Mathematical formulation of quantum mechanics

This article represents the mathematical structure of quantum mechanics

Formulation by von Neumann

The fundamental basis for the mathematically rigorous formulation of quantum mechanics were formulated by John von Neumann in 1932. Thus, a physical system can be generally described by three essential ingredients: Its states, its observables and its dynamics ( that is, by its temporal evolution ). The von Neumann 's postulates are here in a slightly updated form ( spin, Pauli principle, see below ), respectively:

Postulates of quantum mechanics ( the Copenhagen interpretation)

As part of the Copenhagen interpretation is based quantum mechanical description of a system on the following postulates:

These statements come across spin and Pauli exclusion principle, which can be justified only in a relativistic extension of quantum mechanics, but this is essential for the non- relativistic quantum mechanics and, for example, determine the periodic table of elements crucial.

Quantum mechanical states

In classical mechanics the state of a physical system with degrees of freedom and its evolution in time by specifying pairs of canonically conjugate variables is completely determined. Because according to two mutually conjugate observables are not simultaneously arbitrary in principle be determined exactly in quantum mechanics, there is the fundamental question of how far a definition of the state of a quantum physical system is useful. The fundamental approach in the framework of quantum mechanics, that a physical system is to be defined solely by simultaneously measurable observables, is one of their essential differences to classical mechanics. Only through the consistent implementation of such a state definition, a large number of quantum physical phenomena describe theoretically.

In the context of quantum mechanics, a physical condition over a maximum sentence simultaneously measurable observables is defined, one speaks in this context of a complete set of commuting observables ( VSKO ). Observables can take very specific values ​​for their respective spectrum depends, usually from the system under consideration and by the respective observables in a measurement. The possible values ​​form the spectrum of the observables. They may be distributed either discretely or continuously. In the discrete case they are called eigenvalues ​​of the observables. Most of it is assumed for the sake that the spectrum is purely discrete, although important observables exist whose spectrum is purely continuous ( for example, position and momentum operator).

The corresponding to the eigenvalues ​​states are called eigenstates of observables. In continuous spectrum is called Generalized eigenfunctions. This is to distributions such as the Dirac function or monochromatic plane waves, that do not actually belong to the state space, because they are not square - integrable, superpose from which but allowed by integrating states can ( wave packet formation, see generalized Fourier series ). Hereinafter referred to as the discrete case, unless otherwise stated, as viewed.

Since affect measurements of the observables of a VSKO not mutually exclusive, can be explained by the use of suitable filters a given quantum physical system to prepare a state that is the eigenstate of each of the observables of the VSKO:

Such a condition is often referred to as pure quantum state. It is defined by its eigenvalues ​​and determined maximum.

It should be emphasized that over such a prepared quantum state - are certainly not all measurable properties of the physical system - in contrast to the state of a classical system! For observables that are incompatible with the VSKO, only a certain probability can be specified for each of its eigenvalues ​​, with the result of a measurement; the measurement result is in each case a proper value of the observable. This basic indeterminacy is related to the above-mentioned uncertainty relation. She is one of the most important statements of quantum mechanics and is both a cause for rejection of many of these over.

For a given quantum physical system belonging to the eigenvalues ​​of an observable eigenstates form a linear state space - mathematically called a Hilbert space. This represents the totality of all possible states of the system and thus has generally been the case of simple systems such as the quantum harmonic oscillator infinitely many dimensions; However, it comes with countably infinite spaces of ( separable Hilbert spaces ). This does not contradict that there are also measurements with a continuous spectrum. It is essential in any case that a linear superposition of several eigenstates is again part of the state space, even if the superposition state is not an eigenstate of the observable. One speaks in this context of a superposition of several states. This characteristic is compared with the vectors in a plane, the superposition is also a vector in the plane.

A simple example of a quantum system is the two-state system, see the article qubit.

Statistical statements of quantum mechanics

From the decomposition of the state in a linear combination of orthonormal eigenstates of the observable result of the square value of the corresponding pre-factor is a measure of the probability of measuring the intrinsic value in such a superimposed state, respectively to make the system in the eigenstate. The coefficients are therefore referred to as " probability amplitudes " for the measured values ​​. They can be as a projection ( = scalar ) of the respective eigenstate directions ( see Figure 2):

Thus resulting in repeated execution of a measurement of an observable A., different results, even if the system before the measurement was always in the same state. Exception: If the system is prepared in an eigenstate of an observable, further measurements of these observables each give the same reading. Experimentally, let the statistical distributions of the measured values ​​by repeatedly performing measurements to determine identically prepared systems. This relationship between the measurement protocol and the mathematical calculus of quantum mechanics is confirmed in all experiments.

Here, for simplicity, assume that you are dealing with a purely discrete spectrum.

Temporal development

The dynamics of quantum states is described by different representations, called " images ", which can be converted into one another by redefinition of the operators and states and thus are equivalent. For all images, the expectation values ​​of operators are the same.

Schrödinger picture

In the Schrödinger picture, the momentum from the following analysis reveals: The condition is defined by a differentiable map of t parameterized by time on the Hilbert space of states. If the state of the system, " t " refers to an arbitrary time, the so-called Schrödinger equation,

With as a densely - defined self - adjoint operator, the Hamiltonian, the imaginary unit "i" and the reduced Planck 's constant. As an observable corresponds to the total energy of the system. In general, the Hamiltonian can be time-dependent (eg interaction of the system with an electromagnetic field ).

