TheInfoList

In
quantum physics Quantum mechanics is a fundamental theory A theory is a reason, rational type of abstraction, abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with ...
, a quantum state is a mathematical entity that provides a
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
for the outcomes of each possible
measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of meas ...
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 In chemistry Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds composed of atoms, ...
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 Ray may refer to: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ...
in a
Hilbert space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
over the
complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, while mixed states are represented by
density matrices 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 in quantum mechanics, measurement performed upon this sy ...
, which are positive semidefinite operators that act on Hilbert spaces. Pure states are also known as state vectors or
wave function 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 number, complex-valued probability amplitude, and the probabilities for the possible results of me ...

s, the latter term applying particularly when they are represented as functions of position or momentum. For example, when dealing with the
energy spectrum In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior thro ...
of the
electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has ma ...

in a
hydrogen atom A hydrogen atom is an atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touch ...

, the relevant state vectors are identified by the
principal quantum numberIn quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 phy ...
, the
angular momentum quantum number The azimuthal quantum number is a quantum number Single electron orbitals for hydrogen-like atoms with quantum numbers (blocks), (rows) and (columns). The spin is not visible, because it has no spatial dependence. In chemistry Chemist ...
, the
magnetic quantum number The magnetic quantum number (symbol ''ml'') is one of four quantum numbers in atomic physics. The set is: principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number. Together, they describe the unique q ...
, and the
spin Spin or spinning may refer to: Businesses * SPIN (cable system) or South Pacific Island Network * Spin (company), an American scooter-sharing system * SPiN, a chain of table tennis lounges Computing * SPIN model checker, Gerard Holzmann's tool fo ...
z-component . For another example, if the spin of an electron is measured in any direction, e.g. with a
Stern–Gerlach experiment The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important ...
, there are two possible results: up or down. The Hilbert space for the electron's spin is therefore two-dimensional, constituting a
qubit In quantum computing, a qubit () or quantum bit (sometimes qbit) is the basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a Two-state quantu ...
. A pure state here is represented by a two-dimensional
complex The UCL Faculty of Mathematical and Physical Sciences is one of the 11 constituent faculties of University College London , mottoeng = Let all come who by merit deserve the most reward , established = , type = Public university, Public rese ...

vector $\left(\alpha, \beta\right)$, with a length of one; that is, with :$, \alpha, ^2 + , \beta, ^2 = 1,$ where $, \alpha,$ and $, \beta,$ are the of $\alpha$ and $\beta$. A mixed state, in this case, has the structure of a $2 \times 2$ matrix that is
Hermitian{{short description, Wikipedia list article Numerous things are named after the French mathematician Charles Hermite Charles Hermite () FRS FRSE Fellowship of the Royal Society of Edinburgh (FRSE) is an award granted to individuals that the R ...
and positive semi-definite, and has
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band) Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leaving ...
1. A more complicated case is given (in
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 ...
) by the
singlet state In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 ph ...
, which exemplifies
quantum entanglement Quantum entanglement is a physical phenomenon that occurs when a pair or group of particles is generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the pair or group cannot be described indepen ...
: : $\left, \psi\right\rang = \frac\big\left(\left, \uparrow\downarrow\right\rang - \left, \downarrow\uparrow\right\rang \big\right),$ which involves superposition of joint spin states for two particles with spin . 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 combinationImage:Convex combination illustration.svg, Given three points x_1, x_2, x_3 in a plane as shown in the figure, the point P ''is'' a convex combination of the three points, while Q is ''not.'' (Q is however an affine combination of the three points, a ...
of pure states. Before a particular
measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of meas ...
is performed on a quantum system, the theory gives only a
probability distribution In probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
for the outcome, and the form that this distribution takes is completely determined by the quantum state and the
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s describing the measurement. Probability distributions for different measurements exhibit tradeoffs exemplified by the
uncertainty principle In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 quant ...

: 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

In the
mathematical formulation of quantum mechanics The mathematical formulations of quantum mechanics are those Formalism (mathematics), mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especiall ...
, pure quantum states correspond to
vectors Vector may refer to: Biology *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism; a disease vector *Vector (molecular biology), a DNA molecule used as a vehicle to artificially carr ...
in a
Hilbert space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, while each observable quantity (such as the energy or momentum of a
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or chemical , chemical properties ...

) is associated with a mathematical operator. The operator serves as a
linear function In mathematics, the term linear function refers to two distinct but related notions: * In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomial ...
which acts on the states of the system. The
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

s 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 corresponding
eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...

