First quantization

The first quantization is a schematic procedure for establishing a quantum mechanical equations of motion for a physical system. It was the first time - introduced in 1925 by Werner Heisenberg and Erwin Schrödinger in 1926 by the so founded the modern quantum mechanics - in two different forms.

The first quantization can be made ​​plausible in specific cases, by examining the motion of wave packets for the classical limit (: reduced Planck constant ).

The term first quantization is due to its relation to the second quantization. Historically, it was not the first attempt of quantization in modern physics (see quantization).

Procedure

Heisenberg and Schrödinger assume that initially, as in classical physics, the Hamiltonian of the system is set up.

According to Schrödinger

After Schrödinger then energy and momentum are replaced by operators that are defined on a Hilbert space:

The result is a differential equation for a time varying state vector, in this illustration a wave equation of the wave function. The stationary solutions of the differential equation. Obtained for constant boundary conditions, have discrete eigenvalues ​​for the energy and some other mechanical quantities.

From the classical Hamiltonian so the Schrödinger equation from a relativistic Hamiltonian arises the Klein-Gordon equation for bosons or the Dirac equation for fermions.

By Heisenberg

Perhaps even less clear, but mathematically equivalent, is the procedure introduced by Heisenberg, the classic sizes location x and momentum p to be considered as matrices (), which must satisfy certain commutation relations: