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%TCIDATA{Created=Sun Sep 01 16:43:57 1996}
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\markright{{\sf Wave packets. Group velocity}}
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\section{7.8 \thinspace \thinspace \thinspace \thinspace \thinspace
\thinspace Wave packets. Group velocity.\thinspace \thinspace \thinspace
\thinspace (covers also 7.9).}

\subsection{7.8.1}

\quad \thinspace \textbf{a. }The main topic of this section is the
propagation of plane waves that are not simply monochromatic in a medium
where the relation between $\omega $ and $k$ is not simply linear, but still 
$k$ and $\omega $ can be regarded as real over the range of frequencies that
contribute appreciably to the wave packet. In the language of optics, such a
medium is dispersive ($n$ is a function of $\omega $) but not dissipative (Im%
$\varepsilon $ is vanishingly small).

The results also apply to weakly dissipative media, with the obvious
modification that the amplitude decays exponentially as a function of time,
or of distance travelled. In the presence of strong absorption, the wave
does not go very far.

\smallskip\ 

\textbf{b. }The discussion is closely parallel to that given in detail for
Schr\"odinger waves. The main difference is that the ``wave function'' $%
u(\xi ,t)$ (one of the components of $\mathbf{E}$ or $\mathbf{B}$) must be
real and thus contains an equal admixture of positive-frequency $(\omega >0) 
$ and negative-frequency components. One should refer back to 7.1.2; but we
shall now use $x$ for the coordinate in the direction of propagation and
will let $k$ run over positive and negative values (it is thus $k_\xi $ in
the notation of 7.1.2). Then

\begin{equation}
u(x,t)=\frac 12\int_{-\infty }^{+\infty }A(k)e^{i(kx-\omega t)}\frac{dk}{%
2\pi }+\frac 12\int_{-\infty }^{+\infty }A^{*}(k)e^{-i(kx-\omega t)}\frac{dk%
}{2\pi }.  \label{1}
\end{equation}
For a monochromatic plane wave, $A(k)=A\delta (k-k_0).$ We have used $k$ as
the independent label and think of $\omega $ as $\omega (k).$ Note that $%
\omega (-k)=\omega (k)$, but $A(-k)\neq A(k)$, in general. $A(-k)$ is called 
$B$ on 7.1.2.

\newpage

\subsection{7.8.2}

\textbf{\quad \thinspace a. }$A(k)$ is determined from the initial
conditions:

\begin{equation}
u(x,0)=\frac 12\int_{-\infty }^{+\infty }\left[ A(k)+A^{*}(-k)\right] e^{ikx}%
\frac{dk}{2\pi }\,,  \label{2}
\end{equation}
\[
\frac{\partial u(x,t)}{\partial t}\mid _{t=0}=-\frac i2\int_{-\infty
}^{+\infty }\left[ A(k)-A^{*}(-k)\right] e^{ikx}\frac{dk}{2\pi }\,. 
\]

Jackson's (7.91) ($\times \sqrt{2\pi }$) is obtained from these by Fourier
inversion.

\smallskip\ 

\textbf{b. }The facts that $u$ is real and obeys a second-order differential
equation (in time) make things harder than for Schr\"odinger waves. However,
if we write $u=\psi +\psi ^{*}$, where $\psi $ is the positive-frequency
part, we can take over many of the results. Note however that we must know
both $\ u(x,0)$ and $\partial u/\partial t$ at $t=0$ to find $\psi (x,0),$
or $A(k).$

\smallskip\ 

\textbf{c. }The intensity is given not by $u^{2},$ but by $2\left| \psi
\right| ^{2}$ (see 7.2.2). This is because the term $\psi \psi ,$ which
contains $\exp (-2i\omega t),$averages to zero over a time of observation
longer than $\omega ^{-1}$. Similarly, $\psi ^{*}\psi ^{*}$contains $\exp
(2i\omega t)$ and also averages to zero. Quantum mechanically, $\psi
^{\dagger }\psi $ and $\psi \psi ^{\dagger }$ have non-zero matrix elements
between states with the same number of photons, while $\psi \psi $ and $\psi
^{\dagger }\psi ^{\dagger }$ require a change of the photon number by two,
and thus describe different processes.

