group of states nearly equal in energy E = E_{i}
+ 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)|W_{Ei}|^{2}δ_{E-Ei,ħω},
where W_{Ei} = <Φ_{E}|W|Φ_{i}>.

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 W_{ni}
= <Φ_{n}|W(t)|Φ_{i}> 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)>|^{2}dα.

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)>|^{2}dE.

If W is a **constant perturbation**, then δP(i,βE) = ∫_{ΔE }dE
ρ(β,E)[|W_{Ei}|^{2}/ħ^{2}] sin^{2}(ω_{Ei}t/2)/(ω_{Ei}/2)^{2}.

The function sin^{2}(ω_{Ei}t/2)/(ω_{Ei}/2)^{2}
peaks at ω_{Ei} = 0 and has an appreciable amplitude only in a small
interval Δω_{Ei} or ΔE about ω_{Ei} = 0. We assume that ρ(β,E)
and |W_{Ei}|^{2} are nearly constant in that small interval and
therefore may be taken out of the integral. Then

δP(i,βE) = ρ(β,E)[|W_{Ei}|^{2}/ħ^{2}]∫ dħω_{Ei}
sin^{2}(ω_{Ei}t/2)/(ω_{Ei}/2)^{2}

= ρ(β,E)[|W_{Ei}|^{2}/ħ^{2}]2ħ ∫dx sin^{2}(xt)/x^{2}.

∫_{-∞}^{∞}dy sin^{2}(y)/y^{2} = π.

We have (πt)^{-1}∫dx sin^{2}(xt)/x^{2} = 1.
Therefore

δP(i,βE) = (2π/ħ) ρ(β,E) |W_{Ei}|^{2} δ(E - E_{i})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) |W_{Ei}|^{2} δ(E - E_{i}).

Similarly, for a **sinusoidal perturbation** W(t) = Wsinωt or W(t) =
Wcosωt we obtain

w(i,βE) = (π/2ħ) ρ(β,E) |W_{Ei}|^{2} δ(E - E_{i}
- ħω).

and for W(t) = Wexp(±iωt) we obtain w(i,βE) = (2π/ħ) ρ(β,E) |W_{Ei}|^{2}
δ(E - E_{i} - ħω).

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 H_{0} = p^{2}/2m + U(r), i.e. if we are neglecting the spin-orbit coupling,

Δl
= ±1, Δm
= 0, ±1.

If H_{0} contains a spin orbit coupling term f(r)**L∙S**,
then he dipole selection rules then become

Δj = 0, ±1, (except j_{i }= j_{f }= 0),
Δl = ±1, Δm_{j }=
0, ±1.