Concepts and formulas

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.


Elastic scattering

In a typical scattering experiment a target is struck by a beam of mono-energetic particles.
Let Fi be the incident flux, i.e. the number of particles per unit area per unit time. 
Fi = npv, where np is the number of particles per unit volume. 
Typically np is very small and we can neglect any interaction between different incident particles.
We measure the number ΔNp of particles scattered per unit time into a solid angle ΔΩ about the direction defined by the spherical coordinates θ and φ. 
We expect ΔNp ∝ Fi, and ΔNp ∝ ΔΩ .
We define ΔNp = σt(θ,φ)FiΔΩ.  Here σt(θ,φ) is the differential scattering cross section of the target.  It has the units of an area.  Commonly used units are cm2 and barn = 10-24 cm2.


The Born approximation

The elastic scattering cross section in the Born Approximation is
σkB(θ,φ) = σkB(k,k') = [μ2/(4п2ħ4)]|∫d3r' exp(-iq∙r')U(r')|2,
where q = k' - k, k = μv0/ħ, k' = μv0/ħ (k'/k'), and μ is the reduced mass.

The differential scattering cross section is proportional to the square of the Fourier transform of the potential.
We often want to know the scattering cross section as a function of the scattering angle and not as a function of the momentum transfer.  Let θ be the angle between k and k'.  Then q = 2ksin(θ/2).

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