Quantum state

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In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in time exhausts all that can be predicted about the system's behavior. A mixture of quantum states is again a quantum state. Quantum states that cannot be written as a mixture of other states are called pure quantum states, while all other states are called mixed quantum states. A pure quantum state can be represented by a ray in a Hilbert space over the complex numbers, [1] [2] while mixed states are represented by density matrices, which are positive semidefinite operators that act on Hilbert spaces. [3] [4]

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Pure states are also known as state vectors or wave functions, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the energy spectrum of the electron in a hydrogen atom, the relevant state vectors are identified by the principal quantum number n, the angular momentum quantum number l, the magnetic quantum number m, and the spin z-component sz. For another example, if the spin of an electron is measured in any direction, e.g. with a Stern–Gerlach experiment, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a qubit. A pure state here is represented by a two-dimensional complex vector , with a length of one; that is, with

where and are the absolute values of and . A mixed state, in this case, has the structure of a matrix that is Hermitian and positive semi-definite, and has trace 1. [5] A more complicated case is given (in bra–ket notation) by the singlet state, which exemplifies quantum entanglement:

which involves superposition of joint spin states for two particles with spin 12. 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 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. The Schrödinger–HJW theorem classifies the multitude of ways to write a given mixed state as a convex combination of pure states. [6] Before a particular measurement is performed on a quantum system, the theory gives only a probability distribution for the outcome, and the form that this distribution takes is completely determined by the quantum state and the linear operators describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the uncertainty principle: a state that implies a narrow spread of possible outcomes for one experiment necessarily implies a wide spread of possible outcomes for another.

Conceptual description

Pure states

Probability densities for the electron of a hydrogen atom in different quantum states. Hydrogen Density Plots.png
Probability densities for the electron of a hydrogen atom in different quantum states.

In the mathematical formulation of quantum mechanics, pure quantum states correspond to vectors in a Hilbert space, while each observable quantity (such as the energy or momentum of a particle) is associated with a mathematical operator. The operator serves as a linear function which acts on the states of the system. The eigenvalues 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 kgm/s if and only if one of the eigenvalues of the momentum operator is 1 kgm/s. The corresponding eigenvector (which physicists call an eigenstate) with eigenvalue 1 kgm/s would be a quantum state with a definite, well-defined value of momentum of 1 kgm/s, with no quantum uncertainty. If its momentum were measured, the result is guaranteed to be 1 kgm/s.

On the other hand, a system in a superposition of multiple different eigenstates does in general have quantum uncertainty for the given observable. We can represent this linear combination of eigenstates as:

The coefficient which 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 the time evolution operator. The symbols and [lower-alpha 1] surrounding the are part of bra–ket notation.

Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a statistical 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. [7] [8]

The expectation value of an observable A 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 which is simultaneously an eigenstate for all observables. For example, we cannot prepare a state such that both the position measurement Q(t) and the momentum measurement P(t) (at the same time t) are known exactly; at least one of them will have a range of possible values. [lower-alpha 2] This is the content of the Heisenberg uncertainty relation.

Moreover, in contrast to classical mechanics, it is unavoidable that performing a measurement on the system generally changes its state. [9] [10] [lower-alpha 3] More precisely: After measuring an observable A, the system will be in an eigenstate of A; thus the state has changed, unless the system was already in that eigenstate. This expresses a kind of logical consistency: If we measure A twice in the same run of the experiment, the measurements being directly consecutive in time, [lower-alpha 4] then they will produce the same results. This has some strange consequences, however, as follows.

Consider two incompatible observables, A and B, where A corresponds to a measurement earlier in time than B. [lower-alpha 5] Suppose that the system is in an eigenstate of B at the experiment's beginning. If we measure only B, all runs of the experiment will yield the same result. If we measure first A and then B in the same run of the experiment, the system will transfer to an eigenstate of A after the first measurement, and we will generally notice that the results of B 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, called entangled states, that show certain statistical correlations between measurements on the two particles which cannot be explained by classical theory. For details, see 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.

