# How do you define the width of a wave packet?

As functions of the form $u(k x - \omega t)$ represent solutions to the wave equation, it seems natural to write a (one-dimensional, non-dispersive) wave packet solution as $$u(x, t) = e^{-(kx-\omega t)^2/(2\sigma^2)}\cos(kx-\omega t)$$ with $\sigma$ defining the width of the packet. Written this way, however, $\sigma$ has no dimensions and seems unconstrained by any function $f(c, k, \omega)$. How, then, does one translate this into a pulse measured by its duration (i.e. "a 5ps pulse")?

• The half-width in time is $\sigma/\omega$. The half-width in space is $\sigma/k$.
– pwf
Dec 8 '16 at 23:53

One defines the width in many different ways and different definitions are good / intuitive / useful in different situations, just as different statistics / parameters are used to describe populations, depending on the application (for example mean versus median to measure a "typical" value). So in general one must take care that one understands the conventions of the author in question. Sometimes this is not easy, and an implicit standards holds in certain, restricted fields of discussion.

Common examples are:

1. Full-width-half-maximum intensity (FWHMI): the distance / duration between the two nearest points on either side of the peak intensity point where the intensity falls to half its peak value;

2. Full-width-half-maximum amplitude (FWHMA): as in 1., but with "intensity replaced by magnitude of amplitude;

3. $1/e$ widths, $1/e^2$ widths: the he distance / duration between the two nearest points on either side of the peak intensity point where the intensity (or amplitude, as applicable) falls to $1/e$ (or $1/e^2$) its peak value. Most often used for Gaussian pulses like yours.

4. Root second moment width, i.e. "standard deviation": the square root of second moment of the pulse's intensity taken about the pulse's intensity centroid ("center of mass"). This is the measure that is used in the statement of the Heisenberg uncertainty principle.

As an exercise, work these out for your pulse. I believe not all the comments give you correct values for the stated times.