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Problems

  1. Poisson's equation $\nabla^2\phi = -
\rho/\epsilon_0$ is usually differenced in 2 dimensions as

    \begin{displaymath}
\phi_{i-1,j} + \phi_{i+1,j} + \phi_{i,j-1} + \phi_{i,j+1} - 4\phi_{ij} =
S_{ij}
\end{displaymath}

    where $S_{ij} = - \delta x^2 \rho_{ij}/\epsilon_0$ and $i,j =
0,1,2,\ldots,N-1$. The difference equation can be written as a matrix equation $\sum_n A_{mn}f_n = \sigma_m$ where $f_{iN+j} = \phi_{ij}$ and $\sigma_{iN+j} = S_{ij}$. Write down the matrix $\bss{A}$ for $N=3$. Assume the boundaries are at zero potential, i.e. $\phi_{ij} = 0$ iff $i\mbox{ or }j=-1$ or $i\mbox{ or
}j=N$.
  2. The equation in question 1 can be solved by the Gauss-Seidel method. If $S_{11}=1$ and $S_{ij}=0$ otherwise, find $\phi_{ij}^{(1)}$ and $\phi_{ij}^{(2)}$ if $\phi_{ij}^{(0)}=0$ for all $i$ and $j$, where $\phi_{ij}^{(n)}$ is the value of $\phi$ after $n$ iterations.
  3. Work out the factors, L and U, in the LU decomposition of the matrix,

    \begin{displaymath}
\bss{A} = \left (
\begin{array}{ccc}
1 & 3 & 5 \\
2 & 5 & 3 \\
1 & 1 & 4 \\
\end{array}
\right ) .
\end{displaymath}

    Hence,
  4. Show that the Jacobi method (3.14) is stable as long as, $\mu$, the eigenvalue of largest modulus of $\bss{D}^{-1}\left(\bss{L} + \bss{U}\right)$ is less than unity.
  5. Find both eigenvalues and the corresponding left and right handed eigenvectors of the matrix

    \begin{displaymath}
\left(\begin{array}{ll} \epsilon & -1\\ 1 & 0\end{array}\right).
\end{displaymath}

  6. The vibrational modes of a certain molecule are described by an equation of motion in the form

    \begin{displaymath}
m_i\omega^2 x_i = K_{ij} x_j,
\end{displaymath}

    where $m_i$ and $x_i$ are the mass and the displacement respectively of the $i$th atom and the real symmetric matrix $\bss{K}$ describes the interactions between the atoms. Show that this problem can be represented in the form of a generalised eigenvalue problem: $\bss{A}\bi{x} = \lambda\bss{B}\bi{x}$ in which the matrix $\bss{B}$ is positive definite. By considering the transformation $x'_i = (m_i)^{\mathchoice{{\textstyle{\frac12}}}{{\textstyle{\frac12}}}{{\scriptstyle{1/2}}}{{\scriptscriptstyle{1/2}}}}x_i$, show how to transform this into a simple eigenvalue problem in which the matrix is real symmetric.
  7. Write down the types of matrices which occur in the following problems:
    1. A simple discretisation (as in question1) of Poisson's equation in 1 dimension.
    2. The same but in more than 1D.
    3. A simple discretisation (as above) of Schrödinger's equation in 3 dimensions.
    4. Schrödinger's equation for a molecule written in terms of atomic basis functions.
    5. Schrödinger's equation for a crystal at a general $\bi{k}$ point in the Brillouin zone.

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