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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 420 \\
    Homework \#3, Rev. A, due Monday, 2015-09-28 }
\end{center}

\vspace{.3 in}

{\bf 1.} Given the {\bf non-orthogonal basis}\quad $\bv b_1 = \pmatrix{4\cr
  1}$ and $\bv b_2 = \pmatrix{2\cr 3}$,

{\bf \quad a)} Write the equation $\bv v=\bv b_1 v^1 + \bv b_2 v^2 =
\pmatrix{2\cr -1}$ in matrix form and use it to solve for the components
$\pmatrix{v^1\cr v^2}$ of $\bv v$ in the basis $\pmatrix{\bv b_1 & \bv b_2}$.

{\bf \quad b)} Calculate the dual vectors $\bt b{}^1=\pmatrix{b^1_x & b^1_y}$
and $\bt b{}^2=\pmatrix{b^2_x & b^2_y}$ so that $\bt b^i(\bv
b_j)=\delta^i{}_j$, ie.
\begin{eqnarray*}
  \bt b{}^1 ( \bv b_1) = 1 && \bt b{}^1 ( \bv b_2) = 0 \\
  \bt b{}^2 ( \bv b_1) = 0 && \bt b{}^2 ( \bv b_2) = 1.
\end{eqnarray*}
What is the relation between the basis and cobasis matrices $\bv B =
\pmatrix{\bv b_1 & \bv b_2}$ and $\bt B = \pmatrix{\bt b{}^1\cr\bt b{}^2}$?
Note the appearance of these matrices in part a).  Use these matrices to
resolve the components in $\bv B$ of an arbitrary vector $\bv
v=\pmatrix{v_x\cr v_y}$ or the components in $\bt B$ of an arbitrary dual $\bt
w= \pmatrix{w_1 & w_2}$.

{\bf \quad c)} Calculate the two projector (matrices) $\bs P_i=\bv b_i \bt
b{}^i$ and show that they project $\bv v$ onto each of the basis vectors $\bv
b_i$. Add them to show that $\sum_{i=1}^2 \bs P_i = I$ (closure).  How does that
follow from part b)?

{\bf \quad d)} Calculate the metric $G=\pmatrix{g_{11} & g_{12} \cr g_{21} &
  g_{22}}$ so that $\bv x\cdot \bv y = \pmatrix{x^1 & x^2}\pmatrix{g_{11} &
  g_{12} \cr g_{21} & g_{22}}\pmatrix{y^1\cr y^2}$.  Verify this formula on
the components of $\bv v\cdot \bv v$ in both the normal and $\bv B$ basis.
Note that $g_{ij}=\bv b_i\cdot\bv b_j$.

{\bf \quad e)} [bonus] In this non-orthonormal basis, write the formula for
the adjoint $\bt v = \pmatrix{v_1 v_2}$ of $\bv v$, defined by $\bt v(\bv x) =
\bv v \cdot \bv x$ for any other vector $\bv x$.

{\bf 2. Circular polarized photons} travelling in the $z$-direction may be
described by the wave function $\Psi(\bv r) = \bh x e^{ikz - i\omega t} + i
\bh y e^{ikz - i\omega t}$.  

{\bf \quad a)} What is physical difference between the $x$ and $y$ components?
Plot the real part of this wave.

{\bf \quad b)} Use the Gram-Schmidt method to normalize $\bh e_1=\pmatrix{1\cr
  i}$ and complete the basis with a second orthonormal vector $\bh e_2$.  Note
that in this context our scalar field is $\mathbb{C}$

{\bf \quad c)} Calculate the adjoints $\bh e_1\ad$, $\bh e_2\ad$, and the
projectors $\bs P_1 = \bh e_1 \bv e_1\ad$ and $\bs P_2 = \bh e_2 \bv e_e2\ad$.  Verify
by direct calculation the {\it orthonormality} $\bh e_i\ad \bh e_j =
\delta_{ij}$ and {\it closure} $\sum_i \bh e_i \bh e_i\ad = I$ relations.

{\bf \quad d)} Use $\bv x = I \bv x = \bs P_1 \bv x + \bs P_2 \bv x = \bh e_1
\bv e_1\ad \bv x + \bh e_2 \bv e_2\ad \bv x = \bh e_1 x_1 + \bh e_2 x_2$ to
find the components $x_1$ and $x_2$ of $\bv x=\pmatrix{1 1}$ and verify that
$\bv x = \bh e_1 x_1 + \bh e_2 x_2$.

{\bf 3. Adjoint} Given $\bs a=\pmatrix{2i\cr 3}$, $\bs b=\pmatrix{1\cr i}$,
and $\bs M = \pmatrix{2 & i \cr -i 1}$, show by direct calculation that $\bs
a\ad \bs b = (\bs b\ad \bs a)^*$, $(\bs M \bs a)\ad = \bs a\ad \bs M\ad$,
$(\bs a\ad\bs M\bs b)^*=\bs b\ad \bs M\ad \bs a$, and $\bs a\ad\bs M = (\bs
M\ad \bs a)\ad$.  Note the last formula implies that $\bs M$ multiplies to
the left in its role as adjoint.

\end{document}
