author hackbard Tue, 19 Jun 2012 08:25:55 +0000 (10:25 +0200) committer hackbard Tue, 19 Jun 2012 08:25:55 +0000 (10:25 +0200)

index 33fe8a4..507de12 100644 (file)
@@ -283,7 +283,41 @@ To obtain a final expression for the matrix elements \eqref{eq:solid:so_me}, the
\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
Y^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
\end{eqnarray}
-All megnetic states $m=-l,-l+1,\ldots,l-1,l$ contribute to the potential for angular momentum $l$.
+and if all megnetic states $m=-l,-l+1,\ldots,l-1,l$ that contribute to the potential for angular momentum $l$ are considered
+
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'}) \text{ ,}
+
+which can be rewritten as
+
+\braket{\vec{r'}}{\delta V_l^{\text{SO}}\Phi_{lm}}
+\braket{\Phi_{lm}\delta V_l^{\text{SO}}}{\vec{r''}}=
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}
+\frac{2l+1}{4\pi}P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)
+
+using the vector addition theorem
+
+P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)=
+\frac{4\pi}{2l+1}\sum_mY^*_{lm}(\Omega_{r''})Y_{lm}(\Omega_{r'})
+
+In total, the matrix elements of the potential for angular momentum $l$ can be calculated as
+\begin{eqnarray}
+\bra{\vec{r'}}V^{\text{KB,SO}}\ket{\vec{r''}}&=&
+\bra{\vec{r'}}(r\times p)\ket{\chi_{lm}}E^{\text{SO,KB}}_l
+\braket{\chi_{lm}}{\vec{r''}}\\
+&=&
+-i\hbar(r'\times \nabla_{\vec{r'}})
+\delta V_l^{\text{SO}}(r')\frac{u_l(r')}{r'}
+P_l\left(\frac{\vec{r'}\vec{r''}}{r'r''}\right)\times\\
+&&\times\frac{E^{\text{SO,KB}}_l\delta V_l^{\text{SO}}(r'')\frac{u_l(r'')}{r''}}
+             {\int_{r}{\delta V_l^{\text{SO}}}^2(r) u_l^2(r) dr} \,
+\frac{2l+1}{4\pi}
+\end{eqnarray}
+
Finally, to evaluate $V^{\text{SO}}_l\ket{\Psi}$, the integral \ldots