# First quantization

The first quantization , also called canonical quantization , is a schematic procedure for setting up a quantum mechanical equation of motion for a physical system . It was first presented - in two different forms - in 1925 by Werner Heisenberg and in 1926 by Erwin Schrödinger , who thus established modern quantum mechanics.

The first quantization can be made plausible in specific cases by examining the movement of wave packets for the classic limit case ( : reduced Planck's quantum of action ). ${\ displaystyle \ hbar \ to 0}$ ${\ displaystyle \ hbar}$ The term first quantization is based on its relationship to the second quantization . Historically, it was not the first attempt at quantization in modern physics (see quantization (physics) ).

## Action

Heisenberg and Schrödinger assume that, as in classical physics, the Hamilton function of the system is set up first.

### According to Schrödinger

According to Schrödinger, energy and impulses are then replaced by operators that are defined on a Hilbert space :

${\ displaystyle E \ rightarrow - {\ frac {\ hbar} {i}} {\ frac {\ partial} {\ partial t}}, \ quad p_ {x} \ rightarrow {\ frac {\ hbar} {i} } {\ frac {\ partial} {\ partial x}},}$ analogous for y and z .

This results in a differential equation for a time-varying state vector , in this representation a wave equation for the wave function . The stationary solutions of the differential equation that are obtained for constant boundary conditions have discrete eigenvalues for the energy and some other mechanical quantities.

The Schrödinger equation emerges from the classical Hamilton function , the Klein-Gordon equation for bosons or the Dirac equation for fermions from a relativistic Hamilton function . ${\ displaystyle H = {\ frac {p ^ {2}} {2m}} + V (r)}$ ### According to Heisenberg

Perhaps even less descriptive, but mathematically equivalent, is the procedure introduced by Heisenberg to understand the classical quantities of position  x and momentum  p as matrices ( ) that must fulfill certain commutation relations: ${\ displaystyle \ mathbf {x, \, p}}$ ${\ displaystyle \ mathbf {xp-px} = i \ hbar.}$ 