\begin{problem}
In the particle-wave duality, localized particles are modeled as wave
packets, with both a group speed and a phase speed.  Between Equations
28.13 and 28.16, the text shows that the group speed $v_g$ of a wave
function $\psi$ is the same as the particle speed $u$.  Treat the
particle as a non-relativistic de Broglie wave, and use $v_p = \lambda
f$ to show that the phase speed $v_p = u/2 \ne u = v_g$.
\end{problem} % based on Thornton and Rex's 3rd edition Equation 5.34.

\begin{solution}
Here's a picture of the wave packet, just as a reminder of what we're
talking about.
\begin{center}
\begin{asy}
import graph;

real u=1cm;
real dp=2;
real f=440;
real v=343;
real plength=5; // packet length scale

real xMax=4plength;
real width=6u;

real lambda = v/f;
real k = 2pi/lambda;

real envelope(real x) { return exp(-(x/plength)**2); }
real amplitude(real x) { return envelope(x) * cos(k*x); }

size(width,width/2,IgnoreAspect);
scale(Linear, Linear);
xlimits(-xMax,xMax);
ylimits(-1.2,1.2);
xaxis("$x$");
//yaxis("$y$");

draw(graph(envelope,-xMax,xMax, n=200, operator ..),blue);
draw(graph(amplitude,-xMax,xMax, n=800, operator ..),red);

real vlength = 8lambda;
real vxpos = 3.625lambda;
pair[] vgs = {(vxpos, envelope(vxpos)), (vxpos+vlength, envelope(vxpos))};
pair[] vps = {(vxpos, amplitude(vxpos)), (vxpos+vlength, amplitude(vxpos))};

real scalex(real x) { return x/xMax*(width/2); }

draw("$v_g$", graph(vgs), N, Arrow);
draw("$v_p$", graph(vps), S, Arrow);
\end{asy}
\end{center}
We start with the given definition of the phase velocity
\begin{equation}
  v_p = \lambda f \;.
\end{equation}
DeBroglie tells us (Equations 28.10 and 28.11)
\begin{align}
  \lambda &= \frac{h}{p} \\
  E &= hf \;,
\end{align}
so
\begin{equation}
  v_p = \lambda f = \frac{hf}{p} = \frac{E}{p} \;.
\end{equation}
Then we use the non-relativistic kinetic energy and momentum
\begin{align}
  E &= \frac{p^2}{2m} \\
  p &= mu \\
  v_p &= \frac{E}{p} = \frac{\frac{p^2}{2m}}{p} = \frac{p}{2m}
   = \frac{mu}{2m} = \frac{u}{2} \;.
\end{align}

As a side note, I have no idea why the text uses $u$ instead of $v$
for Equation 28.16, but I though I should stick with it here to avoid
adding to the confusion.
\end{solution}
