# What is formal decohaerence

## Wave function collapse

If a measurement is carried out on a quantum mechanical system, the system goes into a state in which the measured variable has a defined value (previously, for example, a superposition of several such states could exist). This is called Condition reduction, or Wave function collapse.

### Collapse of the wave function within the axiomatic version of quantum mechanics

The collapse of the wave function is described by one of the axioms of quantum mechanics:

 After measuring the condition , the a as a non-degenerate eigenvalue (eigenvector: ), the system is in its own state to a. Is the eigenvalue a degenerate, the system is in any state of the to a belonging Eigenraum.

This means the following: In quantum mechanics, the possible measured values ​​of an observable are its eigenvalues ai to the eigenvectors . If any measurement is carried out on a system, only the eigenvalues ​​can be seen as the result ai receive. After the measurement, the system is then in a defined state, namely the eigenstate of the measured eigenvalue.

### Example: double slit experiment

In the double slit experiment, an interference pattern is observed when the exact path of the particles through the slit is not known. The wave functionψ (z) of the system is then the sum of the wave functions for a system with only the upper slit ψ1(z) and a system with only the lower gap ψ2(z). If you now measure the position of the particles at the gap (e.g. covering a gap), only one of the two wave functions is observed ψ1(z) or ψ2(z). The wave function has thus collapsed onto one of its components.

Before the measurement, the probability was a particle in place z to be found right now

P(z) = | ψ (z) | 2 = | ψ1(z) + ψ2(z) | 2.

After the measurement, the following applies: ### Formal version

As Wave function collapse one denotes their probabilistic transition into a ensemble of (possible) other wave functions, formed from the components of the original wave function with probabilities which, according to Max Born, are equal to the squares of their coefficients. The collapse was explicitly postulated by John von Neumann as a second dynamic in addition to the Schrödinger equation, around well-defined measurement results or a psycho-physical parallelism to enable - but it corresponds to Born's original formulation of the probability interpretation.

According to the decoherence theory appears the ensemble of wave functions to the local observer only because of its entanglement with the environment.