Transitions among states: the Fermi Golden Rule

We consider the form of the perturbation H'(t) corresponding to the interaction of electromagnetic radiation with a molecule. This treatment ignores the spatial dependence of the wave function. The justification for this is that the dimensions of a wavelength of light are much larger than the size of a molecule. The time dependence of an electromagnetic wave, however, is important. One intuitive way to see this is from the well-known Einstein relation which states that the energy of a transition from an initial state |iñ to a final state |fñ must equal hn, where n is the frequency of the photon inducing the transition, Ef - Ei = hn. Since n is frequency with units of s-1 we can see that time-dependence must enter into the transition probability. Recall that the angular frequency w = 2pn and the Bohr frequency can also be written

The derivation that follows considers an interaction of the form H'(t) = -m×E(t) where E(t) is a time-varying electromagnetic field.

The perturbation is

Substituting this perturbation hamiltonian into the expression for the time-dependent coefficient

and recalling that

we have

and

There are two integral terms depending on wfi + w and wfi - w, respectively. The integrals are standard exponential integrals.

Thus, we obtain

The amplitude of the final state is large for the condition w » wfi or w » - wfi. These correspond to transitions to final states higher in energy than |iñ (wfi positive) and lower in energy than |Iñ (wfi negative). A transition i à f to higher energy corresponds to an absorptive transition. A transition to a lower energy final state corresponds to stimulated emission. For now, we consider only absorption (w » wfi). Then only the second term contributes. The coefficient cf(t) is a probability amplitude. To obtain the time-dependent probability we square this term.

If wfi - w = Dw then

For the final step of this transformation we use the sum angle formula

The transition probability from i à f becomes

The form of the transition probability is the square of a matrix element that represents the transition dipole times a factor that resembles sin2x/x2. The Dw term represents the mismatch between the frequency of radiation w and the Bohr frequency for the transition wfi. To understand the functional form of sin2x/x2 we plot this function below.

The area under the sin2(x)x2 function is

If we wish to set x = Dwt/2 then we must divide the function

by t2. In other words

Now, if we consider the integral over frequency we have

Note that this is not exactly the same as sin2x/x2 because the terms in the integrand increase as time increases. We consider the effect of increasing a factor multiplying x in sin2x/x2 in the Figure below.

To consider how this affects the integral we rewrite sin2x/x2 to include a factor a so that our new function is sin2(ax)/(ax)2. Using a standard substitution x' = ax and therefore dx = dx'/a. We can write

Note that the functional form is becoming narrower and narrower as a increases. The area gets smaller as 1/a as seen in the integrated value and in the Figure above.

If we wish to set x = Dwt/2 then we must divide the function

by t2. In other words

Now, if we consider the integral over frequency we have

Note that this is not exactly the same as sin2x/x2 because the terms in the integrand increase as time increases.

Using the substitution Dw' = Dwt/2, dw = 2dw'/t the integral becomes

The last step is a kind of trick used often in quantum mechanics. Functions such as a Lorentzian, Gaussian, sin2(x)/x2 etc. can all be turned into delta functions in the limit that their full-width at half maximum goes to zero. Here we assume the long time limit will perform that function since the width of function decreases as time increases. To see this point consider the Figure above and imagine that the factor multiplying x is time. As the factor gets larger then function gets narrower.

Having established that we can write

we can now write the transition probability as

The transition rate is the transition probability per unit time.

This expression is known as the Fermi Golden Rule. It states that the transition rate is not time dependent. The transition rate depends on the square of the transition moment |ái|m×E0|fñ| = |ái|m|fñ|×E0 = mfi×E0. In these expression m is the dipole operator. For example, along the z axis (for z-polarized radiation) m = -ez, where e is an electronic charge. The delta function d(wfi - w) is an energy matching term. It is supports the Bohr frequency condition hypothesis since it requires that the energy of a photon be exactly matched to the energy of a transition. In practice, the delta function will be replaced with a lineshape function (e.g. a Lorentzian, Gaussian, or a convolution of the two). The reason for this is that the energies of states are not precisely defined in any but the infinite time limit. This is due to the uncertainty principle.

DEDt > h/2p

Clearly if Dt is finite, then DE (the energy width) > 0. At very short times the Fermi Golden Rule is not valid. At sufficiently short times we will consider explicitly the oscillation of population from the initial state to the final state and back again. This is a quantum mechanical effect due to the time-dependent evolution of the wavefunction. In optical spectroscopy we would observe this type of effect for an infinitely narrow wavelength radiation source interacting with a molecular system that had infinitely narrow energy level widths. In practice this type of effect is difficult to observe because of the finite lifetimes of states that give rise to energy broadening of their levels.

The Fermi Golden Rule is valid for short pulses as well as continuous wave experiments. It is value for rotational, vibrational, and electronic transitions as well as nuclear transitions in NMR (radiowave excitation). With a change in nature of the time-dependent perturbation the above treatment is also valid for non-radiative dynamic phenomena such as internal conversion, intersystem crossing, energy transfer, electron transfer, atom transfer etc.