Quantum state

In the quantum-mechanical treatment of a physical system, the instantaneous state of the system is a mathematical and complicated object such as a so-called " state of operation " or a special so-called " state operator ", which for each of the system possible ( fault-free ) measurement and, for each thereby the possible measurements determines the probability with which the respective measurement result is obtained.

• 2.1 state
• 2.2 probability amplitude
• 2.3 Measurement Size ( observable )
• 2.4 eigenvalue single measured value, expected value
• 2.5 phase factor and superposition
• 2.6 state and mixture density operator

Basic concepts

Contrast to classical physics

The introduction of probabilities of different results instead of a unique prediction represents a fundamental departure from classical physics. There is clearly established (always error-free measurement provided ) with the indication of the current state of the system, the result of any possible measurement. Generally meets the macroscopic systems (eg from everyday life ) very well, but is increasingly wrong for ever smaller systems. For example, can be in the macroscopic everyday life of a pellet or a grain of sand in every moment with virtually clear accuracy ascribe a particular place and a particular speed. For an ensemble of quantum mechanical particles, this is ruled out, however. The strictly valid Heisenberg uncertainty principle states namely: is the abode clearly established, then a measurement of the speed with equal probability yield any value, and vice versa as well ( Heisenberg uncertainty principle of 1927). This uncertainty can not be eliminated by precise preparation of the state of the system. It is mathematically rigorous, relatively easy to prove and a central conceptual basis of physics.

Pure state and state mixture

Additional uncertainty about the expected measurement result arises when the state of the system is not clearly defined. This applies, for example, for the common case where the observed system is extracted from a number of similar systems, which are not all prepared in the same state. The states in which the attentional system can be located ( with possibly different probabilities ), form a state mixture. Here, the uncertainty about the expected measurement results could still decrease by only systems are selected in the same condition for measurement. To illustrate the difference to the state mixture is called a uniquely prepared state sometimes called pure state. The following state here means more pure state.

Eigenstate

A condition in which certain for a measure of the expected measurement value that is unique, ie, eigenstate of the measure. Examples are

Here, strictly speaking, for a mathematical subtlety ( the presence of a so-called " continuous spectrum " ) Examples 1 and 2 only in the limit of permitted ( in Example 2 as the " monochromatic limit case " of a wave packet, while the Example 1 thereof by means of a Fourier transform can get ).

Both examples in any case play a significant role in the theoretical description. Example 3 is a state in which a physical quantity (energy) is a certain value, while both the location and the pulse only probabilities for different test results can be given ( for example, by the location of the orbital for pulse by the magnitude-squared of the Fourier transform of the respective local wave function).

Superposition of states

For a (potentially point-like ) particles of the state is given by its position and momentum in classical mechanics, ie by a point in six-dimensional state space, called the phase space. However, the observed in beams of such particles interference effects (so-called wave -particle duality ) enforce the possibility that the superposition (or superposition, a linear combination of complex factors ) of several states a possible state forms ( see, eg, matter waves ). Each state can be predicted for the different to a certain quantity Measuring values ​​with different probabilities, is a superposition of the corresponding to these measured values ​​eigenstates. The probability of obtaining a certain of these eigenvalues ​​as the measurement result is determined by the absolute square of the (generally complex ) factor by which the eigenstate in question occurs in this overlay. This factor is called the probability amplitude.

There is no fundamental difference between a superposition states, and the base states and eigenstates: For a given system, but rather, each state can be superimposed with each other, and each state can be represented as a superposition of the other states. The superpositioned states are so pure states in the above sense.

States, which are formed by superposition are sometimes inaccurately spoken of as mixed states, which should be avoided, however, because confusion with the term state mixture could occur.

State and statistical weight

The set of possible states of the extended by the possibility of superposition phase space is of considerably greater thickness than the original space of classical mechanics. As a measure of this extended space is considered in quantum statistical physics is not the size of this amount themselves but their dimension, which is the condition number of the smallest possible subset, so that might be subject to all at all possible states of the system from it by superposition. Since none of the states can be represented as a superposition of the other within this range, they are linearly independent and form a basis of the extended phase space. Compared with the density of states in the classical statistical physics shows that each quantum-mechanical state of such a base, the " phase space volume " occupied. ( The physical dimension of this " volume " is an " action " = energy times time or place times = pulse. 's Planck's constant. )

Mathematical representation

State

For the mathematical representation of the physical state defined above, two forms are:

• A location-and time-dependent matter wave or
• A state vector in an infinite - dimensional vector space, called a Hilbert space.

