Editing Canonical quantization (section)

===Many-particle systems===

When turning to N-particle systems, i.e., systems containing N [[identical particles]] (particles characterized by the same [[quantum numbers]] such as [[mass]], [[Electric charge|charge]] and [[Spin (physics)|spin]]), it is necessary to extend the single-particle state function <math>\psi(\mathbf{r})</math> to the N-particle state function <math>\psi(\mathbf{r}_1,\mathbf{r}_2,\dots,\mathbf{r}_N)</math>. A fundamental difference between classical and quantum mechanics concerns the concept of [[Identical particles|indistinguishability]] of identical particles. Only two species of particles are thus possible in quantum physics, the so-called [[bosons]] and [[fermions]] which obey the following rules for each kind of particle:

* for bosons: <math display="block">\psi(\mathbf{r}_1,\dots,\mathbf{r}_j,\dots,\mathbf{r}_k,\dots,\mathbf{r}_N)=+\psi(\mathbf{r}_1,\dots,\mathbf{r}_k,\dots,\mathbf{r}_j,\dots,\mathbf{r}_N),</math>
* for fermions: <math display="block">\psi(\mathbf{r}_1,\dots,\mathbf{r}_j,\dots,\mathbf{r}_k,\dots,\mathbf{r}_N)=-\psi(\mathbf{r}_1,\dots,\mathbf{r}_k,\dots,\mathbf{r}_j,\dots,\mathbf{r}_N),</math>

where we have interchanged two coordinates <math>(\mathbf{r}_j, \mathbf{r}_k)</math> of the state function. The usual wave function is obtained using the [[Slater determinant]] and the [[identical particles]] theory. Using this basis, it is possible to solve various many-particle problems.