Scattering rate

The interaction picture

Define the unperturbed Hamiltonian by $H_0$, the time dependent perturbing Hamiltonian by $H_1$ and total Hamiltonian by $H$.

The eigenstates of the unperturbed Hamiltonian are assumed to be

$H=H_0+H_1\$

$H_0 |k\rang = E(k)|k\rang$

In the interaction picture, the state ket is defined by

$|k(t)\rang _I= e^{iH_0 t /\hbar} |k(t)\rang_S= \sum_{k'} c_{k'}(t) |k'\rang$

By a Schrödinger equation, we see

$i\hbar \frac{\partial}{\partial t} |k(t)\rang_I=H_{1I}|k(t)\rang_I$

which is a Schrödinger-like equation with the total $H$ replaced by $H_{1I}$.

Solving the differential equation, we can find the coefficient of n-state.

$c_{k'}(t) =\delta_{k,k'} - \frac{i}{\hbar} \int_0^t dt' \;\lang k'|H_1(t')|k\rang \, e^{-i(E_k - E_{k'})t'/\hbar}$

where, the zeroth-order term and first-order term are

$c_{k'}^{(0)}=\delta_{k,k'}$

$c_{k'}^{(1)}=- \frac{i}{\hbar} \int_0^t dt' \;\lang k'|H_1(t')|k\rang \, e^{-i(E_k - E_{k'})t'/\hbar}$

The transition rate

The probability of finding $|k'\rang$ is found by evaluating $|c_{k'}(t)|^2$.

In case of constant perturbation,$c_{k'}^{(1)}$ is calculated by

$c_{k'}^{(1)}=\frac{\lang\ k'|H_1|k\rang }{E_{k'}-E_k}(1-e^{i(E_{k'} - E_k)t/\hbar})$

$|c_{k'}(t)|^2= |\lang\ k'|H_1|k\rang |^2\frac {sin ^2(\frac {E_{k'}-E_k} {2 \hbar}t)} { ( \frac {E_{k'} -E_k} {2 \hbar} ) ^2 }\frac {1}{\hbar^2}$

Using the equation which is

$\lim_{\alpha \rightarrow \infty} \frac{1}{\pi} \frac{sin^2(\alpha x)}{\alpha x^2}= \delta(x)$

The transition rate of an electron from the initial state $k$ to final state $k'$ is given by

$P(k,k')=\frac {2 \pi} {\hbar} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k)$

where $E_k$ and $E_{k'}$ are the energies of the initial and final states including the perturbation state and ensures the $\delta$-function indicate energy conservation.

The scattering rate

The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by

$w(k)=\sum_{k'}P(k,k')=\frac {2 \pi} {\hbar} \sum_{k'} |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k)$

The integral form is

$w(k)=\frac {2 \pi} {\hbar} \frac {L^3} {(2 \pi)^3} \int d^3k' |\lang\ k'|H_1|k\rang |^2 \delta(E_{k'}-E_k)$

References

• C. Hamaguchi (2001). Basic Semiconductor Physics. Springer. pp. 196–253.
• J.J. Sakurai. Modern Quantum Mechanics. Addison Wesley Longman. pp. 316–319.