State (quantum mechanics)
A quantum mechanical state is the description of the state of a physical system according to the rules of quantum mechanics . It differs fundamentally from the description of the state according to the rules of classical physics , so that the observations made on quantum physical systems can be recorded. Different concepts of state belong to the different interpretations of quantum mechanics . This article deals with the concept of state of the widely held Copenhagen interpretation .
In contrast to the classic term, the state in the Copenhagen interpretation of quantum mechanics does not define a measured value that can be expected with certainty for every measurement that can be carried out on the system , but only the probability that this value will occur for every possible measured value. The borderline case for a measured value (and thus for all others), i.e. the reliable prediction of a measured value, only applies to those states that are eigenstates of the respective measured variable. Also in contrast to the classical state, the time development of the quantum mechanical state is not continuously determined deterministically. Instead, a measurement is generally used to change the state of the system in a way that cannot be influenced and can only be predicted with a certain degree of probability.
The so-called “preparation” of a system in a certain state takes place through the simultaneous measurement of a maximum set of commensurable physical quantities. After this measurement, the system is in a well-defined common eigenstate of all these measured variables, so that they have certain values. If the system was not already in such a common eigenstate, the measurement suddenly causes a state reduction , also known as a collapse, so that afterwards all other possible measured values of these variables have the probability zero. The state reduction is not a physical process, but rather describes the more precise information that the observer receives from the measurement. Between two measurements, the time development of the state is determined deterministically by an equation of motion ; in the non- relativistic case by the Schrödinger equation , in the relativistic case, depending on the spin and mass of the particle, by the Klein-Gordon equation (spin 0), the Dirac equation (massive, spin ½), the Weyl equation (massless , Spin ½), the Proca equation (massive, spin 1) or the Maxwell equations (massless, spin 1).
Mathematically, the quantum mechanical state is usually described by a normalized state vector in the Hilbert space . With the help of a basis of the Hilbert space with a discrete index, this state vector can be written as a linear combination of the basis vectors, or in the case of a basis with a continuous index as a wave function . The state vector has at least one component for each of the possible measured values of a physical variable. The strength of a component (more precisely: the square of the magnitude of its amplitude) determines the probability with which the relevant measured value occurs as a result of a measurement.
The assignment of state and state vector is irreversibly unambiguous, because state vectors that differ only by a constant complex phase factor describe the same physical state. The linear combination of the state vectors of two states is a possible state vector; it describes a state that is physically different from the two superimposed states, whereby the relative complex phase of the two superimposed state vectors is also important. The theoretical basis of the description as a linear combination was developed in 1925 by Werner Heisenberg in matrix mechanics , which was developed as a wave function in 1926 in the position or momentum basis by Erwin Schrödinger in wave mechanics . The two descriptions are based on the same underlying mathematical structure, in which a state is understood as a mapping that assigns a real number to each of the operators that represent a measured variable, which indicates the expected value of the measurement results possible in this state. This was worked out by John von Neumann in 1931 .
In the case of incomplete preparation of an initial state or in quantum statistics , a distinction is made between pure and mixed states . To describe them, the state vector must be extended to become a density operator (also called a state operator ). This formalism also avoids the above-mentioned indeterminacy of the complex phase, but complicates the notion of a wave function, which is sometimes helpful for the visualization.
Difference to classical physics
The introduction of the probabilities of different results instead of a clear prediction means a fundamental departure from classical physics . The result of every possible measurement is clearly defined there with the specification of the current system status (provided that the measurement is always error-free). This generally applies very well to macroscopic systems (e.g. from everyday life). For example, a shotgun or a grain of sand can be assigned a specific location and speed at any moment with practically unambiguous accuracy .
For ever smaller systems, however, this becomes increasingly wrong, for an ensemble of quantum mechanical particles it is impossible. The strictly valid Heisenberg uncertainty principle of 1927 states: if the location is clearly established, then a measurement of the speed can give any value with equal probability , and vice versa; d. H. Only one of the two quantities can be clearly determined at any time. This indeterminacy can not be eliminated even with the most precise preparation of the system state. It is mathematically rigorous, relatively easy to prove, and forms a central conceptual basis of physics.
