The real meaning of probability density?

[jeebs]

In QM people always talk about probability amplitudes and probability densities, but I've never really given these terms much thought and have always just ignored the density/amplitude part and focussed on the probability part.

So what is meant by a probability density - is it what it sounds like, ie. a probability per unit volume, or something different? And how is this distinct from a probability amplitude?

 

[mathman]

Amplitude is a term used by physicists.
http://en.wikipedia.org/wiki/Probability_amplitude

Density is a precise mathematical term. In quantum mechanics it is used for the absolute value of the magnitude squared.

 

[PatrickPowers]

In QM people always talk about probability amplitudes and probability densities, but I've never really given these terms much thought and have always just ignored the density/amplitude part and focussed on the probability part.

So what is meant by a probability density - is it what it sounds like, ie. a probability per unit volume, or something different? And how is this distinct from a probability amplitude?

A probability density function is any function that is everywhere non-negative and integrates over all space to the number one. The basic problem is that when you have infinitely many possibilities and the probability of each possibility is zero. So you have a probability density function and to calculate a probability you integrate over a positive interval.

A probability amplitude is (I think, maybe not always) a complex-valued function that has the property that the integral of its norm over all space is one. The famous Shroedinger Wave Equation is like this.

 

[jfy4]

A probability density function is any function that is everywhere non-negative...

everywhere... where?

 

[PatrickPowers]

everywhere... where?

The pdf is nonnegative over the domain on which it is defined. This is just a fancy way of saying that a pdf cannot have a negative value anywhere over its domain.