Two State Model for Transitions

We are most frequently concerned with the transition between two levels. We can call these the levels the ground and excited state and denote them level 1 and 2, respectively. For example, when we calculate the transition probability using the Einstein B coefficient, only two levels will contribute. These states have energy eigenvalues E1 and E2 shown as follows

The corresponding time-dependent wavefunctions are

The transition frequency is called w0 in the following. This correponds to wif in McHale's treatment. The transition frequency is related to the energy difference between the states by

Assume that at some instant of time the atom is in its ground state y1. Because of the presence of the perturbation H'(t) due to the electromagnetic radiation the time-dependent hamiltonian is

H0 + H'(t)

and the atom is not in a stationary state. There is a finite probability that the atom will be found in state y2. If the frequency of the light is close to w0, only the two selected atomic states are involved in the radiative processes. This means that the wavefunction at any instant of time is a linear superposition of the two wavefunctions for the individual states.

The wavefunction must be normalized at all times

The individual state wavefunctions y1 and y2 have been assumed to be normalized and orthogonal.

Substitution of the wavefunction into the time-dependent Schroedinger equation yields

Multiplication by the complex conjugate Y1* followed by integration over all space yields

Using the definitions

we can express the time evolution as two coupled differential equations.

In order to proceed further we must include an explicit time-dependent hamiltonian H'(t). The interaction that results in transitions is the interaction of dipole with an electric field. This has the form

You might ask what happened to the spatial dependence of the electric field. Since the wavelength of the light is much larger than a typical molecular dimension this dependence is not important for the absorption of radiation. Also, the dipole can be the permanent dipole moment (for vibrational and rotational transitions) or the transition dipole (for electronic transitions). We will consider these details later. At present we will write a generic dipole as m12. With this notation we can write the time-dependent perturbation as

Note that this implies that J11 = J22 = 0. This is because of the symmetry of the problem. Since the electric vector is polarized (e.g. along x, y, or z) the perturbation is an odd function. Since either the ground state Y1 or the excited Y2 multiplied times itself is even this means that the integrals represented by J11 and J22

are odd and must vanish. However, the integrals

do not vanish and are related by J12 = J21*. Consider an explicit example of x-polarized radiation. Then

We now write the coupled equations for states 1 and 2 as 

Assuming that the atom or molecule is in the ground state at t = 0 we have c1(0) = 1 and c2(0) = 0. Keep in mind that |c2(t)|2 is the probability of finding population in state 2 (the excited state) at time t and |c2(t)|2/t is the rate of at which probability in state 2 grows.

We now make the approximation that c1(t) = 1. This leads immediately to

We will use the fact that

to write this expression as

The next step is to integrate

The integral is just the sum of two exponential integrals. However, the integration is not carried out over all time, but from t=0 to the time t. Thus, the coefficient is evaluated at time following the initiation of the time-dependent perturbation.

Note that the factor i cancels. Since we are interested in cases where the incident frequency w is close to the resonant frequency w0 we can apply the rotating wave approximation:

This allows us to neglect the first term in the expression for c2(t). This is known as the rotating wave approximation. The coefficient is

The probability of finding the molecule in state 2 at time t is the square of this coefficient, but since coefficient is complex we must compute c2*(t)c2(t). Thus,

This expression can be considered for narrow band and broad band radiation.

Case I: Narrow band radiation. If w = w0 exactly then we can expand sin(x) = x - x3/6 + x5/120 - …» x. Using sin(x) » x we have

Thus,

Case II: Broad band radiation. For broad band radiation we must integrate over the resonance function sin2(x)/x2. In the extreme broadband limit we integrate from w0 - infinity to w0 + infinity. This integral is:

Thus,

and the probability per unit time is 

This approximation is only valid for large values of (w - w0)t.

For an extremely narrow (infinitesimally narrow) bandwidth we can express the function sin2(x)/x2 function as a delta function:

Therefore,

This is known as the Fermi Golden Rule.