Pure state and mixed state
Additional uncertainty about the expected measurement result arises if the state of the system is not clearly defined. This applies e.g. B. for the frequent case that the observed system is selected from a number of similar systems, which are not all prepared in the same state. The different states in which the observed system can be (with possibly different probability) then form a mixture of states .
The uncertainty about the expected measurement results could be reduced here by only selecting systems in the same state for measurement. To clarify the difference to the mixed state, a clearly prepared state is sometimes referred to as a pure state .
In the following, state always means pure state .
- the particle is localized in one place ( local intrinsic state)
- the particle has a certain speed or momentum (momentum eigenstate)
- the particle is in a bound state of certain energy ( energy intrinsic state ).
Examples 1 and 2 are, strictly speaking (due to a mathematical subtlety: the presence of a " continuous spectrum ") allowed (in Example 2, for instance in "only in the limit of monochromatic limiting case of" an infinitely extended wave packet , while the Example 1 therefrom by a Fourier transformation obtained becomes). Both examples play an important role in the theoretical description.
Example 3 is a state in which a physical quantity (namely the energy) has a certain value, while probabilities for different measurement results can only be given for the location and for the momentum (e.g. for the location through the orbital , for the momentum by the square of the amount of the Fourier transform of the spatial wave function concerned ).
Superposition of states
For a particle in the form of a mass point , in classical mechanics the state is given by the location and the momentum , i.e. by a point in the six-dimensional phase space . Since interference effects are also observed with particle beams ( wave-particle dualism ), the possibility that the superposition (or superposition , linear combination with complex factors) of several states forms a possible state must be allowed (see matter waves ). Every state for which quantum mechanics predicts several possible measured values for a measured variable, each with its own probabilities, is a superposition of those states that are the eigen-states belonging to these measured values. The probability of obtaining a certain one of these eigenvalues as a measurement result is determined by the square of the magnitude of its probability amplitude. The probability amplitude is the (generally complex) factor with which the eigenstate in question occurs in this superposition.
There is no fundamental difference between the properties of being a superposition state or a base or eigenstate: Every state of a system can be viewed as a base state of a suitably chosen basis, but also as a superposition state of the basis vectors of another basis. Any state can be overlaid with any other state of the same system, and any state can be represented as an overlay of other states. States that have been defined as superposition are also pure states in the above sense. Occasionally, however, they are addressed imprecisely as mixed states , but this should be avoided because confusion with the term mixed state could occur.
Condition and statistical weight
The quantum mechanical phase space becomes considerably more powerful than the phase space of classical mechanics for the same system due to the possibility of superposition . In statistical quantum physics, the measure of this expanded space is not the size of this set itself, but its dimension ; that is the smallest possible number of states from which all possible states of the system can result through superposition. Within this smallest possible subset, therefore, none of the states can be represented as a superposition of the others, therefore they are linearly independent and form a basis for the entire phase space.
A comparison with the density of states in classical statistical physics shows that every quantum mechanical state of such a basis occupies the “phase space volume” , whereby the number of independent spatial coordinates is and Planck's quantum of action . The physical dimension of this "volume" is for one effect = energy times time, or = place times momentum.
In the mathematical formalism of quantum mechanics and quantum field theory , a state is a mapping rule that assigns its expected value to every physical quantity. This definition includes a mixture of states. Since the physical quantities are represented by linear operators that form a subset of a C * -algebra , a state in mathematically strict naming is a linear functional that maps from the C * -algebra to the complex numbers , and applies to: and . The one as the argument of the functional is the one element of algebra, and the one on the right is the one of the complex numbers.
The set of these states is a convex set , that is, if and are states and then is also a state. A state is called pure if it can only be trivially dismantled, that is, if or is. These pure states are precisely the extreme points of this set; every mixed state can be written as an integral over pure states.
