The transition probability P_if(t,w) for an atom placed into an electromagnetic plane wave with and angular frequency w near w_fi is in the electric dipole approximation

Here

The incident radiation, however is often not monochromatic.  Let I(w)dw be the intensity of the radiation in the interval between w and w+dw.  Assume the incident radiation is incoherent. Then

The total transition probability can now be found by integrating the transition probability induced by the radiation in each frequency interval.

If I(w) varies little in the interval and then we may evaluate at w=w_fi and treat it as a constant.  We then obtain

The transition probability per unit time is given by  The transition probability per unit time is proportional to the intensity of the radiation for induced transitions.

We can calculate induced transition probabilities using time-dependent perturbation theory, but as long as we treat the electromagnetic field classically, we cannot calculate the probability for spontaneous emission of a photon this way.

We can however avoid this problem by making statistical arguments.  Consider two atomic states of energy E₁ and E₂.  Let E₂>E₁.  Assume the atom is bathed in radiation of energy density u(w).  Transitions between these states can take place in three different ways.

• (1)  spontaneous, 2 to 1, with probability A₂₁ per unit time;
• (2)  induced, 2 to 1, with probability B₂₁u(w₁₂) per unit time;
• (3)  induced, 1 to 2, with probability B₁₂u(w₁₂) per unit time.

A₂₁, B₂₁, and B₁₂ are called the Einstein coefficients.

In a cavity in thermal equilibrium the probabilities that states 1 and 2 are occupied are exp(-E₁/(KT)) and exp(-E₂/(kT)) respectively, and in equilibrium the probability of up transitions must exactly balance the probability of down transitions. We therefore need

since B₁₂=B₂₁ is given by C₁₂ from above, averaged over all polarizations and all directions of incidence of the plane wave.

Since

(Planck’s law),

we have

Problem:

Use this information to derive the relations among Einstein's A and B coefficients which measure the spontaneous and the induced transition probabilities, respectively, between energy levels E_a and E_b.

(b) Now consider an atom in an electromagnetic field

with Hamiltonian

where -q is the charge of the electron.  The energy density in the field is Neglecting terms in A², calculate the probability for absorption of radiation in the dipole approximation.  Express your answer in terms of matrix elements of the form <b|r|a>.

(c) Relate the lifetime of an excited state to the A coefficient and calculate the shape of the intensity distribution ( as a function of w ) observed as a consequence of spontaneous emission.

• Solution:

  (a)  (This was shown above.)

  (b) 

  

  The component of p in the direction of A commutes with A.  A is perpendicular to the direction of propagation but depends on the coordinate in the direction of propagation.

  

  ( dipole approximation ).

  

   

  Therefore

  

  with i = b and f = a.  (See previous module.)

  This is what is required for part (b) of the problem.  Let us expand and assume that the incident radiation is not monochromatic.  Let I(w)dw be the intensity of the radiation in the interval between w and dw.  Assume the incident radiation is incoherent and Then and the transition probability per unit time is

  

  Let us expand even further and assume that the radiation is isotropic.  In order to find the total rate induced by an isotropic field, it is necessary to sum this expression over the two possible states of polarization for each direction of propagation and to integrate over all directions of propagation.  The two polarization vectors and the unit vector form a triplet of orthogonal unit vectors.  For any fixed vector r we have

  

  

  Now let us relate I(w_fi) to u(w_fi) for the isotropic field.  For a particular polarization

  

  The factor of ½ is needed because u(w) contains two linearly independent polarizations.  We now have

  

  (c) 

  A₂₁ is the spontaneous transition rate.

  is the lifetime,

  Since the energies of excited states are not infinitely sharp, the emission lines have Lorentzian shapes. We write

  

  Therefore