Assume that at *t=-*∞ a system is in an eigenstate |*φ _{i}*> of the Hamiltonian

to first order in the perturbation
*W*.

We often write

This is the result of **first
order time dependent perturbation theory**.

Assume
that W(t) = W⋅sinωt, i.e. that we have a sinusoidal perturbation starting at t=0. Then |

Similarly,
if W(t) = W⋅cosωt, then |

If ω = 0 we have a constant perturbation and |

For the harmonic perturbation
*W(t) = W⋅*sin*ωt*, we find that

has an appreciable amplitude only if the denominator of one of the two terms is approximately zero, i.e. if

The first order effect of a perturbation that varies sinusoidally with time is to receive from or transfer to the system a quantum of energy

If the system is initially
in the ground state, then *E _{f}>E_{i}*, and only
the second term needs to be considered. Then

Let *β = ω _{fi} - ω,* and plot

If *β* = 0, i.e. * ω _{fi} = ω *, then

Therefore, if *ω _{fi} = ω*, then the probability of finding the system in the state |

In the above figure the
height of increases
proportional to *t ^{2}*, and the width of the peak is proportional
to 1/

For a first order approximation to be valid, we need

On the other hand, to justify
neglecting the first term in the above formula, we need *2ω _{fi}>>Δω. 2ω_{fi}* is the difference in the positions of the peaks due to the first term and the second term in the above formula,

Combining these two conditions we obtain

Assume there is a group
of states *n*, nearly equal in energy *E*, and that *W _{ni} = <φ_{n} |W| φ_{i} >* is nearly independent of

If *W* is a constant
perturbation, then

The function peaks
at *ω _{Ei}* = 0 and has an appreciable amplitude only in a small interval

Therefore,

It is understood that the
expression is integrated with respect to *dE*.

The **transition
probability per unit time** is the given by **Fermi’s
golden rule**,

Similarly, for a sinusoidal
perturbation *W(t)=W*sin*ωt* or *W(t)=W*cos*ωt*
we obtain

and for *W(t)=Wexp(**±iωt)* we obtain

We have (Coulomb or radiation gauge)

We have a plane electromagnetic
wave propagating in the *y*-direction.

Let and with
*E _{0}* and

** S** is the Poynting
vector.

The Hamiltonian of an atomic electron interacting with this plane wave is

In this expression we consider only one independent electron, and we neglect the spin orbit interaction. We have

[*p _{z},A_{z}*]=0
since

To find induced transition
probabilities, we have to evaluate the matrix elements of *W(t)* between
unperturbed bound states. An order of magnitude estimate reveals

in the optical domain.
We assume that the intensity of the wave is low enough so that *W _{III}*,
the term containing

since
*y* is on the order of atomic dimensions.

Let *W _{DE}(t)*
be the zeroth order term in the expansion.

*W _{DE}* is called the

Note: This form of *W _{DE}*
is equivalent to the form we would get starting with the energy of an electric
dipole in an electric field.

This equivalence can be shown via a gauge transformation

The matrix elements of *W _{DE}*
are proportional to the matrix elements of

The integrant is a product of three spherical harmonics and the integral can be given in terms of Clebsch Gordan coefficients.

These are coefficients

The integral is zero unless

i) | |

ii) | |

iii) |

since

We therefore have that unless
If we choose another direction for the polarization of ** E**,
i.e. then
we find

The **dipole
transition selection rules**therefore
are

These selection rules result
as a **consequence of the properties of the spherical
harmonics**.

An electromagnetic field
is most likely to induce a transition between an initial and a final 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**.

When deriving the dipole
transition selection rules *Δl=±*1*,
**Δm=0*,
±1, we assumed that the Hamiltonian was
perturbed by *W _{DE}(t).
*We neglected the spin orbit interaction.
If

{|*l,s;j,m _{j}*>}
and not {|

(** p** or

if we take the first order
term in the expansion of exp(±*iky*). We may write

If in the expression for
*W _{II}(t)* we only keep the zeroth order term in the expansion
of exp(±

called
the **magnetic dipole Hamiltonian**, and

called
the **electric quadrupole Hamiltonian**.

Transition induced by *W _{DM}*
obey the

or, if *H _{0}
*includes a spin orbit coupling term,

Transition induced by *W _{QM}*
obey the

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