Each state, by means of the GNS construction a Hilbert space representation to be assigned. Every normalized vector in the Hilbert space,, corresponds to a state in, and vice versa, a vector can be assigned to every state. It applies
where the scalar product in Hilbert space denotes and . The pure states form the irreducible representations in the Hilbert space.
Two forms that are equivalent to each other are suitable for the mathematical representation of the pure state physically defined above:
- a state vector in a countable-infinite -dimensional vector space with a scalar product called Hilbert space . This representation goes back to Werner Heisenberg and Paul Dirac and often has the advantage of clear representation in algebraic equations . A state is noted here as a vector , for which Dirac has adopted the ket symbol .
- a state function, e.g. B. the time and place dependent matter wave . A state is noted here as a wave function , e.g. B. . This representation goes back to Erwin Schrödinger and is often easier to illustrate, especially if it is only a single particle.
State vector and co-vector
The state vector in Hilbert space is, like a position vector , a mathematical, abstract object. Just like the position vector in a basic representation
can be written with three mutually orthogonal vectors in three-dimensional Euclidean space, the state vector can be expanded in any complete orthonormal basis. For this development it is necessary to introduce the covector , which is located as a Bra vector in the dual space to the Hilbert space. From a mathematical point of view, a Bra vector is a linear functional that operates on the Hilbert space in the complex numbers. As for vectors in Euclidean space, development is analogous
with . Since the basis vectors form an orthonormal basis, the following applies
with the Kronecker Delta and
with the infinite dimensional identity matrix . Since in quantum mechanics - in contrast to Euclidean vector space - continuous bases can also occur, the same applies to development in a continuous base
with the Dirac distribution respectively
In order not to have to differentiate between continuous and discrete bases in the notation, the symbol is sometimes used.
If the state vector is represented in a basis, then it is mostly in the eigen basis of a Hermitian operator, which is identified with a physical measured variable. The eigenstates of such an operator are often denoted by the symbol of the corresponding physical quantity:
- denotes the local state of a particle,
- the impulse eigenstate,
- the energy eigenstate. It can have both discrete values (e.g. in the case of bound states) and continuous values (e.g. in the case of unbound states).
- If a quantum number is assigned to an eigenvalue (e.g. quantum number for the -th energy level , quantum numbers for the amount and z-component of the angular momentum ), the associated eigenstate is specified by specifying the quantum number (s) or by a specially agreed symbol (examples :) .
So that the wave function can be understood as a probability amplitude according to Born's rule , it is necessary to normalize the state vector. That is, for a physical state must
be valid. However, this does not define the vector unambiguously in a reversible manner , but only up to a constant factor , i.e. a complex number with a magnitude of one. This is also known as the quantum mechanical phase of the state or state vector. The vectors , which all describe the same state, span a one-dimensional subspace ( ray ).
The wave functions or are the expansion coefficients of the state vector in the position or momentum base:
A measurable physical quantity is represented by an operator that effects a linear transformation in the Hilbert space . The measurand and associated operator are collectively called observables . The possible measurement results are the eigenvalues of the operator. That is, it applies to an eigenstate of the operator
In the case of a state that is not an eigenstate of the operator concerned, measurement results cannot be predicted with certainty, but only with probabilities. These probabilities are calculated for each eigenvalue as an absolute square from the scalar product of the respective eigenvector of the measurand with the state vector of the system:
After the measurement, the state vector has collapsed onto the subspace associated with the corresponding eigenvalue , that is to say
As a result, the system is also prepared in its own state , because after this measurement it is exactly in this state. A renewed measurement of this observable taking place instantaneously therefore results in the same value again.
The mean value of many individual measurements of the observable on always the same systems in the same state is called the expected value . From the spectrum of all possible individual results and their probabilities results:
Phase factor and superposition
Linear combinations of two state vectors, e.g. B. with complex numbers that fulfill the condition , also describe allowed states (see above superposition of states ). Here, unlike with a single state vector, the relative phase of the factors, i.e. H. the complex phase in the quotient , no longer arbitrary; depending on the phase, the superposition state has different physical properties. This is why we speak of coherent superposition because, as in the case of optical interference with coherent light, it is not the squares of the magnitude, but the "generating amplitudes " themselves , i.e. and , that are superposed.