Both representations are equivalent. The first is due to Erwin Schrödinger is often easier to illustrate. The second, which goes back to Werner Heisenberg and Paul Dirac, often has the advantage of a clearer representation in algebraic equations. A state can either be listed as a wave function, for example, or as a vector, for which according to the Dirac ket symbol has naturalized. If it is an eigenstate of a physical measurand, the eigenvalue in question is usually indicated in the icon. Examples:

However, the physical state from which the probabilities of all possible measurement results are shown the function or the vector defines not clearly reversible, but only up to a constant factor. All functions or vectors denote the same state when a constant complex number. In Hilbert space, therefore, a state is not a vector, but the one - dimensional subspace ( beam ), which is spanned by it. This is expressed mathematically by the projection operator.

Probability amplitude

The probability amplitude, with which a condition is contained in a state, represented by the convolution integral or the inner product:

It is to the dual vector, which is referred to as bra vector. (bra and ket combine to bracket together, Eng. clamp for. ) Note that the bra is given by the vector. Furthermore, it is assumed here that the functions or vectors are normalized:

So that the probability of observing the state of the same state, right 1 comes out. ( Eigenstates of measured variables with continuous eigenvalues ​​as position and momentum must be normalized by the Dirac δ - distribution, see below) If a system is in an eigenstate of a measurable size, it is certainly not in an eigenstate ( the same measurand ) to meet with another eigenvalue, the probability amplitude thereof has a value of zero:

Two states with the scalar zero are called orthogonal. Because of their ease of clarity often used wave function of the state is, ie the component of the state vector along the basis vector that describes the state of the fixed particles at the site (even if this localized state is not realized in nature ).

Measured quantity ( observable )

A measurable physical quantity is represented by an operator, which causes a linear transform in a Hilbert space. The associated symbol consists of the commonly used letters with a roof over it. Examples:

Physical variable and associated operator are called observables summarized. Since all possible outcomes are real numbers, the operator must be Hermitian, that is, the condition

. meet

Eigenvalue, single measured value, expected value

Does an observable on one of their own eigenstates ( where the subscript indicates the eigenvalue ), the state vector multiplied by the eigenvalue ( eigenvalue equation):

The chance to get to any state in a measurement of observables just the value that is the absolute square of the probability amplitude to meet the system in the corresponding eigenstate:

This one has at the same time prepares the system in the eigenstate, because after this measurement, it is straight ahead in this state.

As expected value of the average of many individual measurements of the observable traffic of the same systems is known in the state. From the spectrum of any individual results and their probabilities gives:

Phase factor and superposition

Two Markets, which differ only by a constant factor, describing the same state, for all to be derived for this condition measurable properties are independent of this factor. The factor may depend on the time, but none of observables such as the place. If the factor the magnitude 1, it is also called phase factor, since it can be written in the form, which is called the quantum mechanical phase.

Linear combinations of two state vectors, eg, with complex numbers, also describe allowable states ( see above superposition of states). Here, the relative phase of the factors is not desired. Depending on the phase of the superimposed state has several measurable properties. One speaks therefore of coherent superposition, because as with optical interference with coherent light is not the absolute squares but the " generating amplitudes " itself, ie, and, are superimposed.

State and mixture density operator

A mixture state in which the system is in the state with probability ( to ) is represented by the density operator, which is the sum of the respective projection operators:

In contrast to the coherent superposition of the density operator remains unchanged when you know the states represented in the mixture with arbitrary phase factors. In state mixture, the states are therefore superimposed incoherent.

The expectation value of a measurement of the observable is therefore the weighted incoherent sum of the expected values ​​of the individual components of the mixture:

This can also be represented simply as a trace of the operator:

The last equation has the advantage that it applies to mixtures and for pure states. (For a pure state is simply the projection operator associated with the state. )

The density operator is also referred to as "a state operator".

Examples

A) The states of a ( one-dimensional ) particle in a potential well of width (from 0 to ) are superpositions of eigenstates of the energy operator, which are described by the wave functions and have the energy eigenvalues ​​. An arbitrary state can be then more than write the normalization condition is observed.

B ) the energy eigenstates are also eigenstates of the angular momentum operator for particles in a central field and therefore carry all three quantum numbers:

Is degenerate with respect to the energy, so that any conditions as can be written again with a suitable normalization condition.

C ) The spin eigenstates of a ( fermionic ) particle is simply written as and.

Formal

In the mathematical formalism of quantum mechanics and quantum field theory is a state of an abstract object. The relation of a state to the real world is represented by special representations, for example, in local coordinates or momenta. The representation in local coordinates is called a wave function. From these representations of the state vectors exact probability statements about the properties of particles or particle systems can be made according to the laws of quantum mechanics. The time variation of the state vectors is described in quantum mechanics by the Schrödinger equation and is thus uniquely determined.

Most is meant an abstract Hilbert space with a state of an element ( vector). States that are represented this way are called vector states. However, most states can be represented only by density matrices, or not in a Hilbert space. To describe general states of the formalism of C *-algebras will be used. A state on a C *-algebra is then a linear functional with and. The amount of these states is a convex set, the pure states are then exactly the extreme points of this set. Each state can be assigned to a Hilbert space representation by means of the GNS construction. More precisely, one finds a Hilbert space representation and a unit vector such that. The irreducible representations of the pure states are allocated accurately.

