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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 521 \\
    Homework \#13, Rev. A, due Thursday, 2017-02-02 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed]
Ch.~3 \#39;~ %p129 generator of translation in x,t
Ch.~4 %\#26,  %pdij + eijk sigmak
      \#27,  %p177 uncertainties in spin
      \#30,  %p178 sigma.n
      \#31,  %p178 Sxyz s=1
      \#49,  %p194 expectation values
      \#52,  %p195 spin matrix s=3/2
      \#53.  %p195 spin matrices arbitrary s

{\bf 1. Clifford algebra}.  The complete 3-vector algebra including dot and
cross products can be implemented using the identity ($I$) as the unit scalar
and Pauli matrices ($\bs\sigma$) as unit vectors:
\begin{equation}
  1           = I           \equiv \pmatrix{1 & 0 \cr 0 & 1 } \quad
  \hat{\bs x} = \bs\sigma_x \equiv \pmatrix{0 & 1 \cr 1 & 0 } \quad
  \hat{\bs y} = \bs\sigma_y \equiv \pmatrix{0 & -i \cr i & 0 } \quad
  \hat{\bs z} = \bs\sigma_z \equiv \pmatrix{1 & 0 \cr 0 & -1 }.
  \label{eq:sigma}
\end{equation}
The dot and cross products can be represented by matrix multiplication, as in
the formula
\begin{equation}
  \bs\sigma_i\bs\sigma_j = I
  (\bs\sigma_i\cdot\bs\sigma_j) + i (\bs\sigma_i\times\bs\sigma_j) = I \delta_{ij} +
  i\epsilon_{ijk}\bs\sigma_k,
  \label{eq:dotcross}
\end{equation}
where the dot and cross products are interpreted in the usual sense of unit
vectors.  Note the difference between the imaginary $i$ and the index $_i$.
Also note that scalars are often implicitly multiplied by $I$ when adding with
other matrices.  This algebra generalizes to a space-time algebra using the
Dirac matrices $\gamma^\mu$ instead of the Pauli matrices $\sigma_i$.

{\bf \quad a)} Verify this formula for all nine products
$\bs\sigma_i\bs\sigma_j$ and show it is equivalent to the expression
\begin{equation}
  (\bs\sigma\cdot\bs a)(\bs\sigma\cdot\bs b)
  = \bs a\cdot\bs b + i \bs\sigma\cdot (\bs a \times \bs b).
\end{equation}
Here, $\sigma_x$, $\sigma_y$, and $\sigma_z$ are considered to be
unit vectors as opposed to components of a vector $\bs \sigma$.

{\bf \quad b)} Which products are symmetric and which are antisymmetric?  A
general matrix product has both symmetric and antisymmetric parts, but this
partition of symmetry into $i=j$ and $i\neq j$ is the defining feature of a
Clifford algebra.

{\bf \quad c)} Show that any linear product $\bs a\circ \bs b$, can be
decomposed into the sum $\bs a\circ \bs b = \{\bs a\circ \bs b\} + \langle \bs
a\circ \bs b\rangle$ of symmetric $\{\bs a\circ \bs b\} \equiv \frac12(\bs
a\circ \bs b + \bs b\circ \bs a)$ and antisymmetric $\langle \bs a\circ \bs
b\rangle \equiv \frac12(\bs a\circ \bs b - \bs b\circ \bs a)$ parts, with
respect to exchange of $\bs a$ and $\bs b$.  Show that $\langle \bs a\circ \bs
a \rangle = 0$ always.  Why are the diagonal elements of an antisymmetric
matrix zero?  Apply this decomposition to the product
$\bs\sigma_i\bs\sigma_j$.

{\bf \quad c)} The imaginary $i$ in the above formula is not present in the
ordinary cross product.  It distinguishes [axial] pseudovectors from [polar]
vectors.  Using part a), calculate the value of the pseudoscalar
$\bs\sigma_i\bs\sigma_j\bs\sigma_k$ and show it is completely antisymmetric in
$i, j, k$.  What is the analog of this triple product in terms of standard
vector products?

{\bf 2. Generators of rotation}.  In Griffiths \#3.39, we showed that
$p_x/\hbar$ is the generator of translation and $\mathcal H/\hbar$ is the
generator of time evolution of the wavefunction: $\exp(-i p_x x_0/\hbar)
\psi(x) = \psi(x-x_0)$ and $\exp(-i \mathcal H t_0/\hbar)\Psi(x,t) =
\Psi(x,t+t_0)$.  We also saw an example of this in Homework 9, where $M_z=\bh
z\times = \scriptsize{\pmatrix{0 & -1\cr 1 & 0}}$ generates $R_\phi=\exp(M_z
\phi) = \scriptsize{\pmatrix{\cos\phi & -\sin\phi\cr \sin\phi & \cos\phi}}$,
which rotates 2-dimensional vectors (spin 1, not two-component $s{=}\frac12$
spinors).  In 3-d, $\bs M=(M_x,M_y,M_z)$ generates the rotation $R_{\bs
\omega} = \exp(\bs M \cdot \bs \omega) = I \cos \omega + \bs M{\cdot} \bh
\omega \sin \omega + \bh \omega \bh \omega^T (1-\cos \omega)$ of 3-vectors
about the axis $\bs \omega$.

{\bf \quad a)} In analogy with $p_x$, show that $L_z$ generates the rotation
of a wave function about the z-axis: $\exp(-i\phi_0 L_z/\hbar)
\psi(r,\theta,\phi) = \psi(r,\theta,\phi-\phi_0)$.  This generalizes to
$\exp(i\bs L\cdot\bs\omega) \psi(\bs r) = \psi(R_{\bs \omega} \bs r)$, where
$R_{\bs \omega}$ is a normal rotation matrix for vectors.

{\bf \quad b)} Show that the generators $\bs M$ are the cartesian equivalent
of the $3\times 3$ spin $s{=}1$ generator matrices $\bs i\bs S/\hbar$ in
spherical tensor components of Griffiths \#4.31.  Hint: The vector $\bs
v=(v_x,v_y,v_z)$ has spherical tensor components $v_{\pm 1} =
\frac1{\sqrt2}(v_x \pm i v_y)$ and $v_0=v_z$, which are not the same as its
spherical components $\bs v = \bh r v_r + \bh \theta v_\theta + \bh \phi
v_\phi$.

{\bf \quad c)} Calculate the $s=\frac12$ spinnor rotation matrices
$R_i(\phi)=\exp(i\sigma_i\phi/2)$ about the $i=x,y,z$ axes.  How do these
relate to the eigenvectors of $\bs \sigma\cdot \bh n$ in Griffiths \#3.30?
Show that a spinnor changes sign after a full revolution and must be rotated
by $4\pi$ to return back to its original value.

\end{document}

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