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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 420 \\
    Homework \#4, Rev. A, due Monday, 2015-10-05 }
\end{center}

\vspace{.3 in}

{\bf 1.} The {\bf complex plane} $\{\bs z=(x,y)\}$ is a vector space of {\it
  real} and {\it imaginary} components of $z=x + iy \in \mathbb{C}$ with the
additional structure of multiplication, analogous to the {\it general linear
  group} GL($n$) of operators.  We explore this analogy using a generalization
\mbox{of the exponential map.}

{\bf \quad a)} Which two complex numbers form a basis the the above components
of $z$?

{\bf \quad b)} Show that the dot and cross product of two points $\bs
z_1=(x_1,y_1)$ and $\bs z_2=(x_2,y_2)$ are the real and imaginary parts of the
complex product $z_1^* z_2 = \bs z_1\cdot \bs z_2 + i (\bs z_1 \times \bs
z_2)_z$, where the $z^*=x-iy$ is the {\it complex conjugate} of $z$.  Identify
the symmetric and antisymmetric parts.  Compare the complex $|z|^2 = z^*z =
\bs z \cdot \bs z$ and vector $|\bv v|^2=\bv v\cdot \bv v$ square magnitudes.
Note that the cross product is not closed over the $xy$-plane: only the
$z$-component is nontrivial!

{\bf \quad c)} Show graphically that the operator ``multiply by $i$'' rotates
a point $z$ 90$^\circ$ \mbox{CCW about the origin.}

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

{\bf \quad e)} Obtain a finite rotation by an infinite composition of
rotations by $d\phi$, as follows: formally integrate the equation $dz = i z\,
d\phi$ with the initial condition $z|_{\phi=0}=z_0$ to obtain the rotation
formula $z(\phi)=R_\phi z_0$, with $R_\phi = e^{i\phi}$.  Use this property 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 $\bh
z\,\times$, such that $M_z \bs r = \bh z \times \bv r$.  Do the same for $M_x$
and $M_y$.  Show that $\bv v\times = \bv v \cdot \bs M = v_x M_x + v_y M_y +
v_z M_z$ for any vector $\bv v$, and derive the matrix representation of $\bv
v\, \times$.  Note that each of these matrices is antisymmetric, and the stack
of three matrices $\bs M$ (cross product tensor) is completely antisymmetric,
\mbox{with components $\varepsilon_{ijk}$.}

{\bf \quad i)} Calculate all nine combinations $M_iM_j$.  Show that
$(M_x,M_y,M_z)$ have the same properties of Hamilton's qua\-ter\-nions
$(i,j,k)$, which extend the rotational properties of complex numbers to 3
dimensions: $i^2=j^2=k^2=ijk=-1$.  Do these matrices {\it commute}?

{\bf \quad j)} Using the same method as above, show that the matrix for a
vector rotation of angle $\phi$ CCW in the $xy$-plane can be written $R_\phi =
e^{M_z \phi} = I \cos\phi + M_z \sin\phi = {\scriptsize\pmatrix{\cos\phi &
    -\sin\phi\cr \sin\phi & \cos\phi}}$, neglecting trivial $z$-components.
The {\it exponential} of matrix $M_z$ is defined in term of its Taylor
expansion.  In general, the matrix for a CCW rotation of angle $v=|\bv v|$
about $\bh v$ can be written $R_{\bs v} = %(e^{\bs M \cdot \bv v} =
I \cos v + \bs M{\cdot} \bh v \sin v + \bh v \bh v^T (1-\cos v)$.  The third
term handles the non-rotating component parallel to $\bh v$.

{\bf \quad k)} 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 j).
Thus real Hermitian matrices have real eigenvalues while antiHermitian
matrices have Tr=0 and imaginary eigenvalues.  The exponential of a Hermitian
matrix remains Hermitian, while the exponential of an antiHermitian matrix is
an unitary rotation with Det=1 and unit modulus eigenvalues.  Amazingly, the
Hermitian/antiHermitian decomposition of matrices is analogous to the
real/imaginary decomposition of complex numbers.

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