State mixture and density operator
A mixture of states, in which the system is likely to be in the (with ) state , is represented by the density operator , which is the sum of the corresponding projection operators:
In contrast to the coherent superposition , the density operator remains unchanged if the states represented in the mixture are provided with any phase factors; In the mixed state, the states are thus superimposed incoherently .
The expected value of a measurement of the observable is accordingly the weighted incoherent sum of the expected values of the individual components of the mixture:
This can also be represented as a trace of the operator :
The last equation has the advantage that it applies equally to mixtures and to pure states. (In the case of a pure state , the projection operator belonging to the state is.)
The density operator is also known as the “state operator”.
- The states of a particle in the (one-dimensional) box of width (from 0 to ) can be written as superpositions of eigenstates of the Hamilton operator. Whose eigen-states are in the spatial space
- and the associated energy eigenvalues to be
- For particles in a central field , the eigen-states of energy can be chosen so that they are also eigen-states of the angular momentum operator. Then they carry all three quantum numbers :
- Due to the energy degeneration with regard to the quantum number, a measurement of the energy is generally not sufficient to clearly determine the state.
- The spin eigenstates of a (fermionic) particle are simply written as and .
- The state of a system that arises from the s-wave decay of a single bound elementary particle system into two spin 1/2 particles is . By measuring the spin of one particle, the state collapses instantaneously, so that an immediately following measurement of the other particle delivers a clearly correlated result (namely the opposite in each case). This is an example of quantum entanglement .
Pure states and mixed states
In quantum mechanics and quantum statistics , a distinction is made between pure states and mixed states. Pure states represent the ideal case of maximum knowledge of the observable properties ( observables ) of the system. Often, however, the state of the system is only incompletely known after preparation or due to measurement inaccuracies (example: the spin of the individual electron in an unpolarized electron beam). Then only probabilities can be assigned to the various possibly occurring pure states or the assigned projection operators (see below). Such incompletely known states are called mixed states. The density operator ρ, which is also called the density matrix or state operator, is used to represent mixtures of states.
A pure state corresponds to a one-dimensional subspace (ray) in a Hilbert space . The associated density matrix is the operator for the projection onto this subspace. It fulfills the condition of idempotence , i.e. H. . In contrast, mixtures of states can only be represented by non-trivial density matrices, i.e. that is , that applies. A description by a ray is then not possible.
Characteristic features of this state description are the superposability ("coherence") of the pure states and the consequent phenomenon of quantum entanglement , while the contributions of the various states involved are incoherently added up in the mixed states.
The result of measurements on a quantum system, when repeated on a system prepared in exactly the same way, gives a non-trivial distribution of measured values even in pure states, which is additionally (incoherently!) Weighted with the quantum statistics . The distribution corresponds in detail to the quantum mechanical state (or ) and the observables for the measurement process (generally represents the measurement apparatus). For pure states it follows from quantum mechanics: The mean value of the measurement series generated by repetition and the quantum mechanical expected value are identical.
In contrast to classical physics, only one probability can be given for the result of the measurements, even with pure (i.e. completely known) quantum mechanical states (therefore, in the following it is not the result but the expected result , see below). Because of the additional (incoherent!) Indeterminacy, the following applies to mixtures of states :
Even the expected result of the outcome of a single measurement can only be predicted with certainty in special cases. Only the (special!) Eigenstates of the observed observable or the associated eigenvalues are possible as measured values at all, and even in the above-mentioned case of a pure state , e.g. H. Even if the wave function is completely known, only probabilities can be given for the various eigenstates given a given, although the state is exactly reproduced in an immediately subsequent measurement with the same apparatus. Unknown states, on the other hand, cannot be determined by measurement (see no-cloning theorem ). It also applies d. This means that now the kets belonging to the projection operators are not superimposed, but the projection operators themselves are given probabilities.
Overall, the following applies: where the index i relates to the (pure) states, whereas the index k relates to the measured variable.
