The time dependent Schrödinger equation is

Derivation of an Expression for the Intensity as Function of the Transition Probability per Unit Time

The formal solution is

where the following definitions apply

We can treat an applied electromagnetic field as a perturbation to the hamiltonian. The zeroth order hamiltonian (that of system without the field) is H₀. The field hamiltonian is H₁ = -m^.E. The total hamiltonian is

We consider two states, an initial state i and a final state f. The transition probability between the two states is

This expression is an approximate solution known as the Fermi Golden Rule. The Golden Rule gives the transition probability per unit time for transitions from i à f. The above expression is also valid for transitions from f à i if we interchange the indices. The same inherent transition rate applies for the absorption probability and for the emission probability. We wish to relate the transition rate to the intensity of absorbed radiation. It is the intensity that we will use to define the time-correlator for spectroscopy. First, we determine the net energy change in the radiation field:

The energy density is u. There are N molecules in volume V. The p_i is the probability that the ith initial state is occupied.

where q is the partition function. (Note that McHale calls the partition function z). w_if is the transition probability per unit time from above and hw_fi is the energy of an absorbed photon.

We now substitute in the explicit Golden Rule transition probability per unit time for w_if.

Since i and f are dummy indices we can interchange them in the second sum. Further we use the fact that w_fi = - w_if.

Note that

so that the change in the energy density per unit time becomes:

We can define the intensity of absorbed or emitted radiation as

Our goal is to relate the transition probability to the intensity of absorbed radiation. Recall that we have shown that the absorption coefficient is obtained from the intensity reduction as electromagnetic radiation travels through a medium:

From the definition of the imaginary part of the dielectric constant

we have the absorption coefficient g expressed in terms of the imaginary part of the permittivity:

We also saw that by comparison with Beer's law

we can express the familiar exctinction coefficient in terms of the imaginary part of the dielectric constant (relative permittivity):

By restablishing the connection between e_r''(w) and I(w) the relative intensity we can see that it is necessary to determine the imaginary part of the relative permittivity before we proceed. We can express e_r''(w) in terms of the energy density u.

Using the above relation we can express the imaginary part of the permittivity in terms of the transition probability. Note that the amplitude of the radiation E₀ has been factored out and instead there remains a unit vector pointed in the direction of the polarization of the radiation. We can choose any direction for the incident radiation (in the absence of applied static fields). Suppose we choose x-polarization. The square of the transition moment (m_if)² can be replaced by 1/3 m_fi×m_if. For a given polarization we have:

Combining the above expressions the intensity is:

It is the expression on the right that is used to derive a time-correlator.