small fixes

diff --git a/solid_state_physics/tutorial/1_01.tex b/solid_state_physics/tutorial/1_01.tex
index b1fdabf..5510a4f 100644 (file)
--- a/solid_state_physics/tutorial/1_01.tex
+++ b/solid_state_physics/tutorial/1_01.tex
@@ -65,7 +65,7 @@ Using these approximations it is sufficient to consider a single electron locate
 Since most materials condense into almost perfect periodic arrays the periodicity should also hold for the potential style.
 
 Within this tutorial even the periodic potential is simplified.
 Since most materials condense into almost perfect periodic arrays the periodicity should also hold for the potential style.
 
 Within this tutorial even the periodic potential is simplified.

-Consider a single particle (mass $m$) enclosed in a box (side length $L=\mathcal{V}^{1/3}$) where the potential is constant ($V_0$) inside the box and infinite at the surface.
+Consider a single particle (mass $m$) enclosed in a box (side length $L=\mathcal{V}^{1/3}$) where the potential is zero inside the box and infinite at the surface.

 
 \begin{enumerate}
  \item Write down the Schr"odinger equation and boundary conditions
 
 \begin{enumerate}
  \item Write down the Schr"odinger equation and boundary conditions

diff --git a/solid_state_physics/tutorial/1_01s.tex b/solid_state_physics/tutorial/1_01s.tex
index 57d822d..83391bb 100644 (file)
--- a/solid_state_physics/tutorial/1_01s.tex
+++ b/solid_state_physics/tutorial/1_01s.tex
@@ -123,9 +123,9 @@
        \]
  \item $n_x,n_y,n_z=1,2,3\ldots$\\
        Allowed $k_{x,y,z}$ values located in positive octant only.
        \]
  \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}
        \includegraphics[width=10cm]{feg_kvals.eps}

-       \end{center}
+       \end{flushleft}

 
 \end{enumerate}
 
 
 \end{enumerate}
 


@@ -139,21 +139,36 @@ Convention:
 Prove:
 \[
 V_{real}=a_1(a_2 \times a_3)
 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) ]
 \]
 \[
        (a_3 \times a_1) \times (a_1 \times a_2) ]
 \]
 \[

-\textrm{hint 1: }
+\textrm{Hint 1: }

 (a_3 \times a_1) \times (a_1 \times a_2) =
 (a_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}

 \]
 \[
 \Rightarrow V_{rec}= \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} 
 \]
 \[
 \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}
 \]
 
 \end{document}