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32 {\LARGE {\bf Materials Physics I}\\}
37 {\Large\bf Tutorial 1 - proposed solutions}
40 \section{Free electron in a box}
45 \item Schr"odinger equation:\\
47 - \frac{\hbar^2}{2m} \nabla^2 \Psi({\bf r}) + V({\bf r}) \Psi({\bf r})
48 = E \Psi({\bf r}) \textrm{ , }
49 V({\bf r}) = 0 \textrm{ for } {\bf r} \in [0,L]
51 \item Boundary conditions: $\Psi({\bf r}) = \Psi(x,y,z)$\\
53 \Psi(0,y,z) = \Psi(L,y,z) = 0 \qquad
54 \Psi(x,0,z) = \Psi(x,L,z) = 0 \qquad
55 \Psi(x,y,0) = \Psi(x,y,L) = 0
62 \item Product ansatz: $\Psi({\bf r})=F_x(x)F_y(y)F_z(z)$, with\\
63 $F_x(x)=0$ for $x=0,L$\\
64 $F_y(y)=0$ for $y=0,L$\\
65 $F_z(z)=0$ for $z=0,L$.
66 \item Schr"odinger equation:\\
67 Use: $\nabla^2=\frac{\partial^2}{\partial x^2} +
68 \frac{\partial^2}{\partial y^2} +
69 \frac{\partial^2}{\partial z^2} \Rightarrow$\\
71 - \frac{\hbar^2}{2m} \Big[
72 F_y(y) F_z(z) \frac{d^2}{dx^2} F_x(x) +
73 F_x(x) F_z(z) \frac{d^2}{dy^2} F_y(y) +
74 F_x(x) F_y(y) \frac{d^2}{dz^2} F_z(z)
76 E F_x(x) F_y(y) F_z(z)
78 \item Schr"odinger equation fullfilled if:
80 - \frac{\hbar^2}{2m} \frac{d^2}{dx^2} F_x(x) = E_x F_x(x), \quad
81 - \frac{\hbar^2}{2m} \frac{d^2}{dy^2} F_y(y) = E_y F_y(y),\quad
82 - \frac{\hbar^2}{2m} \frac{d^2}{dz^2} F_z(z) = E_z F_z(z).
85 \Rightarrow \Big[E_x + E_y + E_z\Big] F_x(x) F_y(y) F_z(z) =
86 E F_x(x)F_y(y)F_z(z) \textrm{, } \quad E = E_x + E_y + E_z
88 Three eigenvalue problems of the same character.
89 Sufficient to examine only one!
90 \item Solution of the Schr"odinger equation:\\
92 \item Ansatz: $F_x = A_x \exp(ik_xx) + B_x \exp(-ik_xx)$
93 \item Boundary conditions:\\
94 $F_x(0)=0 \Rightarrow B_x=-A_x$, \quad
95 let ${A}_x = \frac{1}{2i}\tilde{A}_x$
96 $\Rightarrow$ $F_x(x) = \tilde{A}_x \sin(k_xx)$\\
97 $F_x(L)=0 \Rightarrow k_x L = n_x \pi$ or rather
98 $k_x=n_x \frac{\pi}{L}$, \quad $n_x=0,\pm1,\pm2,\ldots$
99 \item Forbidden values for $n_x$:\\
100 $n_x \ne 0$: otherwise wave function zero for all $x$\\
101 $n_x > 0$: wave functions for $+n_x$ and $-n_x$
102 not linearly independent (same quantum sate)
106 \Psi_{n_x n_y n_z} = A \sin(\frac{n_x \pi}{L}x)
107 \sin(\frac{n_y \pi}{L}y)
108 \sin(\frac{n_z \pi}{L}z),
109 \quad A=\tilde{A}_x\tilde{A}_y\tilde{A}_z
113 E_{n_x n_y n_z} = \frac{\hbar^2 \pi^2}{2m L^2}(n_x^2+n_y^2+n_z^2)
119 \Psi_{111} = A \sin(\frac{\pi}{L}x) \sin(\frac{\pi}{L}x)
120 \sin(\frac{\pi}{L}x) \qquad
121 E_{111} = \frac{\hbar^2 \pi^2}{2m L^2} (1+1+1)
122 = \frac{3 \hbar^2 \pi^2}{2m L^2}
124 \item $n_x,n_y,n_z=1,2,3\ldots$\\
125 Allowed $k_{x,y,z}$ values located in positive octant only.
127 \includegraphics[width=10cm]{feg_kvals.eps}
132 \section{Reciprocal lattice}
136 \item basis of unit cell in real space: $a_1,a_2,a_3$
137 \item basis of unit cell in reciprocal space: $b_1,b_2,b_3$
141 V_{real}=a_1(a_2 \times a_3)
143 b_1=\frac{2\pi(a_2 \times a_3)}{a_1(a_2 \times a_3)}
145 b_2=\frac{2\pi(a_3 \times a_1)}{a_1(a_2 \times a_3)}
147 b_3=\frac{2\pi(a_1 \times a_2)}{a_1(a_2 \times a_3)}
150 V_{rec}=b_1 ( b_2 \times b_3)=
151 \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3} (a_2 \times a_3) [
152 (a_3 \times a_1) \times (a_1 \times a_2) ]
156 (a_3 \times a_1) \times (a_1 \times a_2) =
157 a_1((a_3 \times a_1)a_2) - \underbrace{a_2((a_3 \times a_1)a_1)}_{=0}
160 \Rightarrow V_{rec}= \frac{(2\pi)^3}{(a_1(a_2 \times a_3))^3}
161 (a_2 \times a_3) (a_1((a_3 \times a_1) a_2))
165 (a_2 \times a_3) (a_1((a_3 \times a_1) a_2)) =
166 (a_2 \times a_3) (a_1((a_2 \times a_3) a_1)) =
167 (a_1 (a_2 \times a_3))^2
170 \Rightarrow V_{rec}=\frac{(2\pi)^3}{a_1(a_2 \times a_3)}=
171 \frac{(2\pi)^3}{V_{real}}