Quantum tunneling is common in the Higgs field

Astro-Lexicon W 2


The wave function, usually symbolized by the Greek letter Ψ, describes a microscopic particle in quantum theory.

From classical mechanics to quantum mechanics

In classical mechanics, particle trajectories are determined by dealing with the problem of motion, namely theEquation of motion, solves. The result is a particle trajectory that may depend on certain setting parameters, but is well defined and discrete.
In quantum mechanics, the wave function replaces the concept of the particle trajectory (it still has to be interpreted appropriately, more on that later) and also solves a quantum mechanical equation of motion: the famous oneSchrödinger equation. As we shall see, the orbit of the quantum mechanical particle, the quantum, is no longer so well determined.

Properties of the wave function

Since the wave functions describe quantum mechanical particles, they must also be able to contain all properties of particles. Quantum physicists call these properties quantum numbers and mean, for example, the mass of the particle, the electrical charge, the spin and the isospin. Furthermore, the wave function depends on the spatial coordinates and time.

Probability wave

The wave function is a scalar distribution function of a certain amplitude (wave amplitude), which generally depends on the location and time and is also parameterized by the particle properties. A quantum mechanical problem is considered to be solved when the characteristic curve of the wave function in the coordinates is known. The classical point particle is no longer localized at a certain point in space and time, but 'smeared' in the form of the wave function. It is not the wave function itself that is important, but you Absolute square. Because the wave function can also be complex-valued. Absolute squares are real-valued. Note: An absolute square is always calculated by multiplying a given complex quantity by the associated complex conjugate quantity (symbolized with an additional asterisk, see equation above). This distribution of the absolute square (not the wave function itself!) Is calledProbability distribution interpreted. A higher probability of finding the quantum is expected where the absolute square is larger.

Wave-particle dualism

In quantum physics, particles can behave as waves or as particles, i.e. in one experiment the wave properties are suitable for clarifying the observation; in another the particle or corpuscular properties are suitable. This phenomenon is called Wave-particle dualism and is characteristic of quanta. This dualism therefore applies, for example, to light, more precisely to photons, but also to electrons, to the neutrino or to the atomic nucleus.

Microworld is basically blurred

In addition, there is another complication in the quantum world: It is not possible for quantum mechanical observers to measure certain properties of a quant at the same time: A quantum observer can measure either the position or the momentum (the speed) of a quantum. However, a quantum observer can only measure either energy or time. This blurring is in the Heisenberg's uncertainty principle summarized.

To the quantum mechanical measurement process

What about the wave function at quantum mechanical measurement process happens, is explained in the entry Copenhagen Interpretation. The interpretation has fundamental consequences for our understanding of the world.

This postulate was made in 1923 by H. Weyl designed and says that the galaxies in the universe should move like 'elementary particles' in a liquid. A space pervading substrate can in this sense be considered an ideal liquid because the geodesics only intersect at a singular point in the past (and possibly also in the future). Matter (galaxies) therefore has a unique speed at every point in space-time.

The Weyl tensor or also called conformal tensor was named after the German mathematicianHermann Klaus Hugo Weyl (1885-1955). This 4th order tensor is relatively complicated and can initially be used for any dimensions n generally be noted. In the general theory of relativity (GTR) the following applies n = 4, because spacetime is characterized by one dimension of time and three dimensions of space.

Calculation of the Weyl tensor

Using the definition equation above, one can see immediately that the Weyl tensor is a rather complicated structure that can only be calculated with some effort, even with simple spacetime. Unless one uses so-called computer-algebraic systems that allow tensor calculations on the computer quickly and easily.
The equation also shows that the Weyl tensor can be derived from the Riemann tensor (curvature tensor;R. with four indices), the Ricci tensor (R. with two indices) and the Ricci scalar (R. without indices).


In addition to the symmetries of the Riemann tensor, the Weyl tensor has an additional symmetry: it is without a trace, that is, the sum of its diagonal elements vanishes. The discussion of its symmetry properties allows a classification of the vacuum-spacetime, which is known under the Petrov classification.

