From: hackbard Date: Mon, 18 Jun 2012 19:45:56 +0000 (+0200) Subject: more on so in real space X-Git-Url: https://hackdaworld.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f47fcf0c7b7c7e2d5adc294d72a0c914289c584c;p=lectures%2Flatex.git more on so in real space --- diff --git a/physics_compact/solid.tex b/physics_compact/solid.tex index c4cf869..1fc7a17 100644 --- a/physics_compact/solid.tex +++ b/physics_compact/solid.tex @@ -200,3 +200,75 @@ V(r)=\sum_l \ket{l}\left[\bar{V}_l(r)+V^{\text{SO}}_l(r)LS\right]\bra{l} \text{ ,} \end{equation} where the first term correpsonds to the mass velocity and Darwin relativistic corrections and the latter is associated with the spin-orbit (SO) coupling. + + +\subsubsection{Excursus: real space representation suitable for an iterative treatment} + +In the following, the spin-orbit part is evaluated in real space. +Since spin is treated in another subspace, it can be treated separately. +The matrix elements of the orbital angular momentum part of the potential in KB form for orbital angular momentum $l$ read +\begin{equation} +\bra{r'}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l\braket{\chi_{lm}}{r''} +\text{ .} +\end{equation} +With +\begin{eqnarray} +\bra{r'}p\ket{\chi_{lm}} & = & -i\hbar\nabla_{r'} \braket{r'}{\chi_{lm}} +=-i\hbar\nabla_{r'}\,\chi_{lm}(r') \\ +r\ket{r'} & = & r'\ket{r'} +\end{eqnarray} +we get +\begin{equation} +-i\hbar(r'\times \nabla_{r'})\braket{r'}{\chi_{lm}}E^{\text{SO,KB}}_l +\braket{\chi_{lm}}{r''} +\text{ .} +\end{equation} +To further evaluate this expression, the KB projectors +\begin{equation} +\chi_{lm}=\frac{\ket{\delta V_l^{\text{SO}}\Phi_{lm}}} +{\braket{\delta V_l^{\text{SO}}\Phi_{lm}} + {\Phi_{lm}\delta V_l^{\text{SO}}}^{1/2}} +\end{equation} +must be known in real space (with respect to $r$). +\begin{equation} +\braket{r'}{\chi_{lm}}= +\frac{\braket{r'}{\delta V_l^{\text{SO}}\Phi_{lm}}}{ +\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} +^{1/2}} +\end{equation} +and +\begin{equation} +\braket{r'}{\delta V_l^{\text{SO}}\Phi_{lm}}= +\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}Y_{lm}(\Omega_{r'}) +\text{ .} +\end{equation} +In this expression, only the spherical harmonics are complex functions. +Thus, the complex conjugate with respect to $r''$ is given by +\begin{equation} +\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{r''}= +\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}Y^*_{lm}(\Omega_{r''}) +\text{ .} +\end{equation} +Using the orthonormality property +\begin{equation} +\int Y^*_{l'm'}(\Omega_r)Y_{lm}(\Omega_r) d\Omega_r = \delta_{ll'}\delta_{mm'} +\label{eq:solid:y_ortho} +\end{equation} +of the spherical harmonics, the norm of the $\chi_{lm}$ reduces to +\begin{eqnarray} +\braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\Phi_{lm}\delta V_l^{\text{SO}}} &=& +\int \braket{\delta V_l^{\text{SO}}\Phi_{lm}}{\vec{r}'} +\braket{\vec{r}'}{\Phi_{lm}\delta V_l^{\text{SO}}} d\vec{r'}'\\ +&=&\int +{\delta V_l^{\text{SO}}}^2(r')\frac{u_l^2(r')}{r'^2}Y^*_{lm}(\Omega_{r'}) +Y_{lm}(\Omega_{r'}) +r'^2 dr' d\Omega_{r'} \\ +&=&\int_{r'} +{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' +\int_{\Omega_{r'}}Y^*_{lm}(\Omega_{r'})Y_{lm}(\Omega_{r'}) d\Omega_{r'}\\ +&=&\int_{r'}{\delta V_l^{\text{SO}}}^2(r') u_l^2(r') dr' \text{ .} +\end{eqnarray} + +Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots +\begin{equation} +\end{equation}