(which physicists call an eigenstate) 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 no
quantum uncertainty In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 quantu ...
. If its momentum were measured, the result is guaranteed to be 1 kg⋅m/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: :$, \Psi\left(t\right)\rangle = \sum_n C_n\left(t\right) , \Phi_n\rang.$ The coefficient which corresponds to a particular state in the
linear combination In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 $\rangle$ surrounding the $\Psi$ are part of
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 ...
. Statistical mixtures of states are a different type of linear combination. A statistical mixture of states is a
statistical ensemble In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
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 $\Phi_n$. A number $P_n$ represents the probability of a randomly selected system being in the state $\Phi_n$. Unlike the linear combination case each system is in a definite eigenstate. The expectation value $\langle A \rangle _\sigma$ 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. This is the content of the . Moreover, in contrast to classical mechanics, it is unavoidable that ''performing a measurement on the system generally changes its state''. 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, 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''. 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 () 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
. (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 $, \Psi(t)\rangle = \sum_n C_n(t) , \Phi_n\rang$.) 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 Quantum mechanics is a fundamental Scientific theory, 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 quan ...
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 In theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict List of natural phenomena, natural phenomena. This is in con ...
. Compare with Dirac picture.

Formalism in quantum physics

Pure states as rays in a complex Hilbert space

Quantum physics is most commonly formulated in terms of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (math ...
, as follows. Any given system is identified with some finite- or infinite-dimensional
Hilbert space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
. The pure states correspond to vectors of
norm Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy) Norms are concepts ( sentences) of practical import, oriented to effecting an action, rat ...
1. Thus the set of all pure states corresponds to the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center (geometry), center. More generally, it is the Locus (mathematics), set of points of distance 1 from a fixed central point, where different norm (mathematics), norm ...
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 $H$ 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 $H$ and also to the same point in the
projective Hilbert spaceIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of $H$.

Bra–ket notation

Calculations in quantum mechanics make frequent use of
linear operator In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s, scalar products,
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by consta ...
s and
Hermitian conjugationIn mathematics, specifically in functional analysis, each bounded linear map , linear operator on a complex Hilbert space has a corresponding Hermitian adjoint (or adjoint operator). Adjoints of operators generalize conjugate transposes of Square m ...
. 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 Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. Dirac made fundamental contributions to the early developme ...

invented a notation to describe quantum states, known as ''
bra–ket notation In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 ...
''. Although the details of this are beyond the scope of this article, some consequences of this are: *The expression used to denote a state vector (which corresponds to a pure quantum state) takes the form $, \psi\rangle$ (where the "$\psi$" can be replaced by any other symbols, letters, numbers, or even words). This can be contrasted with the usual ''mathematical'' notation, where vectors are usually lower-case latin letters, and it is clear from the context that they are indeed vectors. *Dirac defined two kinds of vector, ''bra'' and ''ket'', dual to each other. *Each ket $, \psi\rangle$ is uniquely associated with a so-called ''bra'', denoted $\langle\psi,$, which corresponds to the same physical quantum state. Technically, the bra is the adjoint of the ket. It is an element of the
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by consta ...
, and related to the ket by the
Riesz representation theorem:''This article describes a theorem concerning the dual of a Hilbert space The mathematical concept of a Hilbert space, named after David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one o ...
. In a finite-dimensional space with a chosen basis, writing $, \psi\rangle$ as a column vector, $\langle\psi,$ is a row vector; to obtain it just take the
transpose In linear algebra, the transpose of a matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matr ...

and entry-wise
complex conjugate In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
of $, \psi\rangle$. *Scalar products (also called ''brackets'') are written so as to look like a bra and ket next to each other: $\lang \psi_1, \psi_2\rang$. (The phrase "bra-ket" is supposed to resemble "bracket".)

Spin

The
angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a close ...

