Calculation of time for electronic transitions

Question

How do we determine the time for electronic transitions in atoms or in semiconductor devices, from one energy level to another?

Answer 1

The lifetime of an atom in an excited state can be estimated from the Fermi Golden rule: $$ \frac{1}{\tau_{i\rightarrow f}} = \Gamma_{i\rightarrow f} =\frac{2\pi}{\hbar}|\langle i|\hat{V}|f\rangle|^2\delta(E_i-E_f) $$ What seems to pose the problem in this expression is the delta-function.

In a semiconductor it is actually not a problem at all, since, if one calculate the probability of transitions between, e.g.,conduction and valence bands (with an emission of photon) this necessarily involves integrating over the initial and/or final states with the appropriate density of states and the distribution function. Of course, once the delta function is under an integral, it does not pose any problem.

In atoms the situation seems a bit more tricky only because the Fermi Golden rule is usually derived in the context of a semiclassical theory, where the electromagnetic field is treated as externally driven and strictly monichromatic. (In fact, this semiclassical theory cannot accomodate spontaneous emission, so one postulates the corresponding Einstein coefficient). However, once the field is considered as having its own degress of freedom, with a continuous spectrum of photon frequencies, which also need to be integrated, the problem disappears.

In other words, the probability that an atom in a state $2$ emits a photon by transitioning to state $1$ is $$ \frac{1}{\tau_{2\rightarrow 1}} = \Gamma_{2\rightarrow 1} = \sum_k\frac{2\pi}{\hbar}|\langle 2|\hat{V}|1, 1_k\rangle|^2\delta(E_2-E_1-\hbar\omega_k) = \int d\omega\rho(\omega)\frac{2\pi}{\hbar}|\langle 2|\hat{V}|1, 1_\omega\rangle|^2\delta(E_2-E_1-\hbar\omega)= \frac{2\pi}{\hbar}|\langle 2|\hat{V}|1, 1_{\frac{E_2-E_1}{\hbar}}\rangle|^2\rho\left(\frac{E_2-E_1}{\hbar}\right), $$ where $\rho(\omega)$ is the density of the photon states and for simplicity I assumed zero temperature (so that I do not have to account for photon distribution function, spontaneous emission, etc.)

It is this latter form of the Fermi Golden rule that is often postulated in the semiclassical theory for practical applications: $$ \frac{1}{\tau_{2\rightarrow 1}} = \Gamma_{2\rightarrow 1} =\frac{2\pi}{\hbar}|\langle 2|\hat{V}|1\rangle|^2\rho(E_f) $$

Answer 2

In some cases one can measure the time, for example with pump-probe experiments with a variable delay between the laser pulses.

In cases where the line width is limited by the lifetime, one can use Heisenberg's indeterminacy principle for energy and time: $\Delta E \ \Delta t > \hbar/2.$

I understand this as the same relation as in the spectral bandwidth of sound pulses: the fewer oscillations there are in the pulse the greater is the uncertainty in frequency.

Answer 3

For an absorption process, the transition time (in order of $femto second$) is usually estimated using uncertainty principle: $$ \Delta t = \frac{\hbar}{\Delta E} = \frac{\hbar}{E_2 - E_1}; $$ Where $E_1$ and $E_2$ are the energies of the two levels.

For emision process, the life time of electron is much longer (in the range of pico seconds ~ nano seconds). The mechanisms cause the emission are very complicated often coupled to phonon, and therefore depends on temperature.