= Solution
A one-dimensional <probability density> has units $L^{-1}$, while $D$ has units $L^2T^{-1}$. With no other length scale, <dimensional analysis> requires
$$
\boxed{n(x,t)=\frac1{\sqrt{Dt}}F\left(\frac{x}{\sqrt{Dt}}\right).}
$$
Normalization gives $\int_{-\infty}^{\infty}F(\xi)\,d\xi=1$ and unbiased <diffusion> gives an even $F$. \b[Dimensional arguments alone do not specify $F$.]
For ordinary Fickian <diffusion>, impose $\partial_t n=D\partial_x^2 n$ with $n(x,0)=\delta(x)$. Substitution of the similarity form yields
$$
F''+\tfrac12(\xi F)'=0.
$$
A decaying symmetric solution obeys $F'=-\xi F/2$, so $F=C e^{-\xi^2/4}$. Normalization fixes $C=1/\sqrt{4\pi}$:
$$
\boxed{n(x,t)=\frac1{\sqrt{4\pi Dt}}
\exp\left(-\frac{x^2}{4Dt}\right),\qquad t>0.}
$$
This <heat kernel> integrates to one, has <variance> $2Dt$, and tends to the initial <Dirac delta distribution> as $t\downarrow0$.
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