The time evolution of a state is given by the unitary time evolution operator ( is the time ordering operator, see below):

( The Zeitordungsoperator ensures that operator products of the form are rearranged in non- commutativity so that applies This results in a causal chain, .. Right cause, effect left )

Heisenberg picture

In the Heisenberg picture of quantum mechanics, the time dependence is instead of temporal changes of the states that remain constant in this picture, described by time-dependent operators for observables. For the time-dependent Heisenberg operators is obtained for the differential equation ( Heisenberg equation of motion)

With the ( already known ) time evolution operator, the states and operators in the Heisenberg picture can be linked to those in the Schrödinger picture by the following relations:

Dirac picture

The so-called Dirac picture or interaction picture has both time-dependent states, as well as time-dependent observables, which apply to states and observables different Hamiltonians. In the Schrödinger picture the Hamiltonian is composed of a time-independent and time-dependent Hermitian operator

^ In the interaction picture for the observables are then used with only the similarity transformation and the expression for the states constituted accordingly. It is generally even, which is indeed not changed by the similarity transformation much " easier " than.

The interaction picture is most useful when the time evolution of the observables exactly solvable (that is, if it is " trivial " ), so that all mathematical complications remain limited to the time evolution of the states. For this reason, the Hamiltonian for the operators is referred to as "free Hamiltonian " and the Hamiltonian for the states as " interaction Hamiltonian ". The dynamic development is therefore now described by the following two equations:

• First, the equation of state ( cf. Schrödinger equation)

• And secondly, the operator equation ( cf. Heisenberg equation of motion)

With the time evolution operator of the time-independent part of the problem

States and operators in Diracbild which are linked in the Schrödinger picture of the following relationships:

Comments

At the time of voting the operators and states of all the images match:

The Heisenberg picture corresponds to the classical Hamiltonian mechanics ( for example, correspond to the commutators of quantum mechanics the classical Poisson brackets ). But Physically, the Schrödinger picture intuitive. The Dirac representation is often used in perturbation theory - especially in quantum field theory and many-body physics.

Some wave functions form probability distributions, which do not change with time. Many systems that need to be described with a dynamic time behavior in classical mechanics that fact in the quantum mechanical description of such "static" wave functions. For example, a single electron in an atom is described in the ground state through a circular trajectory around the atomic nucleus, whereas in quantum mechanics is described by a static, spherically symmetric wavefunction surrounding the nucleus. (Note that only the smallest angular momentum states of the "S" waves, are spherically symmetric ).

The Schrödinger equation is as the closely related Heisenberg equation and the equations of the interaction picture a partial differential equation that can be solved analytically only for a few model systems ( the most important examples include the quantum mechanical harmonic oscillator and the electron in the Coulomb potential ). Even the electron structure of the helium atom, which has only one electron more than hydrogen is already no longer be calculated analytically. However, there are a number of different techniques for calculating approximate solutions. An example is the already mentioned perturbation theory, simplified model systems are used as starting point for the calculation of more complex models in the existing analytical solutions. This method is particularly successful when can the interactions of the complex model formulated as a "small" disturbances of the simple model system. Another method is the so-called "semi -classical approximation ", which may be applied to systems which have only small quantum effects. The quantum- mechanically induced effects can then be calculated assuming classical motion trajectories. This approach is used for example in the study of quantum chaos as a basis.

Spin

In addition to their other properties all particles have a kind of intrinsic angular momentum, the spin, for which there is no counterpart in classical physics. The spin is quantized in units of full-or half-integer, therefore does not apply in the coordinate representation, but where ms, which is often referred to, one of the following discrete values:.

A distinction bosons (S = 0 or 1 or 2, or ...) and fermions (S = 1/ 2 or 3/ 2 or 5 /2, or ... )

Pauli principle

Linked to this is for systems of N identical particles, the so-called Pauli exclusion principle, which states, for example, in the coordinate representation, that in exchange of two of the N particles, must apply for the N-particle wave function Permutationsverhalten following:

That is, for bosonic must prefactor 1, arise for fermions on the other hand (-1). In two spatial dimensions can be replaced by an arbitrary complex number of magnitude one ( see anyons ). In so-called supersymmetric theories, as discussed in high-energy physics, the state would be a linear combination of a bosonic and fermionic one share.

Electrons are fermions with S = 1 /2; Photons are bosons with S = 1

Newer formalisms

An alternative approach to calculating quantum mechanical systems is the path integral formalism Richard Feynman, in which a quantum mechanical amplitude is represented as the sum of the probability amplitudes for all theoretically possible paths of a particle as it moves from an initial state to a target state. This formulation is the quantum mechanical analogue of the classical action principle.

Only recently has a more general mathematical description of observables by positive - operator-valued probability measures (positive operator valued probability measures, POVM ) is developed, which is hardly covered by traditional textbooks yet. Operations on quantum systems can be described in a comprehensive and mathematically by completely positive maps in the modern, but still little-known version of quantum mechanics. This theory is generalized for both the unitary -time development as well as the above-described traditional von Neumann description of the modification of a quantum system in a measurement. Concepts that can be described in traditional image is difficult, such as continuously running unsharp measurements, fit easily into this newer description. Mentioned here is the so-called method of C * - algebras.