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\begin{document}

\section{Wave Functions and Expectation Values}

In this section we consider how to extract quantitative information
from wave functions.  This information is obtained in the form of
so-called {\em expectation values} of variables.  We first explore
expectation values related to the position of a particle.  We then
investigate how to obtain information about particle momentum from
the wave function.

\subsection{Expected Position}

The wave function as predicted by the Schr\"{o}dinger equation yields
probabilistic information about the position of a particle, in that
the absolute square of the wave function, $| \psi^2 (x) |$, is the
relative probability of finding the particle at location $x$.  The
{\em expectation value} of the particle position is the
probability-weighted average of all the possible particle positions,
$x_i$:
\begin{equation}
	< x > \equiv \sum_i x_i | \psi (x_i ) |^2 \left/
			\sum_i | \psi (x_i ) |^2 \right. .
\end{equation}
Since the possible particle positions are continuously distributed
in $x$, we may wish to replace sums by integrals:
\begin{equation}
	< x > = \int x | \psi (x) |^2 dx \left/ 
			\int | \psi (x) |^2 dx \right. .
\end{equation}
For normalized wave functions the denominators in the above two
equations are unity.  Furthermore, since $| \psi |^2 = \psi^{*} \psi$,
we can write
\begin{equation}
	< x > = \sum_i \psi_i^{*} x_i \psi_i
\end{equation}
or
\begin{equation}
	< x > = \int \psi^{*} x \psi dx .
\end{equation}

Sometimes the expectation value of $x$ is insufficient information.
An estimate of the size of a wave packet can be obtained by finding
the expectation value
\begin{equation}
	\sigma^2 = <(x - <x> )^2 > = <x^2 > - 2 <x <x>> + <x>^2 .
\end{equation}
Since $<x <x>> = <x><x> = <x>^2$, we find that
\begin{equation}
	\sigma = ( <x^2> - <x>^2 )^{1/2} .
\end{equation}
The variable $\sigma$ is a measure of the half-width of the wave
packet.  In order to obtain $\sigma$, the only new thing that needs
to be calculated is
\begin{equation}
	< x^2 > = \sum_i \psi_i^{*} x_i^2 \psi_i .
\end{equation}
It is clear how the expectation value of any function of $x$, $f(x)$,
is calculated:
\begin{equation}
	< f(x) > = \sum_i \psi_i^{*} f(x_i ) \psi_i .
\end{equation}

\subsection{Momentum Representation}

We know from the work of de Broglie that a particle has a definite
momentum if the associated wave is a plane wave.  Thus, for a wave
function of the form
\begin{equation}
	\psi = \exp (ikx) ,
\end{equation}
the momentum is $p = \hbar k$.  In this case there is no question
that the expectation value of the momentum is just $p$.

For a wave function of the form
\begin{equation}
	\psi = \phi_1 \exp (ik_1 x) + \phi_2 \exp (ik_2 x)
\end{equation}
with $|\phi_1 |^2 + |\phi_2 |^2 = 1$, the probability of the
wavenumber being $k_1$ is $|\phi_1 |^2$, while the probability of it
being $k_2$ is $|\phi_2 |^2$.  The expectation value of the wavenumber
is therefore $< k > = k_1 |\phi_1 |^2 + k_2 |\phi_2 |^2$.
Generalizing from this, the expectation value of the wavenumber from
the superposition of an arbitrary number of plane waves with different
wavenumbers,
\begin{equation}
	\psi = \sum_i \phi_i \exp (i k_i x) , ~~~~~ \sum_i |\phi_i |^2 = 1 ,
\end{equation}
is
\begin{equation}
	< k > = \sum_i k_i |\phi_i |^2 = \sum_i \phi_i^{*} k_i \phi_i .
\end{equation}
The expectation value for wavenumber written in terms of the $\phi_i$
is thus in the same form as the expectation value for position written
in terms of the $\psi_i$.

\end{document}
