# Many particles systems

For a single particle, we know the meaning of and . For many particles systems a natural extention of these concepts holds. We do distinguish two type of particles:

- Entire spin particles, are called
*bosons*; - Semi-entire spin particles, are called
*fermions*.

The global w.f. of the system has to be:

- Completely symmetrical within respect the switch of coordinates for bosons;
- Completely anti-symmetrical within respect the switch of coordinates for fermions.

The global w.f. of the system can also be seen as the determinant of that matrix that has the energy levels as rows and the particle states as columns. This determinant is called *Slater determinant* Since bosons have symmetrical w.f., all the negative signs have to be changed into positive signs when calculating this determinant in order to find a boson-system w.f.

**Example**

Let us assume that we are dealing with three fermions. Then we have:

And the w.f. of the system is .