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\begin{document}
\begin{center}
  {\bf University of Kentucky, Physics 520 \\
    Homework \#6, Rev. A, due Friday, 2015-10-07 }
\end{center}

\vspace{.3 in}

{\bf 0.} Griffiths [2ed] Ch.~2
% \#5, %p38, infsq mixed state
\#9, %p40, <H>
\#10, %p50, SHO psi_2
\#11, %p50, SHI <x,p,x2,p2>
\#36, %p85, centered infsq well
\#41. %p86, SHO mixed state

{\bf 1.}  The {\bf harmonic oscillator} is described using ladder opperators
$a_\pm \equiv \frac1{\sqrt{2\hbar m\omega}}(m\omega x \mp ip)$, which act on
the stationary states of the TISE as follows: $a_+\psi_n =
\sqrt{n+1}\psi_{n+1}$, $a_-\psi_n = \sqrt{n}\psi_{n-1}$.

{\bf \quad a)} Write out the matrices for $a_-$ and $a_+$ in the hamonic
oscillator energy eigenbasis $|n\rangle$ with components $\psi = c_0 \psi_0 +
c_1 \psi_1 + c_2 \psi_2 + c_3 \psi_3 + \ldots$, (up to $n=3$). {\quad \it
  Hint: act $a_\pm$ on $\psi$ to determine the new components.  Then figure
  out the matrix that acts on $[c_0, c_1, c_2, c_3]^T$ in the same way.}

{\bf \quad b)} Verify that $a_+={a_-}\ad$ and $[a_-,a_+]=1$.

{\bf \quad c)} Solve for $x$ and $p$ in terms of $a_\pm$ and calculate their
matrices.

{\bf \quad d)} Calculate $\sigma_x$ and $\sigma_p$ to verify the Heisenberg
uncertainty principle \mbox{for the states $\psi_0$ and $\psi_1$.}

{\bf \quad e)} Show that $[x,p]=i\hbar$, both algebraically and by matrix
calculation.  This the formal expression of the Heisenberg uncertainty
principle.

{\bf \quad f)} Calculate the matrix for $H=p^2/2m + \frac12 mw^2x^2$.  Why is
it diagonal?  Show that the uncertainty in energy $\sigma_E$ is zero for any
stationary state $\psi_n$.  Note that by the Heisenberg uncertainty principle,
an unperturbed particle will stay in this state forever.

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