Curvature properties indicator

Physically, the Weyl tensor is particularly important because it is suitable for investigating the curvature properties of spacetime. From the Riemann tensor and the Weyl tensor, Riemann and Weyl invariants determine who Not depend on the coordinate system. This also includes the Kretschmann angelfish. A discussion of such quantities makes it clear where the curvature is particularly strong or particularly weak. This makes it easier to find curvature singularities and to characterize them as asymptotically flat spacetime.

The time-dependent Schrödinger equation of quantum theory clearly describes the dynamics of the wave function, i.e. its development over time. Solutions of the Schrödinger equation thus reveal the state of the quantum system at a point in time t and in one place r (Vector).

from Schrödinger to Wheeler-DeWitt

A relativistic formulation of the stationary Schrödinger equation is known as the Wheeler-DeWitt equation. She is the subject of Quantum cosmology. This equation is obtained by using the theory of relativity in Hamiltonian form circumscribes. This approach is already known in classical mechanics, but has such a general formulation that it can also be transferred to other theories. It is important to note that the Wheeler-DeWitt equation Notis covariant. This in itself is not surprising when one takes into account that the Schrödinger equation in the context of the non-relativistic Quantum mechanics is derived. The violation of covariance manifests itself in the fact that certain space-like hypersurfaces are distinguished. In addition, the Wheeler-DeWitt equation only applies pointwise. Any renormalization procedures are therefore necessary.
Technically, one has to break up the space-time symmetry of the general theory of relativity and separate it into space and time. This as an ADM formalism (according to the relativists Arnowitt, Deser and Misner) known technology distinguishes space-like hypersurfaces, the foliage of which (Foliation) represents the passage of time: the canonical time parameter changes from one hypersurface to the other. In the usual Hamiltonian formalism, one now calculates the generalized canonical momenta and the canonical conjugate variable from the derivatives of the Lagrange density. Integration finally yields the Hamilton function. It is the essential operator of the Wheeler-DeWitt equation, which is formally very similar to the (stationary) Schrödinger equation (with zero energy eigenvalue). The wave function is now called the 'Wave function of the universe' designated. She is on an infinitely dimensional Super room the spacetime geometries and all matter fields.

This is how it goes on: Fix boundary conditions

For this hyperbolic, partial differential equation, one now has to set boundary conditions, such as, for example Vilenkin and Linden tree suggested. They derived an analogy to the quantum mechanical tunnel effect, which one 'Quantum tunneling out of (or into) nothing'called. This is explained by the fact that a probability stream (with the usual quantum mechanical definition) flows out of (or into) the superspace.
Hartle and Hawking indicated an alternative boundary condition that 'no-boundary-condition'. Here the edge of the four-dimensional manifold is de facto always the same. The problem with this approach via path integrals is that the wave function of the universe then not clear can be determined because an evaluation of the path integral in the complex plane leads to different results, depending on which integration path one follows.

Creation and destruction of entire universes!

In quantum cosmology there is also the familiar apparatus of canonical quantization. So one can have many-particle states, Baby universes called, from vacuum states, voids called, create by using creation operators. However, the associated Wheeler-DeWitt equation becomes even more complex and even non-linear, because interactions between these states have to be taken into account. The quantum mechanical particle generation and annihilation corresponds in this application to the cosmos of Creation and destruction of universes! The epistemological content of this theory is immense.

The time aspect

The Wheeler-DeWitt equation is in its fundamental formulation regardless of the parameter time! There is a time parameter that determines the foliation of space-time in hypersurfaces. However, the foliation is completely arbitrary! Therefore the resulting concept of time is not unambiguous either. It must now be investigated whether the quantum theories of different foliations ('gauges') unitarily equivalent are. If this were the case, the selected foliation would be irrelevant. Only special solutions of the Wheeler-DeWitt equation (if the local observer has to be reintroduced) lead to an order parameter that can be identified over time. In the past this led to the question of a 'Physics without time'.

Note to the reader

Wien's radiation formula

See in connection under Planck's radiator.



© Andreas Müller, August 2007