has the same dimension ( M··) as the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant A physical constant, sometimes fundamental physical constant or universal constant, is a physical quantity that is generally believed to be both universal in nature an ...
and, at quantum scale, behaves as a ''discrete'' 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 with
spinor In geometry and physics, spinors are elements of a complex numbers, complex vector space that can be associated with Euclidean space. Like Euclidean vector, geometric vectors and more general tensors, spinors linear transformation, transform line ...
s. In non-relativistic quantum mechanics the group representations of the
Lie group In mathematics, a Lie group (pronounced "Lee") is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operati ...
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 The Planck constant, or Planck's constant, is the quantum of Electromagnetism, electromagnetic action (physics), action that relates a photon's energy to its frequency. The Planck constant multiplied by a photon's frequency is equal to a photo ...
''ħ'', is either an
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of t ...
(0, 1, 2 ...) or a
half-integer In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(1/2, 3/2, 5/2 ...). For a
massive Massive is an adjective related to mass. Massive may refer to: although the term massive is commonly referred to a large or big object, in this instance it does actually refer to the automotive YouTuber Auto Alex “ kersten “.. Arts, entertain ...
particle with spin ''S'', its
spin quantum number In atomic physics, the spin quantum number is a quantum number that describes the intrinsic angular momentum In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational equivalent of linear momentum. It ...
''m'' always assumes one of the 2''S'' + 1 possible values in the set :$\$ As a consequence, the quantum state of a particle with spin is described by a
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
-valued wave function with values in C2''S''+1. Equivalently, it is represented by a
complex-valued function Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematic ...
of four variables: one discrete
quantum number Single electron orbitals for hydrogen-like atoms with quantum numbers (blocks), (rows) and (columns). The spin is not visible, because it has no spatial dependence. In chemistry Chemistry is the scientific discipline involved with Chemi ...
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 Spin or spinning may refer to: Businesses * SPIN (cable system) or South Pacific Island Network * Spin (company), an American scooter-sharing system * SPiN, a chain of table tennis lounges Computing * SPIN model checker, Gerard Holzmann's tool fo ...
, e.g. :$, \psi \left(\mathbf r_1,\, m_1;\; \dots;\; \mathbf r_N,\, m_N\right)\rangle.$ Here, the spin variables ''mν'' assume values from the set :$\$ where $S_\nu$ is the spin of ''ν''th particle. $S_\nu = 0$ for a particle that does not exhibit spin. The treatment of
identical particles In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 ...
is very different for
boson In quantum mechanics, a boson (, ) is a particle that follows Bose–Einstein statistics. Bosons make up one of two classes of elementary particles, the other being fermions. The name boson was coined by Paul Dirac to commemorate the contributio ...

s (particles with integer spin) versus
fermion In particle physics, a fermion is a particle that follows Fermi–Dirac statistics and generally has half odd integer spin: spin 1/2, Spin (physics)#Higher spins, spin 3/2, etc. These particles obey the Pauli exclusion principle. Fermions include ...
s (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,
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

s (quanta of light) are bosons with ''S'' = 1 (although in the
vacuum A vacuum is a space devoid of matter. The word is derived from the Latin adjective ''vacuus'' for "vacant" or "Void (astronomy), void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressure. Ph ...

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 product In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s of one-particle spaces, to which we will return later.

Basis states of one-particle systems

As with any
Hilbert space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
, if a
basis Basis may refer to: Finance and accounting *Adjusted basisIn tax accounting, adjusted basis is the net cost of an asset after adjusting for various tax-related items. Adjusted Basis or Adjusted Tax Basis refers to the original cost or other ba ...
is chosen for the Hilbert space of a system, then any ket can be expanded as a
linear combination In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of those basis elements. Symbolically, given basis kets $, \rang$, any ket $, \psi\rang$ can be written :$, \psi \rang = \sum_i c_i , \rangle$ where ''ci'' are
complex number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s. In physical terms, this is described by saying that $, \psi\rang$ has been expressed as a ''quantum superposition'' of the states $, \rang$. If the basis kets are chosen to be
orthonormalIn linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quanti ...
(as is often the case), then $c_i=\lang , \psi \rang$. One property worth noting is that the ''normalized'' states $, \psi\rang$ are characterized by :$\lang\psi, \psi\rang = 1,$ and for orthonormal basis this translates to :$\sum_i \left , c_i \right , ^2 = 1.$ Expansions of this sort play an important role in measurement in quantum mechanics. In particular, if the $, \rang$ are
eigenstates In quantum physics 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 qua ...

(with
eigenvalue In linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces an ...

s ''ki'') of an observable, and that observable is measured on the normalized state $, \psi\rang$, 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 $, \mathbf\rang$ with eigenvalues $\mathbf$ of the observable which corresponds to measuring position. If these eigenstates are nondegenerate (for example, if the system is a single, spinless particle), then any ket $, \psi\rang$ is associated with a complex-valued function of three-dimensional space :$\psi\left(\mathbf\right) \equiv \lang \mathbf , \psi \rang.$ This function is called the
wave function 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 number, complex-valued probability amplitude, and the probabilities for the possible results of me ...

corresponding to $, \psi\rang$. Similarly to the discrete case above, the probability ''density'' of the particle being found at position $\mathbf$ is $, \psi\left(\mathbf\right), ^2$ and the normalized states have :$\int \mathrm^3 \mathbf \, , \psi\left(\mathbf\right), ^2 = 1$. In terms of the continuous set of position basis $, \mathbf\rang$, the state $, \psi \rang$ is: :$, \psi \rang = \int \mathrm^3 \mathbf \, \psi \left(\mathbf\right) , \mathbf\rang$.

