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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#9, Rev. A, due Wednesday, 2016-11-08 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed] App.~A
\#4, %p440, Gram-Schmidt
%\#5, %p440, Schwartz
\#9, %p446, matrix calc
\#10. %p446 symmetry decomp

{\bf 1.} The {\bf complex plane} $\{\bs w=(x,y)\}$ is the vector space of {\it
  real} and {\it imaginary} components of $w=x + iy \in \mathbb{C}$, with the
additional operation of multiplication, similar to the {\it general linear
  group} GL($n$) of $n\times n$ matrices.  We explore this analogy by
generalizing the exponential map $e^{i \phi}$.

{\bf \quad a)} Identify the two basis elements of the complex plane in the
above representation of $w$.

{\bf \quad b)} Show that the dot and cross product of two points $\bs
w_1=(x_1,y_1)$ and $\bs w_2=(x_2,y_2)$ are given by the real and imaginary
parts of the complex product $w_1^* w_2 = \bs w_1\cdot \bs w_2 + i (\bs w_1
\times \bs w_2)_z$, where $w^*=x-iy$ is the {\it complex conjugate} of $w$.
Identify the symmetric and antisymmetric terms of this product.  Thus the
complex product $|w|^2 = w^*w$ equals the vector product $\bs w \cdot \bs w$.

{\bf \quad c)} Show graphically that the operator $w\to iw$ rotates the point
$w$ 90$^\circ$ \mbox{CCW about the origin.}

{\bf \quad d)} Show graphically that the operator $1+i\, d\phi: w \mapsto w +i
w\, d\phi$ preserves the magnitude of $w$ (assuming $d\phi\,^2=0$), but
rotates it CCW by the infinitesimal angle $d\phi$.

{\bf \quad e)} Obtain a finite rotation from an infinite number of $d\phi$
rotations as follows: formally integrate the equation $dw = i w\, d\phi$ with
the initial condition $w|_{\phi=0}=w_0$ to obtain the rotation formula
$w(\phi)=R_\phi w_0$, where $R_\phi = e^{i\phi}$.  Use this result to justify
the identity {$\displaystyle
  e^x=\lim_{n\to\infty}\textstyle(1+\frac{x}{n})^n$}.

{\bf \quad f)} Separate the Taylor expansion of $e^{i\phi}$ into $x+iy$ to
prove Euler's formula, \mbox{$e^{i\phi}=\cos\phi + i \sin\phi$}.

{\bf \quad g)} Show that complex multiplication by $i$ is equivalent to the
vector operator $\bh z\, \times$.

{\bf \quad h)} Determine the matrix representation $M_z$ of the operator $\bh
z\,\times$, where $M_z \bs r = \bh z \times \bs r$.  Do the same for $M_x$ and
$M_y$ to show that $\bs v\times = \bs v \cdot \bs M = v_x M_x + v_y M_y + v_z
M_z$ is the matrix representation of $\bs v\, \times$ for any vector $\bs v$.
{\it Hint:} You should find that the vector of matrices $\bs M \sim
(M_i)_{jk}=\varepsilon_{ijk}$ (cross product tensor) is completely
antisymmetric in indices $i,j,k$.

{\bf \quad i)} Restricting to the $xy$-plane, show that the $2\times 2$ matrix
$M_z^2=-I$ analogous to $i^2=-1$, and the matrix for a CCW rotation $\phi$ is
$R_\phi = e^{M_z \phi} = I \cos\phi + M_z \sin\phi =
\scriptsize{\pmatrix{\cos\phi & -\sin\phi\cr \sin\phi & \cos\phi}}$.  {\it
  Hint:} the {\it exponential} of a matrix $M_z$ is defined by its Taylor
expansion as in part f).  Note that in general, the matrix $R_{\bs v}$ for a
CCW rotation by angle $v=|\bs v|$ about the $\bh v$-axis can be written as
$R_{\bs v} = %(e^{\bs M \cdot \bs v} =
I \cos v + \bs M{\cdot} \bh v \sin v + \bh v \bh v^T (1-\cos v)$, where the
third term corrects for the non-rotating projection along $\bs v$.

{\bf \quad j)} Calculate the eigenvalues and eigenvectors of $M_z=\scriptsize{
  \pmatrix{0&-1\cr 1&0}}$ to show that $M_z = VWV\ad = \frac12
\scriptsize{\pmatrix{1&1 \cr i&-i}} \scriptsize{\pmatrix{-i&0\cr0&i}}
\scriptsize{\pmatrix{1&-i\cr1&i}}$ and $e^{M_z \phi} = V e^{W\phi}V\ad =
\frac12 \scriptsize{\pmatrix{1&1 \cr i&-i}}
\scriptsize{\pmatrix{e^{-i\phi}&0\cr0&e^{i\phi}}}
\scriptsize{\pmatrix{1&-i\cr1&i}}$.  Multiply this out to verify part i).
Thus real Hermitian matrices have real eigenvalues while antiHermitian
matrices have Tr=0 and imaginary eigenvalues.  The exponential of a Hermitian
matrix is positive definite with real positive eigenvalues, while the
exponential of an antiHermitian matrix is unitary with Det=1 and unit modulus
eigenvalues.  This is the normal matrix analogy: Hermitian/antiHermitian
matrices are analogous to the real/imaginary numbers.

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