What's the difference between probability and probability density

[Newbie says Hi]

Main Question or Discussion Point

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?

 

[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.

 

[mathman]

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).

 

[Hurkyl]

Please correct me if this is wrong. SO what exactly does the "density" refer to?

Probability density : Probability :: (mass) density : mass

 

[dextercioby]

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?

Nope, that's incorrect. It's the probability ascribed to a finite/infinite domain (in any # of dimensions, 1,2,3,...), not to a point of the domain.

 

[Newbie says Hi]

Thanks to everyone who answered :)

NOW, I think understand why they call it probability density -- it is the probability of finding said particle in a given region (interval) of space.