\smallskip\ 

\textbf{d. }A simple and general picture is obtained if $A(k)$ is strongly
peaked at $k=k_0,$ so that we can expand

\begin{equation}
\omega (k)\simeq \omega _0+\omega _0^{\prime }(k-k_0)+\frac 12\omega
_0^{\prime \prime }(k-k_0)^2  \label{3}
\end{equation}

and

\begin{equation}
\psi (x,t)\simeq \frac 12e^{i\left( k_0x-\omega _0t\right) }\int \frac{dk}{%
2\pi }A(k)e^{i(k-k_0)(x-\omega _0^{\prime }t)}e^{-\frac 12\omega _0^{\prime
\prime }(k-k_0)^2t}.  \label{4}
\end{equation}

\newpage

\subsection{7.8.3}

\textbf{a. }The integral in (\ref{4}) can be evaluated explicitly if 

\begin{equation}
A(k)=\frac{\left( 2\pi \right) ^{1/4}}{\sqrt{\Delta k}}\exp \left[ -\frac{%
\left( k-k_{0}\right) ^{2}}{4\left( \Delta k\right) ^{2}}\right] \exp \left(
ikx_{0}\right) .  \label{5}
\end{equation}
so that $\left| A(k)\right| ^{2}$ is a gaussian. This corresponds to $u(x,0)$
being a Gaussian-modulated wave packet centered at $x=x_{0,}$ with $\Delta
x\Delta k=1/2:$

\begin{equation}
u(x,0)=\frac{\left( 2\pi \right) ^{-1/4}}{\sqrt{\Delta x}}\exp \left[ -\frac{%
\left( x-x_{0}\right) ^{2}}{4\left( \Delta x\right) ^{2}}\right] \cos \left(
k_{0}(x-x_{0})\right) .  \label{6}
\end{equation}
Here

\begin{equation}
\left( \Delta k\right) ^2=\frac{\int \left| A(k)\right| ^2\left(
k-k_0\right) ^2dk/2\pi }{\int \left| A(k)\right| ^2dk/2\pi }\quad \,\text{and%
}\quad \left( \Delta x\right) ^2=\frac{\int \left| \psi \right| ^2\left(
x-x_0\right) ^2dx}{\int \left| \psi \right| ^2dx}\,  \label{7}
\end{equation}
Note that $\left( \Delta x\right) ^2$ is defined using $\ \left| \psi
\right| ^2$ and not $u^2.$ See 7.8.2c.

\smallskip\ 

\textbf{b.} The definitions (\ref{7}) are entirely general, if $k_0$ and $%
x_0 $ are interpreted as $\left\langle k\right\rangle $ and $\left\langle
x\right\rangle .$ The uncertainty relation states that $\Delta x\Delta k\geq
1/2,$ with the $=$ sign holding only if $A(k)$ is of the form (8.5).

\smallskip\ 

\textbf{c. }One finds that $\left| \psi \right| ^2$ at time $t$ is a
gaussian centered at $x_0+\omega _0^{\prime }t$; thus $\omega _0^{\prime }$
represents the \textsl{group velocity} of the wave packet: $\omega
_0^{\prime }=\left( d\omega /dk\right) \mid _{k=k_0}.$ The spread is given
by $\left[ \Delta x(t)\right] ^2=\left( \Delta x\right) ^2+t^2\left( \omega
_0^{\prime \prime }\Delta k\right) ^2;$ thus $\omega _0^{\prime \prime
}\Delta k$ determines the rate of spreading. Physically the spreading arises
because the different k-components of the packet travel with different group
velocities $\omega _0^{\prime }:\Delta \omega _0^{\prime }\simeq \omega
_0^{\prime \prime }\Delta k.$

\smallskip\ 

\textbf{d. }The explicit form of $u(x,t)$ is given by $\psi +\psi ^{*}$ with

\[
2\left| \psi (x,t)\right| =\frac{\left( 2\pi \right) ^{-1/4}}{\sqrt{\Delta
x(t)}}\exp \left[ -\frac{\left[ x-x_0(t)\right] ^2}{4\left( \Delta
x(t)\right) ^2}\right] ,\quad \quad \text{and}\,\quad x_0(t)=x_0+\omega
_0^{\prime }t. 
\]

\newpage

\subsection{7.8.4}

\quad \thinspace \textbf{a. }The results of 7.8.3c can be extended to more
general $A(k)$ in two ways given below in (b), (c). We are restricted to
real $\omega .$

\smallskip\ 

\textbf{b.} If $\left| A(k)\right| ^2$ is peaked near $k_0,$ application of
stationary-phase arguments to the integral (\ref{4}) shows that $\left| \psi
(x,t)\right| ^2$ has a peak that moves with the group velocity $v_g=\omega
_0^{\prime }=\left( d\omega /dk\right) \mid _{k=k_0}.$

\smallskip\ 

\textbf{c. }For any $A(k)$ and the corresponding $\psi $ we can define
averages such as

\[
\left\langle \frac{d\omega }{dk}\right\rangle =\frac{\int \left| A(k)\right|
^{2}\left( d\omega /dk\right) dk}{\int \left| A(k)\right| ^{2}dk}%
\,;\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\langle
x(t)\right\rangle =\frac{\int \left| \psi (x,t)\right| ^{2}xdx}{\int \left|
\psi (x,t)\right| ^{2}dx}\,. 
\]
We then have the exact result:

\begin{equation}
\frac d{dt}\left\langle x(t)\right\rangle =\left\langle \frac{d\omega }{dk}%
\right\rangle .  \label{8}
\end{equation}
[Compare with $d\left\langle x(t)\right\rangle /dt=\left\langle
p\right\rangle /m$ for a particle of mass $m$]