(If the one or those were only known "roughly", they would have to be multiplied by two corresponding probability factors, or .)
The information entropy of the state or the Von Neumann entropy multiplied by the Boltzmann constant is a quantitative measure of the ignorance that exists with regard to the possible statement about the existence of a certain pure state. The Von Neumann entropy,, is the same for a mixture of states. For pure states it is zero (note for ). In this Boltzmann 'sche units used in particular is the Boltzmann constant . In Shannon 's units, however, this constant is replaced by one and the natural logarithm by the binary logarithm .
- Quantum mechanical measurement
- Quantum tomography
- Heisenberg's uncertainty principle
- Einstein-Podolsky-Rosen paradox
References and footnotes
- Wolfgang Nolting: Basic Course Theoretical Physics 5/1; Quantum Mechanics - Basics . 5th edition. Springer, Berlin Heidelberg 2002, ISBN 3-540-42114-9 , pp. 119 .
- FH Fröhner: Missing Link between Probability Theory and Quantum Mechanics: the Riesz-Fejér Theorem. In: Journal for Nature Research. 53a (1998), pp. 637-654 ( online )
- For a single electron in a particle beam , a simultaneous "sharp" recording measurement of momentum and location is possible using one and the same measuring apparatus ("counter"). In a magnetic spectrometer z. B. even the point of impact is used as the measured variable from which the impulse can be calculated. A prediction of which counter from a given arrangement, which covers all possibilities, will respond to the subsequent electron, or at least the simultaneity of “sharp” mean values of location and momentum in a series of measurements , are excluded . See Feynman lectures on physics . 3 volumes, ISBN 0-201-02115-3 (German lectures on physics . Oldenbourg Wissenschaftsverlag, Munich 2007, ISBN 978-3-486-58444-8 ), first published in 1963/1965 by Addison / Wesley. In Volume 3, Quantum Mechanics, Chap. 16, the terminology of Heisenberg's uncertainty principle is dealt with in detail .
- See article Heisenberg's uncertainty principle or for example Albert Messiah Quantenmechanik , de Gruyter 1978, Volume 1, pp. 121ff
- In the case of unbound eigenstates of the energy operator , limit value problems analogous to those in Examples 1 and 2 (see below) occur.
- This dimension can be finite or countable-infinite (as in the standard case of the Hilbert space ) or even uncountable-infinite (as in Gelfand's space triples , a generalization of the Hilbert space for better recording of continuous spectra).
- Walter Thirring: Quantum Mechanics of Atoms and Molecules . In: Textbook of Mathematical Physics . 3. Edition. tape 3 . Springer, Vienna 1994, ISBN 978-3-211-82535-8 , pp. 26 .
- W. Heisenberg: About quantum theoretical reinterpretation of kinematic and mechanical relationships . In: Journal of Physics . Volume 33, 1925, pp. 879-893.
- PAM Dirac : On the theory of quantum mechanics . In: Proceedings of the Royal Society of London A . Volume 112, 1926, pp. 661-677.
- E. Schrödinger : " Quantization as Eigenwertproblem I ", Annalen der Physik 79 (1926), 361–376. E. Schrödinger: " Quantization as Eigenwertproblem II ", Annalen der Physik 79 (1926), 489-527. E. Schrödinger: " Quantization as Eigenwertproblem III ", Annalen der Physik 80 (1926), 734–756. E. Schrödinger: " Quantization as Eigenwertproblem IV ", Annalen der Physik 81 (1926), 109-139
- Torsten Fließbach: Quantum Mechanics . 4th edition. Spektrum, Munich 2005, ISBN 3-8274-1589-6 , pp. 231 .
- Example: If the eigenstates for spin are “up” or “down” in the z-direction, then the eigenstate is “up” in the x-direction, but the eigenstate is “up” in the y-direction. (The normalization factor was omitted.)
- Imagine the practically impossible task of determining the many-particle state of a system of N = 10 23 electrons.
- "Incoherent" because they are weighted with a square expression in the
- This means, among other things, that they can not be determined by specifying such and such.