scattering rate

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

:$H=H\_0+H\_1\backslash$

:$H\_0\; |k\backslash rang\; =\; E(k)|k\backslash rang$

In the interaction picture, the state ket is defined by

:$|k(t)\backslash rang\; \_I=\; e^\{iH\_0\; t\; /\backslash hbar\}\; |k(t)\backslash rang\_S=\; \backslash sum\_\{k\text{'}\}\; c\_\{k\text{'}\}(t)\; |k\text{'}\backslash rang$

By a Schrödinger equation, we see

:$i\backslash hbar\; \backslash frac\{\backslash partial\}\{\backslash partial\; t\}\; |k(t)\backslash rang\_I=H\_\{1I\}|k(t)\backslash rang\_I$

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.

:$c\_\{k\text{'}\}(t)\; =\backslash delta\_\{k,k\text{'}\}\; -\; \backslash frac\{i\}\{\backslash hbar\}\; \backslash int\_0^t\; dt\text{'}\; \backslash ;\backslash lang\; k\text{'}|H\_1(t\text{'})|k\backslash rang\; \backslash ,\; e^\{-i(E\_k\; -\; E\_\{k\text{'}\})t\text{'}/\backslash hbar\}$

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

:$c\_\{k\text{'}\}^\{(0)\}=\backslash delta\_\{k,k\text{'}\}$

:$c\_\{k\text{'}\}^\{(1)\}=-\; \backslash frac\{i\}\{\backslash hbar\}\; \backslash int\_0^t\; dt\text{'}\; \backslash ;\backslash lang\; k\text{'}|H\_1(t\text{'})|k\backslash rang\; \backslash ,\; e^\{-i(E\_k\; -\; E\_\{k\text{'}\})t\text{'}/\backslash hbar\}$

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

In case of constant perturbation,$c\_\{k\text{'}\}^\{(1)\}$ is calculated by

:$c\_\{k\text{'}\}^\{(1)\}=\backslash frac\{\backslash lang\backslash \; k\text{'}|H\_1|k\backslash rang\; \}\{E\_\{k\text{'}\}-E\_k\}(1-e^\{i(E\_\{k\text{'}\}\; -\; E\_k)t/\backslash hbar\})$

:$|c\_\{k\text{'}\}(t)|^2=\; |\backslash lang\backslash \; k\text{'}|H\_1|k\backslash rang\; |^2\backslash frac\; \{sin\; ^2(\backslash frac\; \{E\_\{k\text{'}\}-E\_k\}\; \{2\; \backslash hbar\}t)\}\; \{\; (\; \backslash frac\; \{E\_\{k\text{'}\}$

-E_k} {2 \hbar} ) ^2 }\frac {1}{\hbar^2}

Using the equation which is

:$\backslash lim\_\{\backslash alpha\; \backslash rightarrow\; \backslash infty\}\; \backslash frac\{1\}\{\backslash pi\}\; \backslash frac\{sin^2(\backslash alpha\; x)\}\{\backslash alpha\; x^2\}=\; \backslash delta(x)$

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

:$P(k,k\text{'})=\backslash frac\; \{2\; \backslash pi\}\; \{\backslash hbar\}\; |\backslash lang\backslash \; k\text{'}|H\_1|k\backslash rang\; |^2\; \backslash delta(E\_\{k\text{'}\}-E\_k)$

where $E\_k$ and $E\_\{k\text{'}\}$ are the energies of the initial and final states including the perturbation state and ensures the $\backslash delta$-function indicate energy conservation.

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

:$w(k)=\backslash sum\_\{k\text{'}\}P(k,k\text{'})=\backslash frac\; \{2\; \backslash pi\}\; \{\backslash hbar\}\; \backslash sum\_\{k\text{'}\}\; |\backslash lang\backslash \; k\text{'}|H\_1|k\backslash rang\; |^2\; \backslash delta(E\_\{k\text{'}\}-E\_k)$

The integral form is

:$w(k)=\backslash frac\; \{2\; \backslash pi\}\; \{\backslash hbar\}\; \backslash frac\; \{L^3\}\; \{(2\; \backslash pi)^3\}\; \backslash int\; d^3k\text{'}\; |\backslash lang\backslash \; k\text{'}|H\_1|k\backslash rang\; |^2\; \backslash delta(E\_\{k\text{'}\}-E\_k)$

• {{cite book | author=C. Hamaguchi | title=Basic Semiconductor Physics | publisher=Springer | year=2001 | pages= 196–253}}
• {{cite book | author=J.J. Sakurai | title=Modern Quantum Mechanics | publisher=Addison Wesley Longman | pages= 316–319}}

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