# Quantization (physics)

In the theoretical description of a physical system, quantization is the step in which results, terms or methods of classical physics are modified in such a way that quantum physical observations on the system are correctly reproduced. Among other things, this is intended to explain the quantification of many measurable quantities, e.g. B. the presence of certain, discrete energy values ​​at the excitation levels of an atom.

From 1900, at the beginning of quantum physics , quantization essentially meant that, with the help of certain rules, from the processes and states possible according to classical physics, those that contradicted the observations were to be excluded. This characterizes the older quantum theories, among them e.g. B. the well-known Bohr atomic model . In 1925/26 Werner Heisenberg and Erwin Schrödinger independently found two ways of modifying the basic concepts and equations of classical mechanics instead of the results of classical mechanics in order to be able to correctly predict quantum physical observations. The development of today's quantum mechanics began . The common ground of these two ways is called canonical quantization . Canonical quantization can also be carried out for physical fields and became the basis of quantum field theory from 1927 .

## development

### Older quantum theory (1900-1925)

The first rule for quantization was given by Max Planck in 1900 in order to be able to calculate the spectrum of thermal radiation using the means of classical statistical physics . This rule, known at the time as the quantum hypothesis , reads: The exchange of energy between matter and electromagnetic radiation of frequency only takes place in quanta of size , i.e. i.e., it is quantized . In this the constant is Planck's quantum of action . ${\ displaystyle \ nu}$${\ displaystyle h \ nu}$${\ displaystyle h}$

The idea that it is a harmonic oscillator to which the electromagnetic field supplies or removes energy leads to the statement that it cannot be excited with any freely selectable energy, but only has states with discrete, equidistant energy levels at a distance . This selection from the continuum of the classically permitted states can be derived from the more general assumption that every state in phase space claims a volume of the size (per space dimension). Equally meaningful is the requirement that the phase integral of a state can only assume integer multiples of for each coordinate (Bohr-Sommerfeld quantum condition): ${\ displaystyle \ Delta E = h \ nu}$${\ displaystyle h}$${\ displaystyle h}$

${\ displaystyle \ oint p \, dq = nh}$, ( )${\ displaystyle n = 0, \, 1, \, 2 \, \ ldots}$

This contains a (generalized) position coordinate and the associated (canonical) momentum , in the sense of classical mechanics in its formulation according to Hamilton or Lagrange . ${\ displaystyle q}$${\ displaystyle p}$

### Quantum Mechanics (from 1925)

The quantum mechanics modifies the Hamiltonian mechanics in that the position and momentum no longer correspond to numerical values ( "c-numbers" for "classical number)", but operators ( "q-number" of "quantum number"). The Hamilton function thus becomes the Hamilton operator . Such quantities are called observables , their possible measured values ​​are given by the eigenvalues ​​of the associated operator, which, depending on the operator, can be continuously or discretely distributed ( quantized ). The deviations from the results of classical mechanics result from the fact that these operators cannot be interchanged in products . In particular, Bohr-Sommerfeld's quantum condition is obtained as an approximation.

The rule of replacing the variables in the classical Hamiltonian, which are called a pair of canonically conjugate coordinates in Hamiltonian mechanics , with suitable operators, is also called 1st quantization or canonical quantization .

### Quantum Electrodynamics (from 1927)

The quantum electrodynamics is based on the classical field equations (in this case the Maxwell equations ) in Hamiltonian form and quantized it along the lines of the first quantization. From the operators for the field strength and the associated canonical momentum, ascending and descending operators can be formed, which change the energy of the field by . This is like the position and momentum operators of the harmonic oscillator, but here has the meaning of an increase or decrease in the number of photons, i.e. H. the field quanta of the electromagnetic field. In a certain sense, the number of particles itself becomes a quantum-theoretical measurable variable ( observable ) with quantized eigenvalues, which is why the term 2nd quantization is used for the entire process . ${\ displaystyle h \ nu \ (= \ hbar \ omega)}$

### Other quantum field theories (from 1934)

Since not only photons , but all particles can be generated and destroyed, they are treated in quantum field theory as field quanta of their respective fields. If there are no classical models for the Hamilton function (or Lagrange function ) of the relevant field, this is placed at the beginning of the theoretical treatment in the form of an approach . The quantization takes place according to the model of quantum electrodynamics, by introducing ascending and descending operators. They are referred to here as the creation or annihilation operator . The commutation rules that they satisfy are either specified as in quantum electrodynamics (as commutators ), or with a change in sign as anti- commutators . In the first case the field quanta result as bosons , in the second case as fermions . This method is called canonical field quantization .

## literature

• Walter Weizel: Textbook of Theoretical Physics . 2nd Edition. Springer-Verlag, Heidelberg 1958.
• Georg Joos: Textbook of Theoretical Physics . 11th edition. Akad. Verlagsgesellsch., Frankfurt am Main. 1959.
• Albert Messiah: Quantum Mechanics . 1st edition. North Holland Publ. Comp., Amsterdam 1958.