Schrödinger picture vs. Heisenberg picture

One can take the observables to be dependent on time, while the state σ was fixed once at the beginning of the experiment. This approach is called the Heisenberg picture. (This approach was taken in the later part of the discussion above, with time-varying observables P(t), Q(t).) One can, equivalently, treat the observables as fixed, while the state of the system depends on time; that is known as the Schrödinger 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-relativistic quantum mechanics is usually formulated in terms of the Schrödinger picture, the Heisenberg picture is often preferred in a relativistic context, that is, for quantum field theory. Compare with Dirac picture. [12] : 65

Formalism in quantum physics

Pure states as rays in a complex Hilbert space

Quantum physics is most commonly formulated in terms of linear algebra, as follows. Any given system is identified with some finite- or infinite-dimensional Hilbert space. The pure states correspond to vectors of norm 1. Thus the set of all pure states corresponds to the unit 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 a scalar 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 are said to correspond to the same "ray" in [1] :50 and also to the same point in the projective Hilbert space of .

Bra–ket notation

Calculations in quantum mechanics make frequent use of linear operators, scalar products, dual spaces and Hermitian conjugation. In order to make such calculations flow smoothly, and to make it unnecessary (in some contexts) to fully understand the underlying linear algebra, Paul Dirac invented a notation to describe quantum states, known as bra–ket notation . Although the details of this are beyond the scope of this article, some consequences of this are:

Spin

The angular momentum has the same dimension (M·L 2·T −1) as the Planck constant and, at quantum scale, behaves as a discrete degree of freedom of a quantum system.[ which? ] 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 with spinors. In non-relativistic quantum mechanics the group representations of the Lie 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 number S that, in units of Planck's reduced constant ħ, is either an integer (0, 1, 2 ...) or a half-integer (1/2, 3/2, 5/2 ...). For a massive particle with spin S, its spin quantum number m always assumes one of the 2S + 1 possible values in the set

As a consequence, the quantum state of a particle with spin is described by a vector-valued wave function with values in C2S+1. Equivalently, it is represented by a complex-valued function of four variables: one discrete quantum number variable (for the spin) is added to the usual three continuous variables (for the position in space).

Many-body states and particle statistics

The quantum state of a system of N particles, each potentially with spin, is described by a complex-valued function with four variables per particle, corresponding to 3 spatial coordinates and spin, e.g.

Here, the spin variables mν assume values from the set

where is the spin of νth particle. for a particle that does not exhibit spin.

The treatment of identical particles is very different for bosons (particles with integer spin) versus fermions (particles with half-integer spin). The above N-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 all N 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 with S = 1/2, photons (quanta of light) are bosons with S = 1 (although in the vacuum they are massless and can't be described with Schrödinger mechanics).

When symmetrization or anti-symmetrization is unnecessary, N-particle spaces of states can be obtained simply by tensor products of one-particle spaces, to which we will return later.

Basis states of one-particle systems

As with any Hilbert space, if a basis is chosen for the Hilbert space of a system, then any ket can be expanded as a linear combination of those basis elements. Symbolically, given basis kets , any ket can be written

where ci are complex numbers. In physical terms, this is described by saying that has been expressed as a quantum superposition of the states . If the basis kets are chosen to be orthonormal (as is often the case), then .

One property worth noting is that the normalized states are characterized by

and for orthonormal basis this translates to

Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the are eigenstates (with eigenvalues ki) of an observable, and that observable is measured on the normalized state , then the probability that the result of the measurement is ki is |ci|2. (The normalization condition above mandates that the total sum of probabilities is equal to one.)

A particularly important example is the position basis, which is the basis consisting of eigenstates with eigenvalues of the observable which corresponds to measuring position. [lower-alpha 9] If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket is associated with a complex-valued function of three-dimensional space

[lower-alpha 11] i.e. (a Dirac delta function ), which means that </ref>

This function is called the wave function corresponding to . Similarly to the discrete case above, the probability density of the particle being found at position is and the normalized states have

.

In terms of the continuous set of position basis , the state is:

.

Superposition of pure states

As mentioned above, quantum states may be superposed. If and are two kets corresponding to quantum states, the ket

is a different quantum state (possibly not normalized). Note that both the amplitudes and phases (arguments) of and will influence the resulting quantum state. In other words, for example, even though and (for real θ) correspond to the same physical quantum state, they are not interchangeable, since and will not correspond to the same physical state for all choices of . However, and will correspond to the same physical state. This is sometimes described by saying that "global" phase factors are unphysical, but "relative" phase factors are physical and important.

