Math 658-600 Current Assignment — Fall 2012
Assignment 9 Not to be handed in.
- Problems
- Let $L_\pm$ be the raising and lowering operators in
the
Notes on Spherical Harmonics. Use them to show that, for $m\ge
0$, if $Y_{\ell,m}(\theta,\varphi) =
Q_{\ell,\,m}(\cos(\theta))\sin^m(\theta)e^{im\varphi}$, where
$Q_{\ell,\,m}(t)$ is a polynomial in $t$, then $Q_{\ell,\,m-1} =
C((1-t^2)Q'_{\ell,\, m}+2m t Q_{\ell,\,m})$ and, in addition, that $
Q_{\ell,\,m+1} = C Q_{\ell,\,m}'(t)$. Use these formulas along with
$Y_{\ell,\,\ell}(\theta,\varphi)=C\sin^\ell(\theta)e^{i\ell \varphi}$
to show that the $Y_{\ell,\, m}$ have the form in Theorem 3.1, up to
normalization constants. ("C" is a generic constant.)
- Use Theorem 3.1
in the
Notes on Spherical Harmonics to expand the function
$f(\theta,\varphi) = 2 - \cos^2(\theta)+ \cos(\theta)\sin^2(\varphi)$
in spherical harmonics.
- Recall that a function $f$ in $\mathbb R^3$ is said to be
harmonic if and only if $\Delta_{\mathbb R^3}f = 0$, and also that a
harmonic polynomial homogeneous of degree $\ell$ is related to a
spherical harmonic $Y_\ell$ via $p(\mathbf x)=r^\ell Y_\ell(\mathbf
x/r)$. (Here $\Delta_{\mathbb S^2}Y_\ell + \ell(\ell+1)Y_\ell =0$.)
Find a basis for harmonic polynomials homogeneous of degree $\ell=3$
using Theorem 3.1 in
the
Notes on Spherical Harmonics.
- Fix $m\ge 1$. Show that the polynomials $\{P^{(m)}_\ell : \ell\ge m \}$ are
orthogonal with respect to the inner product $\langle f,g\rangle =
\int_{-1}^1 f(t)\overline{g(t)}(1-t^2)^m dt$. (Hint: use Theorem
3.1.)
- For the rotation matrix $R$ below, find these: the Euler angles, the angle of rotation, and the axis of rotation.
\[
R =\begin{pmatrix}
\frac{2}{3} & \frac{2}{3} &-\frac{1}{3}\\
-\frac{1}{3} & \frac{2}{3} & \frac{2}{3} \\
\frac{2}{3} & -\frac{1}{3} & \frac{2}{3}
\end{pmatrix}.
\]
- Let $A$ be an n×n matrix such that $A^3=-A$. Show that $e^{\theta A} = I + \sin(\theta) A + (1-\cos(\theta))A^2$. Using the axis of rotation $\mathbf \omega$ and angle of rotation $\theta$ for $R$ found in the previous problem, find $A$ such that $R=e^{\theta A}$. (Hint: recall that $A\mathbf x = \mathbf \omega \times \mathbf x$ and that $R\mathbf x = \mathbf x + \sin(\theta)\mathbf \omega \times \mathbf x+ (1-\cos(\theta))\mathbf \omega\times (\mathbf \omega \times \mathbf x)$.)
- Let $\mathbf 1 = \begin{pmatrix} 1&0\\0&1\end{pmatrix}$, $\mathbf i = \begin{pmatrix} i&0\\0&-i\end{pmatrix},\ \mathbf j = \begin{pmatrix} 0&1\\ -1&0\end{pmatrix},\ \mathbf k = \begin{pmatrix} 0&i\\i&0\end{pmatrix}.\ $ For $q_0\in \mathbb R$ and $\mathbf q\in \mathbb R^3$, a quaternion $Q = q_0\mathbf 1 + \mathbf q$, where $\mathbf q = q_1\mathbf i+q_2\mathbf j + q_3\mathbf k$ is the "pure" part of $Q$. The set of quaternions is denoted by $\mathbb H$. Also, $SU(2)$ comprises all $U\in \mathbb H$ such that $u_0^2 + |\mathbf u|^2 = 1$.
- Show that $PQ = (p_0q_0 - \mathbf p\cdot \mathbf q)\mathbf 1 +
q_0\mathbf p + p_0\mathbf q+\mathbf p \times \mathbf q$.
- Let $U=u_0\mathbf 1 + \mathbf u$ be in $SU(2)$ and let $X=\mathbf x$ be a pure quaternion representing a point in $\mathbb R^3$. In addition, let $\overline{U} = u_0\mathbf 1 - \mathbf u$. Show that
\[
UX\overline{U} = \mathbf x + 2u_0\mathbf u \times \mathbf x+ 2\mathbf u \times (\mathbf u\times x).
\]
- If $u_0=\cos(\psi)$ and $\mathbf u = \sin(\psi)\mathbf \omega$, where $\mathbf \omega \in S^2$, then show that
\[
UX\overline{U} = \mathbf x + \sin(2\psi)\mathbf \omega \times \mathbf x+ (1-\cos(2\psi))\mathbf \omega \times (\mathbf \omega\times \mathbf x).
\]
- Find all quaternions $U$ that correspond to the rotation matrix $R$ in problem 5. For each of them, what is $\psi$?
Updated 12/11/2012 (fjn)