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Inquantum physics, aquantum state is a mathematical entity that embodies the knowledge of a quantum system.Quantum mechanics specifies the construction, evolution, andmeasurement of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system.
Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are
Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide intopure versusmixed states, or intocoherent states and incoherent states. Categories with special properties includestationary states for time independence andquantum vacuum states inquantum field theory.
As a tool for physics, quantum states grew out of states inclassical mechanics. A classical dynamical state consists of a set of dynamical variables with well-definedreal values at each instant of time.[1]: 3 For example, the state of a cannon ball would consist of its position and velocity. The state values evolve under equations of motion and thus remain strictly determined. If we know the position of a cannon and the exit velocity of its projectiles, then we can use equations containing the force of gravity to predict the trajectory of a cannon ball precisely.
Similarly, quantum states consist of sets of dynamical variables that evolve under equations of motion. However, the values derived from quantum states arecomplex numbers, quantized, limited byuncertainty relations,[1]: 159 and only provide aprobability distribution for the outcomes for a system. These constraints alter the nature of quantum dynamic variables. For example, the quantum state of an electron in adouble-slit experiment would consist of complex values over the detection region and, when squared, only predict the probability distribution of electron counts across the detector.
The process of describing a quantum system with quantum mechanics begins with identifying a set of variables defining the quantum state of the system.[1]: 204 The set will containcompatible and incompatible variables. Simultaneous measurement of acomplete set of compatible variables prepares the system in a unique state. The state then evolves deterministically according to theequations of motion. Subsequent measurement of the state produces a sample from a probability distribution predicted by the quantum mechanicaloperator corresponding to the measurement.
The fundamentally statistical or probabilisitic nature of quantum measurements changes the role of quantum states in quantum mechanics compared to classical states in classical mechanics. In classical mechanics, the initial state of one or more bodies is measured; the state evolves according to the equations of motion; measurements of the final state are compared to predictions. In quantum mechanics, ensembles of identically prepared quantum states evolve according to the equations of motion and many repeated measurements are compared to predicted probability distributions.[1]: 204
Measurements, macroscopic operations on quantum states, filter the state.[1]: 196 Whatever the input quantum state might be, repeated identical measurements give consistent values. For this reason, measurements 'prepare' quantum states for experiments, placing the system in a partially defined state. Subsequent measurements may either further prepare the system – these are compatible measurements – or it may alter the state, redefining it – these are called incompatible or complementary measurements. For example, we may measure the momentum of a state along the axis any number of times and get the same result, but if we measure the position after once measuring the momentum, subsequent measurements of momentum are changed. The quantum state appears unavoidably altered by incompatible measurements. This is known as theuncertainty principle.
The quantum state after a measurement is in aneigenstate corresponding to that measurement and the value measured.[1]: 202 Other aspects of the state may be unknown. Repeating the measurement will not alter the state. In some cases, compatible measurements can further refine the state, causing it to be an eigenstate corresponding to all these measurements.[2] A full set of compatible measurements produces apure state. Any state that is not pure is called amixed state as discussed in more depthbelow.[1]: 204 [3]: 73
The eigenstate solutions to theSchrödinger equation can be formed into pure states. Experiments rarely produce pure states. Therefore statistical mixtures of solutions must be compared to experiments.[1]: 204
The same physical quantum state can be expressed mathematically in different ways calledrepresentations.[1] The position wave function is one representation often seen first in introductions to quantum mechanics. The equivalent momentum wave function is another wave function based representation. Representations are analogous to coordinate systems[1]: 244 or similar mathematical devices likeparametric equations. Selecting a representation will make some aspects of a problem easier at the cost of making other things difficult.
In formal quantum mechanics (see§ Formalism in quantum physics below) the theory develops in terms of abstract 'vector space', avoiding any particular representation. This allows many elegant concepts of quantum mechanics to be expressed and to be applied even in cases where no classical analog exists.[1]: 244
Wave functions represent quantum states, particularly when they are functions of position or ofmomentum. Historically, definitions of quantum states used wavefunctions before the more formal methods were developed.[4]: 268 The wave function is a complex-valued function of any complete set of commuting or compatibledegrees of freedom. For example, one set could be the spatial coordinates of an electron.Preparing a system by measuring the complete set of compatible observables produces apure quantum state. More common, incomplete preparation produces amixed quantum state. Wave function solutions ofSchrödinger's equations of motion for operators corresponding to measurements can readily be expressed as pure states; they must be combined with statistical weights matching experimental preparation to compute the expected probability distribution.[1]: 205
Numerical or analytic solutions in quantum mechanics can be expressed aspure states. These solution states, calledeigenstates, are labeled with quantized values, typicallyquantum numbers.For example, when dealing with theenergy spectrum of theelectron in ahydrogen atom, the relevant pure states are identified by theprincipal quantum numbern, theangular momentum quantum numberℓ, themagnetic quantum numberm, and thespinz-componentsz. For another example, if the spin of an electron is measured in any direction, e.g. with aStern–Gerlach experiment, there are two possible results: up or down. A pure state here is represented by a two-dimensionalcomplex vector, with a length of one; that is, withwhere and are theabsolute values of and.
Thepostulates of quantum mechanics state that pure states, at a given time t, correspond tovectors in aseparablecomplexHilbert space, while each measurable physical quantity (such as the energy or momentum of aparticle) is associated with a mathematicaloperator called theobservable. The operator serves as alinear function that acts on the states of the system. Theeigenvalues of the operator correspond to the possible values of the observable. For example, it is possible to observe a particle with a momentum of 1 kg⋅m/s if and only if one of the eigenvalues of the momentum operator is 1 kg⋅m/s. The correspondingeigenvector (which physicists call aneigenstate) with eigenvalue 1 kg⋅m/s would be a quantum state with a definite, well-defined value of momentum of 1 kg⋅m/s, with noquantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/s.
On the other hand, a pure state described as asuperposition of multiple different eigenstatesdoes in general have quantum uncertainty for the given observable. Usingbra–ket notation, thislinear combination of eigenstates can be represented as:[5]: 22, 171, 172 The coefficient that corresponds to a particular state in the linear combination is a complex number, thus allowing interference effects between states. The coefficients are time dependent. How a quantum state changes in time is governed by thetime evolution operator.
A mixed quantum state corresponds to a probabilistic mixture of pure states; however, different distributions of pure states can generate equivalent (i.e., physically indistinguishable) mixed states. Amixture of quantum states is again a quantum state.
A mixed state for electron spins, in the density-matrix formulation, has the structure of a matrix that isHermitian and positive semi-definite, and hastrace 1.[6] A more complicated case is given (inbra–ket notation) by thesinglet state, which exemplifiesquantum entanglement:which involvessuperposition of joint spin states for two particles with spin 1/2. The singlet state satisfies the property that if the particles' spins are measured along the same direction then either the spin of the first particle is observed up and the spin of the second particle is observed down, or the first one is observed down and the second one is observed up, both possibilities occurring with equal probability.
A pure quantum state can be represented by aray in aprojective Hilbert space over thecomplex numbers, while mixed states are represented bydensity matrices, which arepositive semidefinite operators that act on Hilbert spaces.[7][3]TheSchrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as aconvex combination of pure states.[8] Before a particularmeasurement is performed on a quantum system, the theory gives only aprobability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and thelinear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by theuncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.
Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is astatistical ensemble of independent systems. Statistical mixtures represent the degree of knowledge whilst the uncertainty within quantum mechanics is fundamental. Mathematically, a statistical mixture is not a combination using complex coefficients, but rather a combination using real-valued, positive probabilities of different states. A number represents the probability of a randomly selected system being in the state. Unlike the linear combination case each system is in a definite eigenstate.[9][10]
The expectation value of an observableA is a statistical mean of measured values of the observable. It is this mean, and the distribution of probabilities, that is predicted by physical theories.
There is no state that is simultaneously an eigenstate forall observables. For example, we cannot prepare a state such that both the position measurementQ(t) and the momentum measurementP(t) (at the same timet) are known exactly; at least one of them will have a range of possible values.[a] This is the content of theHeisenberg uncertainty relation.
Moreover, in contrast to classical mechanics, it is unavoidable thatperforming a measurement on the system generally changes its state.[11][12][13]: 4 More precisely: After measuring an observableA, the system will be in an eigenstate ofA; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measureA twice in the same run of the experiment, the measurements being directly consecutive in time,[b] then they will produce the same results. This has some strange consequences, however, as follows.
Consider twoincompatible observables,A andB, whereA corresponds to a measurement earlier in time thanB.[c] Suppose that the system is in an eigenstate ofB at the experiment's beginning. If we measure onlyB, all runs of the experiment will yield the same result.If we measure firstA and thenB in the same run of the experiment, the system will transfer to an eigenstate ofA after the first measurement, and we will generally notice that the results ofB are statistical. Thus:Quantum mechanical measurements influence one another, and the order in which they are performed is important.
Another feature of quantum states becomes relevant if we consider a physical system that consists of multiple subsystems; for example, an experiment with two particles rather than one. Quantum physics allows for certain states, calledentangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, seeQuantum entanglement. These entangled states lead to experimentally testable properties (Bell's theorem)that allow us to distinguish between quantum theory and alternative classical (non-quantum) models.
One can take the observables to be dependent on time, while the stateσ was fixed once at the beginning of the experiment. This approach is called theHeisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observablesP(t),Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as theSchrödinger picture. (This approach was taken in the earlier part of the discussion above, with a time-varying state.) Conceptually (and mathematically), the two approaches are equivalent; choosing one of them is a matter of convention.
Both viewpoints are used in quantum theory. While non-relativisticquantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, forquantum field theory. Compare withDirac picture.[14]: 65
Quantum physics is most commonly formulated in terms oflinear algebra, as follows. Any given system is identified with some finite- or infinite-dimensionalHilbert space. The pure states correspond to vectors ofnorm 1. Thus the set of all pure states corresponds to theunit sphere in the Hilbert space, because the unit sphere is defined as the set of all vectors with norm 1.
Multiplying a pure state by ascalar is physically inconsequential (as long as the state is considered by itself). If a vector in a complex Hilbert space can be obtained from another vector by multiplying by some non-zero complex number, the two vectors in are said to correspond to the sameray in theprojective Hilbert space of. Note that although the wordray is used, properly speaking, a point in the projective Hilbert space corresponds to aline passing through the origin of the Hilbert space, rather than ahalf-line, orray in thegeometrical sense.
Theangular momentum has the same dimension (M·L2·T−1) as thePlanck constant and, at quantum scale, behaves as adiscrete degree of freedom of a quantum system. Most particles possess a kind of intrinsic angular momentum that does not appear at all in classical mechanics and arises from Dirac's relativistic generalization of the theory. Mathematically it is described withspinors. In non-relativistic quantum mechanics thegroup representations of theLie group SU(2) are used to describe this additional freedom. For a given particle, the choice of representation (and hence the range of possible values of the spin observable) is specified by a non-negative numberS that, in units of thereduced Planck constantħ, is either aninteger (0, 1, 2, ...) or ahalf-integer (1/2, 3/2, 5/2, ...). For amassive particle with spinS, itsspin quantum numberm always assumes one of the2S + 1 possible values in the set
As a consequence, the quantum state of a particle with spin is described by avector-valued wave function with values inC2S+1. Equivalently, it is represented by acomplex-valued function of four variables: one discretequantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).
The quantum state of a system ofN particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3spatial coordinates andspin, e.g.
Here, the spin variablesmν assume values from the setwhere is the spin ofνth particle. for a particle that does not exhibit spin.
The treatment ofidentical particles is very different forbosons (particles with integer spin) versusfermions (particles with half-integer spin). The aboveN-particle function must either be symmetrized (in the bosonic case) or anti-symmetrized (in the fermionic case) with respect to the particle numbers. If not allN particles are identical, but some of them are, then the function must be (anti)symmetrized separately over the variables corresponding to each group of identical variables, according to its statistics (bosonic or fermionic).
Electrons are fermions withS = 1/2,photons (quanta of light) are bosons withS = 1 (although in thevacuum they aremassless and can't be described with Schrödinger mechanics).
When symmetrization or anti-symmetrization is unnecessary,N-particle spaces of states can be obtained simply bytensor products of one-particle spaces, to which we will return later.
A state belonging to aseparablecomplexHilbert space can always be expressed uniquely as alinear combination of elements of anorthonormal basis of.Usingbra–ket notation, this means any state can be written aswithcomplexcoefficients and basis elements. In this case, thenormalization condition translates toIn physical terms, has been expressed as aquantum superposition of the "basis states", i.e., theeigenstates of an observable. In particular, if said observable is measured on the normalized state, then is the probability that the result of the measurement is.[5]: 22
In general, the expression for probability always consist of a relation between the quantum state and aportion of the spectrum of the dynamical variable (i.e.random variable) being observed.[15]: 98 [16]: 53 For example, the situation above describes the discrete case aseigenvalues belong to thepoint spectrum. Likewise, thewave function is just theeigenfunction of theHamiltonian operator with corresponding eigenvalue(s); the energy of the system.
