# What's the difference between probability and probability density

Newbie says Hi
So the integral of $$|Psi|$$ squared represents the probability of finding a particle at a certain position at a certain time. Please correct me if this is wrong. SO what exactly does the "density" refer to?

## Answers and Replies

masudr
The density of anything is the quantity itself per unit length/area/volume.

If the $|\psi|^2$ is defined for a 3 dimensional system, then the probability density will be the probability divided by volume. And for 2 dimensional and 1 dimensional systems, we have area and length respectively.

The best way to see this is that for 1 dimensional systems, for example, the probability of a particle being between points A and B is given by

$$\int^B_A |\psi(x)|^2\ dx.$$

Here, we have multiplied the prob. density by a length, namely $dx,$ to get a probability. Therefore the density itself is a probability divided by a length. The extension to higher dimensions is easy.

In mathematics, probability density is the derivative of the cumulative probability. Specifically, let F(x)=Prob.(X<=x), where X is some real valued random variable. Then the density is F'(x).

Staff Emeritus
Gold Member
Please correct me if this is wrong. SO what exactly does the "density" refer to?
Probability density : Probability :: (mass) density : mass

So the integral of $$|Psi|$$ squared represents the probability of finding a particle at a certain position at a certain time. Please correct me if this is wrong. SO what exactly does the "density" refer to?