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Probability Density

Probability density is a concept that naturally arises whenever you talk about probability in connection with a continuous variable, such as position of a particle. In contrast to discrete probability (such as idealized coin-flipping or dice-rolling), we can't directly assign a probability to each individual outcome. Probabilities have to sum to one, but there are infinitely many possible positions for a particle, so any individual position would have to have probability zero. So how can we talk about probability spread over a continuous domain?

Well, we can talk about the probability of finding the particle within some region. If the region covers a nonzero length/area/volume/whatever, we can expect it to have a nonzero probability. But the exact probability will generally depend on the size/shape/position of the region in a complicated way.

Suppose we consider a very small region. To be concrete let's suppose we're talking about a 1D position and our small region is some interval $\left[x,x+dx\right]$. The interval has width $dx$. It stands to reason that a small interval will have a small probability. Moreover, if probability is "spread smoothly" in a sense that I won't bother making precise (hey, this is Physics.SE, not Math.SE!), then for small intervals the probability will be proportional to the width of the interval. If you halve the size of the interval you get half as much probability. Therefore

$dP=p\left(x\right)\phantom{\rule{thinmathspace}{0ex}}dx$
where $dP$ is the small probability of finding the particle in this small region, and $p\left(x\right)$ is some proportionality constant. I've written it as a function of $x$ because the proportionality could vary from point to point.

Well, $p\left(x\right)$ is called the probability density, and it has the units of probability per unit length. (Or unit area, or volume, depending on how many dimensions we do this in.) It's a lot like mass density, which is mass per unit length/area/volume, or charge density is charge per unit length/area/volume, etc...

So, now you can imagine calculating the probability to find the particle in any arbitrary region, by slicing up that region into many small ones and summing up $dP$, which is $p\left(x\right)\phantom{\rule{thinmathspace}{0ex}}dx$, for all of them. Well, this is just integrating:

In particular, the particle has to be somewhere, so $p\left(x\right)$ must integrate to one: