Scattering rate

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The interaction picture[edit]

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[edit]

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[edit]

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[edit]

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