One practical example of superposition is the double-slit experiment, in which superposition leads to quantum interference. The photon state is a superposition of two different states, one corresponding to the photon travel through the left slit, and the other corresponding to travel through the right slit. The relative phase of those two states depends on the difference of the distances from the two slits. Depending on that phase, the interference is constructive at some locations and destructive in others, creating the interference pattern. We may say that superposed states are in coherent superposition, by analogy with coherence in other wave phenomena.

Another example of the importance of relative phase in quantum superposition is Rabi oscillations, where the relative phase of two states varies in time due to the Schrödinger equation. The resulting superposition ends up oscillating back and forth between two different states.

Mixed states

A pure quantum state is a state which can be described by a single ket vector, as described above. A mixed quantum state is a statistical ensemble of pure states (see quantum statistical mechanics). Mixed states inevitably arise from pure states when, for a composite quantum system with an entangled state on it, the part is inaccessible to the observer. The state of the part is expressed then as the partial trace over .

A mixed state cannot be described with a single ket vector. Instead, it is described by its associated density matrix (or density operator), usually denoted ρ. Note that density matrices can describe both mixed and 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 a purification) 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 form

where is the fraction of the ensemble in each pure state The density matrix can be thought of as a way of using the one-particle formalism 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 the trace of ρ2 is equal to 1 if the state is pure, and less than 1 if the state is mixed. [lower-alpha 12] [13] Another, equivalent, criterion is that the von 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 observable A is given by

where are eigenkets and eigenvalues, respectively, for the operator A, and "tr" denotes trace. It is important to note that two types of averaging are occurring, one being a weighted quantum superposition over the basis kets of the pure states, and the other being a statistical (said incoherent) average with the probabilities ps of those states.

According to Eugene Wigner, [14] the concept of mixture was put forward by Lev Landau. [15] [16] :3841

Mathematical generalizations

States can be formulated in terms of observables, rather than as vectors in a vector space. These are positive normalized linear functionals on a C*-algebra, or sometimes other classes of algebras of observables. See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.

See also

Notes

  1. Sometimes written ">"; see angle brackets.
  2. To avoid misunderstandings: Here we mean that Q(t) and P(t) are measured in the same state, but not in the same run of the experiment.
  3. Dirac (1958), [11] p. 4: "If a system is small, we cannot observe it without producing a serious disturbance."
  4. i.e. separated by a zero delay. One can think of it as stopping the time, then making the two measurements one after the other, then resuming the time. Thus, the measurements occurred at the same time, but it is still possible to tell which was first.
  5. For concreteness' sake, suppose that A = Q(t1) and B = P(t2) in the above example, with t2 > t1 > 0.
  6. Dirac (1958), [11] p. 20: "The bra vectors, as they have been here introduced, are quite a different kind of vector from the kets, and so far there is no connexion between them except for the existence of a scalar product of a bra and a ket."
  7. Dirac (1958), [11] p. 19: "A scalar product B|A now appears as a complete bracket expression."
  8. Gottfried (2013), [12] p. 31: "to define the scalar products as being between bras and kets."
  9. Note that a state is a superposition of different basis states , so and are elements of the same Hilbert space. A particle in state is located precisely at position , while a particle in state can be found at different positions with corresponding probabilities.
  10. Landau (1965),<ref name='Landau (1965)' group=''>
  11. In the continuous case, the basis kets are not unit kets (unlike the state ): They are normalized according to [lower-alpha 10] p. 17: "∫ ΨfΨf* dq = δ(f′ − f)" (the left side corresponds to f|f′〉), "∫ δ(f′ − f) df′ = 1".
  12. Note that this criterion works when the density matrix is normalized so that the trace of ρ is 1, as it is for the standard definition given in this section. Occasionally a density matrix will be normalized differently, in which case the criterion is

Related Research Articles

In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "" and "", and a vertical bar "", to construct "bras" and "kets".

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert space which is a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces, and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.

Quantum mechanics Branch of physics describing nature on an atomic scale

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Schrödinger equation Linear partial differential equation whose solution describes the quantum-mechanical system.

The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

In quantum mechanics, a density matrix is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. 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 a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation. More importantly, this is not enough to explain wave function collapse, as decoherence does not reduce it to a single eigenstate.

Wave function Mathematical description of the quantum state of a system

A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. The most common symbols for a wave function are the Greek letters ψ and Ψ.

