# Quantization (physics)

Quantization is to be modified in the theoretical description of a physical system of the step in which results, concepts or methods of classical physics so that quantum physical observations are reproduced correctly on the system. Among other things, by the quantization of many measurable parameters are explained, such as the presence of certain discrete energy values at the excitation levels of an atom.

From 1900 to the beginning of quantum physics, quantization meant essentially that using certain rules according to classical physics, the possible processes and those states were excluded from the that contradicted the observations. This identifies the older quantum theories, among them such as the famous Bohr model of the atom. Werner Heisenberg and Erwin Schrödinger found 1925/26, independently of one another in two ways, as has been held the results of classical mechanics to modify their basic concepts and equations to predict the resulting quantum physical observations correctly. It began the development of modern quantum mechanics. The common basis of these two paths is called the canonical quantization. The canonical quantization can be carried out also for physical fields and was from 1927 to the framework of quantum field theory.

## Development

### Older Quantum Theory (1900 - 1925)

The first rule for quantization was given in 1900 by Max Planck in order to calculate the means of classical statistical physics, the spectrum of thermal radiation can. This time, referred to as quantum hypothesis rule is: The energy exchange between matter and electromagnetic radiation of frequency occurs only in quanta of size, that is, it is quantized. It is the constant of Planck's constant.

The idea that there is a harmonic oscillator, which supplies the electromagnetic field energy or decreases, passes to the statement that he can not be excited with any selectable energy but has only states with discrete equidistant energy levels in the distance. This selection from the continuum of classically allowed states can be derived from the more general assumption of each condition stressful in the phase space volume of size (per spatial dimension ). Equivalent is the claim which the phase-integral of a state for each coordinate can only integer multiples of accept ( Bohr - Sommerfeld quantum condition ):

This is a ( generalized) spatial coordinate and the corresponding ( canonical ) momentum, in the sense of classical mechanics in their formulation to Hamilton or Lagrange.

### Quantum mechanics (from 1925)

Quantum mechanics modifies the Hamiltonian mechanics to the effect that the position and momentum coordinates no longer numerical values ("c - number" for " classical number) " correspond, but operators ( "q - number" for "quantum number" ). The Hamiltonian becomes the Hamiltonian. Such variables are called observables, their possible values are given by the eigenvalues of the corresponding operator, the distributed continuously or discretely, depending on the operator ( quantized ) can be. The deviations from the results of classical mechanics arise from the fact that these operators do not commute with each other in products. In particular, the Bohr - Sommerfeld quantum condition is obtained as an approximation. The requirement to replace the classical Hamiltonian of the variables that are referred to in the Hamiltonian mechanics as a pair of canonically conjugate coordinates by suitable operators is also called the first canonical quantization or quantization.

### Quantum Electrodynamics ( 1927 )

The quantum electrodynamics is based on the classical field equations (in this case the Maxwell equations ) in Hamiltonian form and quantizes them according to the model of first quantization. From the operators of the field strength and the associated canonical momentum can be formed up and lowering operators that change the energy of the field at a time. This is like the position and momentum operators of the harmonic oscillator, but has here the importance of an increase or decrease of the number of photons, that is, the field quanta of the electromagnetic field. In a sense here so the particle itself becomes a quantum theoretical measurement variable ( observable ) with quantized eigenvalues , which is why the term second quantization is used for the whole process.

### Other quantum field theories (from 1934)

Since not only photons, but all particles can be created and destroyed, they are treated in quantum field theory as a field quantum of their respective fields. If the Hamiltonian (or Lagrangian ) of the field concerned no classical models exist, this is placed at the beginning of the theoretical treatment in the form of an approach. The quantization is performed on the model of quantum electrodynamics by introducing up-and- lowering operators. They are referred to here as creation and annihilation operators. The commutation relations, where they meet are either determined as in quantum electrodynamics ( as commutators ), or with a change of sign as anticommutators. In the first case, the field quanta arise as bosons, in the second case as fermions. This method is referred to as the canonical field quantization.