\[
- \frac{\hbar^2}{2m} \frac{d^2}{dx^2} F_x(x) = E_x F_x(x), \quad
- \frac{\hbar^2}{2m} \frac{d^2}{dy^2} F_y(y) = E_y F_y(y),\quad
- - \frac{\hbar^2}{2m} \frac{d^2}{dz^2} F_z(z) = E_x F_z(z).
+ - \frac{\hbar^2}{2m} \frac{d^2}{dz^2} F_z(z) = E_z F_z(z).
\]
\[
\Rightarrow \Big[E_x + E_y + E_z\Big] F_x(x) F_y(y) F_z(z) =
\]
\item $n_x,n_y,n_z=1,2,3\ldots$\\
Allowed $k_{x,y,z}$ values located in positive octant only.
- \begin{center}
+ \begin{flushleft}
\includegraphics[width=10cm]{feg_kvals.eps}
- \end{center}
+ \end{flushleft}
\end{enumerate}
Prove:
\[
V_{real}=a_1(a_2 \times a_3)
+\]\[
+b_1=\frac{2\pi(a_2 \times a_3)}{a_1(a_2 \times a_3)}
+\]\[
+b_2=\frac{2\pi(a_3 \times a_1)}{a_1(a_2 \times a_3)}
+\]\[
+b_3=\frac{2\pi(a_1 \times a_2)}{a_1(a_2 \times a_3)}
\]
\[
-V_{rec}=b_1 ( b_2 \times b_3)
- =\frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} (a_2 \times a_3) [
+V_{rec}=b_1 ( b_2 \times b_3)=
+ \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} (a_2 \times a_3) [
(a_3 \times a_1) \times (a_1 \times a_2) ]
\]
\[
-\textrm{hint 1: }
+\textrm{Hint 1: }
(a_3 \times a_1) \times (a_1 \times a_2) =
-a_1((a_3 \times a_1)a_2) - a_2((a_3 \times a_1)a_1) =
-a_1((a_3 \times a_1)a_2)
+a_1((a_3 \times a_1)a_2) - \underbrace{a_2((a_3 \times a_1)a_1)}_{=0}
\]
\[
\Rightarrow V_{rec}= \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3}
-(a_2 \times a_3) (a_1(a_3 \times a_1) a_2)
+(a_2 \times a_3) (a_1((a_3 \times a_1) a_2))
+\]
+\[
+\textrm{Hint 2: }
+(a_2 \times a_3) (a_1((a_3 \times a_1) a_2)) =
+(a_2 \times a_3) (a_1((a_2 \times a_3) a_1)) =
+(a_1 (a_2 \times a_3))^2
+\]
+\[
+\Rightarrow V_{rec}=\frac{(2\pi)^3}{a_1(a_2 \times a_3)}=
+\frac{(2\pi)^3}{V_{real}}
\]
\end{document}