First quantization

A first quantization of a physical system is a semi-classical treatment of quantum mechanics, in which particles or physical objects are treated using quantum wave functions but the surrounding environment (for example a potential well or a bulk electromagnetic field or gravitational field) is treated classically. First quantization is appropriate for studying a single quantum-mechanical system being controlled by a laboratory apparatus that is itself large enough that classical mechanics is applicable to most of the apparatus.

One-particle systems

In general, the one-particle state could be described by a complete set of quantum numbers denoted by $\nu$. For example, the three quantum numbers $n,l,m$ associated to an electron in a coulomb potential, like the hydrogen atom, form a complete set (ignoring spin). Hence, the state is called $|\nu \rangle$ and is an eigenvector of the Hamiltonian operator. One can obtain a state function representation of the state using $\psi _{\nu }({\mathbf {r}})=\langle {\mathbf {r}}|\nu \rangle$. All eigenvectors of a Hermitian operator form a complete basis, so one can construct any state $|\psi \rangle =\sum _{\nu }|\nu \rangle \langle \nu |\psi \rangle$ obtaining the completeness relation:

$\sum _{\nu }|\nu \rangle \langle \nu |={\mathbf {{\hat 1}}}$

All the properties of the particle could be known using this vector basis.

Many-particle systems

When turning to N-particle systems, i.e., systems containing N identical particles i.e. particles characterized by the same physical parameters such as mass, charge and spin, is necessary an extension of single-particle state function $\psi(\bold{r})$ to the N-particle state function $\psi(\bold{r}_1,\bold{r}_2,...,\bold{r}_N)$.^[1] A fundamental difference between classical and quantum mechanics concerns the concept of indistinguishability of identical particles. Only two species of particles are thus possible in quantum physics, the so-called bosons and fermions which obey the rules:

$\psi(\bold{r}_1,...,\bold{r}_j,...,\bold{r}_k,...,\bold{r_N})=+\psi(\bold{r}_1,...,\bold{r}_k,...,\bold{r}_j,...,\bold{r}_N)$ (bosons),

$\psi(\bold{r}_1,...,\bold{r}_j,...,\bold{r}_k,...,\bold{r_N})=-\psi(\bold{r}_1,...,\bold{r}_k,...,\bold{r}_j,...,\bold{r}_N)$ (fermions).

Where we have interchanged two coordinates $(\bold{r}_j, \bold{r}_k)$ 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 any many-particle problem.

See also

• Quantization
• Canonical quantization
• Second quantization

References

1. Merzbacher, E. (1970). Quantum mechanics. New York: John Wiley & sons. ISBN 0471887021.