Particle density vs. Probability Density in Quantum Mechanics I am currently reading trough "Bose-Einstein Condensation and Superfluidity" by Pitaevksii and Stringari and noticed some inconsistencies in my reasoning.
In Chapter 5 (Non-uniform Bose gases at zero temperature) the authors introduce the condensate wave function $\Psi$.
It is futher stated that the normalization of $\Psi$ is given by $N = \int d\vec{r} |\Psi(\vec{r})|^2$, where N is the total number of atoms in the condensate. Up until this point, I think of $\Psi$ as a probability density, as I have been doing when dealing with Quantum Mechanics for the past few years.
The following sentence then really confuses me:

The modulus  $|\Psi(\vec{r})|$ determines the particle density $n(\vec{r}) = |\Psi(\vec{r})|^2$ of the condensate.

My question is: How can something that describes a probability density be a quantity that represents a particle density?
 A: 
The following sentence then really confuses me:

The modulus  $|\Psi(\vec{r})|$ determines the particle density $n(\vec{r}) = |\Psi(\vec{r})|^2$ of the condensate.



My question is: How can something that describes a probability density be a quantity that represents a particle density?

For an N-particle system, the particle density operator is:
$$
\hat n(\vec r) \equiv \sum_i^N \delta(\vec r - \hat{\vec r_i} )
$$
The expectation value of this is:
$$
n(\vec r) = \langle\Phi|\hat n(\vec r)|\Phi\rangle\;,
$$
where $\Phi(\vec r_1, \ldots, \vec r_N)$ is the N-body wave function.
The expression for $n$ can be re-written as:
$$
n(\vec r) = \sum_i^N\int d^3r_1\ldots d^3r_N |\Phi(\vec r_1, \ldots, \vec r_N)|^2\delta(\vec r - \vec r_i)\;.
$$
Due to the symmetry or antisymmetry of the many-body wave fucntion, this can be written more simply as:
$$
n(\vec r) = N\int d^3r_2\ldots d^3r_N |\Phi(\vec r, \vec r_2, \ldots, \vec r_N)|^2
$$
You can choose to define:
$$
|\Psi(\vec r)|^2 = N\int d^3r_2\ldots d^3r_N |\Phi(\vec r, \vec r_2, \ldots, \vec r_N)|^2
$$
That is all that $|\Psi(\vec r)|^2$ means.
A: The wave function describing a BEC is in nature quite different to that of a single quantum particle.
Usually in quantum mechanics you have that $|\psi(t,x)|^2$ is the probability density of the particle being around $x$ at time $t$.
However in BECs the description is different, since quantum phenomena are now apparent macroscopically. This requires us to use a different kind of wave function, describing the condensate as a whole. So in this context, $|\Psi(t,x)|^2$ is the spatial distribution of the BEC cloud at time $t$.
A: 
How can something that describes a probability density be a quantity
that represents a particle density?

That is open for extensive qm interpretations, and your question is related to "what is the physical property of a qm object before its measured".
I know that in quantum chemistry, that a working assumption is that the probability density also is the particle density, and if relevant also the charge density. This gives a good picture of how things look in average in results, but the assumption is also used in midway calculations - contradicting the Copenhagen interpretation that the physical properties arises at measurement.