Quantum decoherence Loss of quantum coherence

Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry. Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.

In physics, an observable is a physical quantity that can be measured. Examples include position and momentum. In systems governed by classical mechanics, it is a real-valued "function" on the set of all possible system states. In quantum physics, it is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value.

In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant with respect to time. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. The Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures.

Probability amplitude Complex number whose squared absolute value is a probability

In quantum mechanics, a probability amplitude is a complex number used in describing the behaviour of systems. The modulus squared of this quantity represents a probability density.

In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.

Two-state quantum system Quantum system that can be measured as one of two values; sought for "quantum bits" in quantum computing

In quantum mechanics, a two-state system is a quantum system that can exist in any quantum superposition of two independent quantum states. The Hilbert space describing such a system is two-dimensional. Therefore, a complete basis spanning the space will consist of two independent states. Any two-state system can also be seen as a qubit.

In quantum mechanics, the momentum operator is the operator associated with the linear momentum. The momentum operator is, in the position representation, an example of a differential operator. For the case of one particle in one spatial dimension, the definition is:

The Born rule is a key postulate of quantum mechanics which gives the probability that a measurement of a quantum system will yield a given result. In its simplest form, it states that the probability density of finding a particle at a given point, when measured, is proportional to the square of the magnitude of the particle's wavefunction at that point. It was formulated by German physicist Max Born in 1926.

Photon polarization is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

In quantum mechanics, the expectation value is the probabilistic expected value of the result (measurement) of an experiment. It can be thought of as an average of all the possible outcomes of a measurement as weighted by their likelihood, and as such it is not the most probable value of a measurement; indeed the expectation value may have zero probability of occurring. It is a fundamental concept in all areas of quantum physics.

This is a glossary for the terminology often encountered in undergraduate quantum mechanics courses.

The Koopman–von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman and John von Neumann in 1931 and 1932, respectively.

References

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  2. Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN   978-0-13-111892-8
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  4. Peres, Asher (1995). Quantum Theory: Concepts and Methods . Kluwer Academic Publishers. ISBN   0-7923-2549-4.
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  6. Kirkpatrick, K. A. (February 2006). "The Schrödinger-HJW Theorem". Foundations of Physics Letters . 19 (1): 95–102. arXiv: quant-ph/0305068 . doi:10.1007/s10702-006-1852-1. ISSN   0894-9875. S2CID   15995449.
  7. Statistical Mixture of States
  8. "The Density Matrix". Archived from the original on January 15, 2012. Retrieved January 24, 2012.
  9. Heisenberg, W. (1927). Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik, Z. Phys.43: 172–198. Translation as 'The actual content of quantum theoretical kinematics and mechanics'. Also translated as 'The physical content of quantum kinematics and mechanics' at pp. 62–84 by editors John Wheeler and Wojciech Zurek, in Quantum Theory and Measurement (1983), Princeton University Press, Princeton NJ.
  10. Bohr, N. (1927/1928). The quantum postulate and the recent development of atomic theory, Nature Supplement April 14 1928, 121: 580–590.
  11. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK.
  12. Gottfried, Kurt; Yan, Tung-Mow (2003). Quantum Mechanics: Fundamentals (2nd, illustrated ed.). Springer. ISBN   9780387955766.
  13. Blum, Density matrix theory and applications, page 39.
  14. Eugene Wigner (1962). "Remarks on the mind-body question" (PDF). In I.J. Good (ed.). The Scientist Speculates. London: Heinemann. pp. 284–302. Footnote 13 on p.180
  15. Lev Landau (1927). "Das Dämpfungsproblem in der Wellenmechanik (The Damping Problem in Wave Mechanics)". Zeitschrift für Physik. 45 (5–6): 430–441. Bibcode:1927ZPhy...45..430L. doi:10.1007/bf01343064. S2CID   125732617. English translation reprinted in: D. Ter Haar, ed. (1965). Collected papers of L.D. Landau. Oxford: Pergamon Press. p.818
  16. Lev Landau; Evgeny Lifshitz (1965). Quantum Mechanics Non-Relativistic Theory (PDF). Course of Theoretical Physics. 3 (2nd ed.). London: Pergamon Press.

Further reading

The concept of quantum states, in particular the content of the section Formalism 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 for Physics 219 at Caltech.

For a discussion of geometric aspects see: