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I have the following Gaussian function:

$$\rho(r) = q_i (\alpha/\pi)^{3/2} \exp(-\alpha r^2)$$

Qualitatively, the "width" of this Gaussian is related to $\frac{1}{\alpha}$: the larger the value of $\alpha$, the smaller the "width" of the Gaussian.

This Wikipedia article uses this definition of a Gaussian function:

$$f(x) = a\exp \left(-\frac{(x-b)^2}{2c^2}\right)$$

and says that one way to define the width is to consider the full width at half maximum:

$$\text{FWHM} = 2 \sqrt{2 \ln 2} c \approx 2.35482c$$

In other words, the width of $f(x)$ is proportional to $c$. But, in my function $\rho(r)$, $\alpha$ appears in two places: in the exponential and as a coefficient of the exponential. How should I define the width of $\rho(r)$?

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A multiplicative factor provides a vertical stretch, that do not change the FWHM value. – enzotib Aug 11 '12 at 20:00
In your example the width of the Gaussian should be related to $\alpha^{-1/2}$ not to $\alpha^{-1}$. – Fabian Aug 11 '12 at 21:25
up vote 5 down vote accepted

The proportionality constant in a Gaussian function does not affect its width. Consider the general form of a Gaussian function (may not be a probability density function):

$f(x) = a \exp\Big(-\frac{(x-b)^2}{c^2}\Big)$

The maxima occurs at $x=b$, and its value is $a$. Hence, the value of $f(x)$ at half maximum is $0.5a$. You can now find the value of $x$ at which $f(x) = 0.5a$:

$f(x^*) = a\exp\Big(-\frac{(x^*-b)^2}{c^2}\Big) = 0.5a$

The proportionality constant $a$ cancels out from both sides, and we end up with $(x^*-b) = \pm c\sqrt{\ln 2}$. The $FWHM$ thus becomes $2c\sqrt{\ln 2}$. Even if you were to define width using any other definition, say full width at $\alpha$ maximum (where $\alpha < 1$), the proportionality constant would always cancel out. Further, the center ($b$) would not matter as well since the Gaussian function is symmetric.

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In the one-dimensional case, the width of $f_c:x\mapsto(2\pi c^2)^ {-1/2}\exp(-(x-b)^2/(2c^2))$ is defined as follows.

One notes that the measure $f_c(x)\mathrm dx$ coincides with $g(z)\mathrm dz$ for some function $g$ independent of $c$, with $z=(x-b)/c$, precisely for the standard gaussian density $g:z\mapsto(2\pi)^ {-1/2}\exp(-z^2/2)$. Since $x=cz+b$, the argument of $f_c$ scales like $c$ hence the width of $f_c$ is proportional to $c$ (the exact coefficient of proportionality, whether one chooses $2\sqrt{2\ln2}$ or another value, being essentially irrelevant).

Likewise, in the three-dimensional case, $\rho_\alpha(r)=C\alpha^{3/2}\exp(-\alpha r^2)$ hence the radial part of the volume measure is $\rho_\alpha(r)r^2\mathrm dr=\sigma(s)\mathrm ds$ for some function $\sigma$ independent of $\alpha$, with $s=\alpha^{1/2}r$, precisely for $\sigma(s)=Cs^2\exp(-s^2)$. Since $r=\alpha^{-1/2}s$, the argument of $\rho_\alpha$ scales like $\alpha^{-1/2}$ hence the width of $\rho_\alpha$ is proportional to $\alpha^{-1/2}$ (the exact coefficient of proportionality being irrelevant).

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