Another example

It was given a two-electron system, one of the two electrons is very far away from the other. Consider further on each of the two places a measuring apparatus that is designed to appeal only to the spin of a single electron and is equally likely that oriented upward or downward. The considered state of the two -electron system is.

Is - mind you! - A single, well-defined pure state. Such conditions may arise from s-wave decay of a single, first elementary bound system in a natural way.

The possible outcomes are still two times ( and have the same probability): spin up at one of the two electrons and down the other. But: Which of the two electrons, 1 or 2, the spin will have upwards, you can not predict.

So because of the large mutual distance is expected independence of the measurement results. Instead, we see a very strong correlation (or rather anti- correlation), although you can only specify probabilities.

This is an example of an entangled state, which also shows the extreme peculiarities of the state concept in quantum mechanics.

Pure states and state mixtures

In quantum mechanics and quantum statistics, a distinction is made ​​between pure states and state mixtures. Pure states represent the ideal case of a maximum knowledge of the observable properties ( observables ) of the system dar. Frequently, however, after preparation or due to measurement inaccuracies of the state of the system only incompletely understood (for example, the spin of a single electron in an unpolarized electron beam). Then the various potentially occurring pure states and the associated projection operators can only probabilities pi are assigned ( see below). Such incompletely known states are referred to as state mixtures. For the presentation of state mixtures used to the density operator ρ, which is also called density matrix or state operator of quantum statistics.

A pure state corresponds to a 1- dimensional subspace (ray) in a Hilbert space. The corresponding density matrix is the operator for the projection onto this subspace. It satisfies the condition of idempotency, that is, ρ2 = ρ. State mixtures are, however, only be represented by non-trivial density matrices, ie that ρ2 < ρ applies. A description by a beam is then not possible.

Characteristic features of this state description, the Superponierbarkeit ( "coherence" ) of the pure states and the consequent phenomenon of quantum entanglement, while the contributions of the various states involved are summed incoherently at the state mixtures.

The results of measurements on a quantum system results when repeated on a prepared exactly the same system also for pure states of a non-trivial distribution of measured values ​​that is weighted in the quantum statistics in addition ( inkohärent! ) with the pi. The distribution corresponds in detail the quantum mechanical state (respectively) and the observables for the measurement process ( iW represents the measuring apparatus ). For pure states follows from quantum mechanics: The mean of the series of measurements generated by repetition and the quantum mechanical expectation value are identical.

For the result of the measurements is therefore in contrast to classical physics, even for pure ( thus completely known ) quantum mechanical states only a probability can be specified ( so it does not mean in the following the result, but the expected result, see below). For mixtures condition applies because of an additional pi ( inkohärente! ) indeterminacy:

So even the expected result of the output of a single measurement can only in special cases (for example ) be predicted. Only the ( speziellen! ) eigenstates of the considered observable and the corresponding eigenvalues ​​are given in general as measured values ​​in question, and even in the above case of a pure state, approximately, ie even when fully known wave function, you can specify for the different eigenstates with given probabilities only, although the state is reproduced at an immediately subsequent measurement result with exactly the same equipment. Unknown states can not, however, be determined by measurement (see no- cloning theorem). It is equally that is, that now is not the belonging to the projection operators Markets are superimposed, but the projection operators are even equipped with probabilities.

Overall, therefore, applies: where the index i on the other hand refers to the ( pure ) states, the index k to the measured variable.

( If the ak or would just known "about", you still would have the pi with two corresponding probability factors, q k and r i, k multiply. )

Information entropy

The information entropy of a condition, or multiplied by the Boltzmann constant, the Von Neumann entropy, is a quantitative measure of its lack of knowledge. The von Neumann entropy is zero for pure states and correspondingly for state mixtures. Thereby Boltzmann units are used, in particular kB is the Boltzmann constant. In Shannon's units, however, this constant is one and the natural logarithm, ln ... ( = e log ... ) by 2 log ... replaced ( change of basis ).

Summary

The quantum mechanical state (" purity " required), state vector that is not simply describes "what is ", but specifies the set of " components " of which can be the projected out by matching the measured size measurement from the vector ( " preparation" ), and, in particular, is the probability with which it selected the k -th component itself must also iA be prepared first by measurements.

In contrast, the state of the measuring apparatus is i.W. " Classic " (although it also depends on the quantum measuring size). Nevertheless, an essential aspect of the measuring apparatus is therefore " classic " because the apparatus (eg pointer position ) finds analogous to a sufficiently large clock clear which eigenvalue ak has come out in the measurement ("Registration "). The newly -prepared state, now obeys anyway until the next measurement is not classical, but the quantum mechanical time evolution (see Mathematical Structure of Quantum Mechanics # Temporal evolution ).

In other words, the quantum mechanical state is " dissection ", the classic contrast " -registering ".