Superposition of pure states

As mentioned above, quantum states may be superposed. If $, \alpha\rangle$ and $, \beta\rangle$ are two kets corresponding to quantum states, the ket :$c_\alpha, \alpha\rang+c_\beta, \beta\rang$ is a different quantum state (possibly not normalized). Note that both the amplitudes and phases (
arguments In logic Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, label=none, lit=possessed of reason, intellectual, dialectical, argumentative, translit=logikḗ)Also related to (''logos''), "word, thought, idea, argument, ...
) of $c_\alpha$ and $c_\beta$ will influence the resulting quantum state. In other words, for example, even though $, \psi\rang$ and $e^, \psi\rang$ (for real ''θ'') correspond to the same physical quantum state, they are ''not interchangeable'', since $, \phi\rang+, \psi\rang$ and $, \phi\rang+e^, \psi\rang$ will ''not'' correspond to the same physical state for all choices of $, \phi\rang$. However, $, \phi\rang+, \psi\rang$ and $e^\left(, \phi\rang+, \psi\rang\right)$ ''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 modern physics Modern physics is a branch of physics either developed in the early 20th century and onward or branches greatly influenced by early 20th century physics. Notable branches of modern physics include quantum physics, special r ...
, in which superposition leads to quantum interference. The
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. Particles currently thought to be eleme ...

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 Coherence, coherency, or coherent may refer to the following: Physics * Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference * Coherence (units of measurement), a derive ...
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 Schrödinger equation is a linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a mac ...
. 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
of pure states (see
quantum statistical mechanics Quantum statistical mechanics is statistical mechanics In physics, statistical mechanics is a mathematical framework that applies Statistics, statistical methods and probability theory to large assemblies of microscopic entities. It does not as ...
). Mixed states inevitably arise from pure states when, for a composite quantum system $H_1 \otimes H_2$ with an entangled state on it, the part $H_2$ is inaccessible to the observer. The state of the part $H_1$ is expressed then as the
partial trace In linear algebra and functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common constru ...
over $H_2$. 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 $H$ can be always represented as the partial trace of a pure quantum state (called a purification) on a larger bipartite system $H \otimes K$ for a sufficiently large Hilbert space $K$. The density matrix describing a mixed state is defined to be an operator of the form :$\rho = \sum_s p_s , \psi_s \rangle \langle \psi_s ,$ where $p_s$ is the fraction of the ensemble in each pure state $, \psi_s\rangle.$ The density matrix can be thought of as a way of using the one-particle
formalism Formalism may refer to: * Form (disambiguation) * Formal (disambiguation) * Legal formalism, legal positivist view that the substantive justice of a law is a question for the legislature rather than the judiciary * Formalism (linguistics) * Scienti ...
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 Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band) Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leaving ...
of ''ρ''2 is equal to 1 if the state is pure, and less than 1 if the state is mixed. Another, equivalent, criterion is that the
von Neumann entropy In quantum statistical mechanics, the von Neumann entropy, named after John von Neumann John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician A mathematicia ...
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 In quantum mechanics Quantum mechanics is a fundamental Scientific theory, 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 p ...
) of a measurement corresponding to an observable ''A'' is given by :$\langle A \rangle = \sum_s p_s \langle \psi_s , A , \psi_s \rangle = \sum_s \sum_i p_s a_i , \langle \alpha_i , \psi_s \rangle , ^2 = \operatorname\left(\rho A\right)$ where $, \alpha_i\rangle, \; a_i$ 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 $, \psi_s\rangle$ of the pure states, and the other being a statistical (said ''incoherent'') average with the probabilities ''ps'' of those states. According to
Eugene Wigner Eugene Paul "E. P." Wigner ( hu, Wigner Jenő Pál, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist and also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his cont ...
, the concept of mixture was put forward by
Lev Landau Lev may refer to: Common uses * Bulgarian lev, the currency of Bulgaria *an abbreviation for Leviticus, the third book of the Hebrew Bible and the Torah People and fictional characters * Lev (given name) * Lev (surname) LEV * Laborious Extra-Orb ...
. English translation reprinted in: p.8–18

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 In mathematics, specifically in functional analysis Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a commo ...
, or sometimes other classes of algebras of observables. See State on a C*-algebra and Gelfand–Naimark–Segal construction for more details.

*
Atomic electron transition Atomic electron transition is a change of an electron The electron is a subatomic particle In physical sciences, subatomic particles are smaller than atom An atom is the smallest unit of ordinary matter In classical physics and ...
*
Bloch sphere In quantum mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacemen ...

* Greenberger–Horne–Zeilinger state * Ground state * Introduction to quantum mechanics * No-cloning theorem * Orthonormal basis * PBR theorem * Quantum harmonic oscillator * Quantum logic gate * State vector reduction, for historical reasons called a ''wave function collapse'' * Stationary state * W state