An example of the continuous case is given by theposition operator. The probability measure for a system in state is given by:[17]where is the probability density function for finding a particle at a given position. These examples emphasize the distinction in charactertistics between the state and the observable. That is, whereas is a pure state belonging to, the(generalized) eigenvectors of the position operator donot.[18]
Though closely related, pure states are not the same as bound states belonging to thepure point spectrum of an observable with no quantum uncertainty. A particle is said to be in abound state if it remains localized in a bounded region of space for all times. A pure state is called a bound stateif and only if for every there is acompact set such thatfor all.[19] The integral represents the probability that a particle is found in a bounded region at any time. If the probability remains arbitrarily close to then the particle is said to remain in.
For example,non-normalizable solutions of thefree Schrödinger equation can be expressed as functions that are normalizable, usingwave packets. These wave packets belong to the pure point spectrum of a correspondingprojection operator which, mathematically speaking, constitutes an observable.[16]: 48 However, they are not bound states.
As mentioned above, quantum states may besuperposed. If and are two kets corresponding to quantum states, the ketis also a quantum state of the same system. Both and can be complex numbers; their relative amplitude and relative phase will influence the resulting quantum state.
Writing the superposed state usingand defining the norm of the state as:and extracting the common factors gives:The overall phase factor in front has no physical effect.[20]: 108 Only the relative phase affects the physical nature of the superposition.
One example of superposition is thedouble-slit experiment, in which superposition leads toquantum interference. Another example of the importance of relative phase isRabi oscillations, where the relative phase of two states varies in time due to theSchrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.
Apure quantum state is a state which can be described by a single ket vector, as described above. Amixed quantum state is astatistical ensemble of pure states (seeQuantum statistical mechanics).[3]: 73
Mixed states arise in quantum mechanics in two different situations: first, when the preparation of the system is not fully known, and thus one must deal with astatistical ensemble of possible preparations; and second, when one wants to describe a physical system which isentangled with another, as its state cannot be described by a pure state. In the first case, there could theoretically be another person who knows the full history of the system, and therefore describe the same system as a pure state; in this case, the density matrix is simply used to represent the limited knowledge of a quantum state. In the second case, however, the existence of quantum entanglement theoretically prevents the existence of complete knowledge about the subsystem, and it's impossible for any person to describe the subsystem of an entangled pair as a pure state.
Mixed states inevitably arise from pure states when, for a composite quantum system with anentangled state on it, the part is inaccessible to the observer.[3]: 121–122 The state of the part is expressed then as thepartial trace over.
A mixed statecannot be described with a single ket vector.[21]: 691–692 Instead, it is described by its associateddensity matrix (ordensity operator), usually denotedρ. Density matrices can describe both mixedand pure states, treating them on the same footing. Moreover, a mixed quantum state on a given quantum system described by a Hilbert space can be always represented as the partial trace of a pure quantum state (called apurification) on a larger bipartite system for a sufficiently large Hilbert space.
The density matrix describing a mixed state is defined to be an operator of the formwhereps is the fraction of the ensemble in each pure state The density matrix can be thought of as a way of using the one-particleformalism to describe the behavior of many similar particles by giving a probability distribution (or ensemble) of states that these particles can be found in.
A simple criterion for checking whether a density matrix is describing a pure or mixed state is that thetrace ofρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed.[d][22] Another, equivalent, criterion is that thevon Neumann entropy is 0 for a pure state, and strictly positive for a mixed state.
The rules for measurement in quantum mechanics are particularly simple to state in terms of density matrices. For example, the ensemble average (expectation value) of a measurement corresponding to an observableA is given bywhere and are eigenkets and eigenvalues, respectively, for the operatorA, and "tr" denotes trace.[3]: 73 It is important to note that two types of averaging are occurring, one (over) being the usual expected value of the observable when the quantum is in state, and the other (over) being a statistical (saidincoherent) average with the probabilitiesps that the quantum is in those states.
States can be formulated in terms of observables, rather than as vectors in a vector space. These arepositive normalized linear functionals on aC*-algebra, or sometimes other classes of algebras of observables.SeeState on a C*-algebra andGelfand–Naimark–Segal construction for more details.
The concept of quantum states, in particular the content of the sectionFormalism in quantum physics above, is covered in most standard textbooks on quantum mechanics.
For a discussion of conceptual aspects and a comparison with classical states, see:
For a more detailed coverage of mathematical aspects, see:
For a discussion of purifications of mixed quantum states, see Chapter 2 of John Preskill's lecture notes forPhysics 219 at Caltech.
For a discussion of geometric aspects see:
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