\smallskip\ 

\textbf{d. }The easiest way to obtain (\ref{8})  is to realize that $k$ is
formally equivalent to the operator $\left( 1/i\right) d/dx=k_{op}$ and that 
$\psi $ obeys a Schr\"{o}dinger-type equation with ``Hamiltonian'' $\hbar
\omega \left( k_{op}\right) .$ One can then pass to a ``Heisenberg''
representation where the time-dependence is associated with the operators,
as in 
\[
x_{op}(t)=e^{\omega \left( k_{op}\right) t}xe^{-i\omega \left( k_{op}\right)
t}.
\]

\smallskip\ 

\textbf{e. }Since $k_{op}$ commutes with $\omega \left( k_{op}\right) ,$ it
is a constant of motion. We then have not only $dx(t)/dt=d\omega \left(
k_{op}\right) /dk_{op},$ leading to (\ref{8}), but also $x_{op}(t)=x+td%
\omega /dk_{op}.$ Then

\begin{eqnarray*}
\left[ \Delta x(t)\right] ^{2} &=&\,\left\langle x^{2}(t)\right\rangle
-\left\langle x(t)\right\rangle ^{2}= \\
&=&\,\left\langle x^{2}\right\rangle -\left\langle x\right\rangle
^{2}+t^{2}\left[ \left\langle \left( \frac{d\omega }{dk}\right)
^{2}\right\rangle -\left\langle \frac{d\omega }{dk}\right\rangle ^{2}\right]
\\
&&+t\left[ \left\langle x\frac{d\omega }{dk}\right\rangle +\left\langle 
\frac{d\omega }{dk}x\right\rangle \right] .
\end{eqnarray*}
We recover the result in 7.8.3c if the last term vanishes and if we expand

\[
\frac{d\omega }{dk}\simeq \left( \frac{d\omega }{dk}\right)
_{k=k_0}+(k-k_0)\left( \frac{d^2\omega }{dk^2}\right) _{k=k_0}. 
\]

\newpage

\subsection{7.8.5}

\quad \thinspace \textbf{a. }The discussion so far has been cast in a
language analogous to ordinary quantum theory, using $\omega (k).$ In
optics, one writes the relation between $\omega $ and $k$ in the form $%
ck=n(\omega )\omega .$ Then $cdk=\left[ \left( dn/d\omega \right) \omega
+n\right] d\omega $ and

\begin{equation}
v_g=\frac{d\omega }{dk}=\frac c{n+\omega \dfrac{dn}{d\omega }}.  \label{9}
\end{equation}

\smallskip\ 

\textbf{b. }Under ``normal'' conditions we have $n>1,\,\,\,\,\,\omega \dfrac{%
dn}{d\omega }>0$ and the \textsl{group velocity }$\left( d\omega /dk\right) $
is less than $c.$

\smallskip\ 

\textbf{c.} An example where $n<1$ but still $v_g<c$ is given by the
high-frequency limit of any dielectric, where $n=\sqrt{1-\omega _p^2/\omega
^2}.$ (See 7.5.7a). Using (\ref{9}), or more easily from the fact that then $%
\omega ^2=\omega _p^2+k^2c^2,$ we find $v_g=c\left( ck/\omega \right) <c.$
In the language of 7.6, the group velocity of the transverse plasmons is
less than $c$, which is of course both the group velocity and the phase
velocity for the free photon.

\smallskip\ 

\textbf{d.} Near the strong absorption line it is possible to have $v_{g}>c$%
, mostly because $dn/d\omega $ is negative. One speaks then of ``anomalous
dispersion''. This is exemplified by $n=\sqrt{\varepsilon }$ with 
\[
\varepsilon =1-\frac{\omega _{p}^{2}}{\omega ^{2}-\omega _{0}^{2}+i\gamma
\omega }\,.
\]
A graph of $n$ is at the end of 7.5. Note that in the ``anomalous'' region $n
$ is complex and the ``exact'' result (\ref{8}) does not hold. It can be
shown quite generally that propagation in a dielectric never violates the
principle that ``nothing that happens in the region $x<0$ for $t<0$ has any
effect in the region $x>x_{0}$ until the time $x_{0}/c$ has elapsed''. Thus
Einstein's formulation of relativity, or causality, are \textsl{not} in
danger. See 7.11 for more on this.

\end{document}
