  Quasi-elastic light scattering spectroscopy of dilute solutions of diffusing
mesoscopic probe particles is regularly used to examine the dynamics of the
fluid through which the probe particles are moving. For probes in a simple
liquid, the light scattering spectrum is a simple exponential; the field
correlation function $g^{(1)}_{P}(q,\tau)$ of the scattering particles is
related to their mean-square displacements $\bar{X^{2}} \equiv < (\Delta
x(\tau))^{2}>$ during $\tau$ via $g^{(1)}(q,\tau) = \exp(- {1/2} q^{2}
\bar{X^{2}})$. However, historical demonstrations of this expression refer only
to ideal Brownian particles in simple liquids, and show that if the form is
correct then it is also true that $g^{(1)}(q,\tau) = \exp(- \Gamma \tau)$, a
pure exponential in $\tau$. In general, $g^{(1)}_{P}(q,\tau)$ is not a single
exponential in time. $g^{(1)}_{P}(q,\tau)$ reflects not only the mean-square
particle displacements but also all higher-order mean displacement moments
$\bar{X^{2n}}$. A correct general form for $g^{(1)}(q,\tau)$, replacing the
generally-incorrect $\exp(- {1/2} q^{2} \bar{X^{2}})$, is obtained. A simple
experimental diagnostic determining when the field correlation function gives
the mean-square displacements is identified, namely $g^{(1)}(q,\tau)$ reveals
$\bar{X^{2}}$ if $g^{(1)}(q,\tau)$ is exponential in $\tau$.
