Many particles systems

For a single particle, we know the meaning of ${\displaystyle \psi ({\overrightarrow {x}})}$ and ${\displaystyle \left|\psi ({\overrightarrow {x}})\right|^{2}}$. 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 ${\displaystyle u_{i}}$ as rows and the particle states ${\displaystyle \left|j\right\rangle }$ 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:

${\displaystyle M={\frac {1}{\sqrt {3!}}}{\begin{bmatrix}u_{a}(1)&u_{a}(2)&u_{a}(3)\\u_{b}(1)&u_{b}(2)&u_{b}(3)\\u_{c}(1)&u_{c}(2)&u_{c}(3)\end{bmatrix}}}$
And the w.f. of the system is ${\displaystyle \psi =\det M}$.