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{{Short description|Mathematical structures that allow quantum mechanics to be explained}}
{{Quantum mechanics|cTopic=Formulations}}
The '''mathematical formulations of quantum mechanics''' are those [[Formalism (mathematics)|mathematical formalisms]] that permit a [[Rigour#Mathematics|rigorous]] description of [[quantum mechanics]]. This mathematical formalism uses mainly a part of [[functional analysis]], especially [[Hilbert space]]s, which are 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 space]]s ([[L2 space|''L''<sup>2</sup> space]] mainly), and [[linear operator|operators]] on these spaces. In brief, values of physical [[observable]]s such as [[energy]] and [[momentum]] were no longer considered as values of [[function (mathematics)|function]]s on [[phase space]], but as [[eigenvalue]]s; more precisely as [[Spectrum (functional analysis)|spectral values]] of linear [[Operator (physics)|operator]]s in Hilbert space.{{sfn | Byron | Fuller | 1992 | page=277}}

These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of ''[[quantum state]]'' and ''quantum observables'', which are radically different from those used in previous [[Mathematical model|models]] of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by [[Heisenberg uncertainty relations|Heisenberg]] through a [[thought experiment]], and is represented mathematically in the new formalism by the [[non-commutative|non-commutativity]] of operators representing quantum observables.

Prior to the development of quantum mechanics as a separate [[theory]], the mathematics used in physics consisted mainly of formal [[mathematical analysis]], beginning with [[calculus]], and increasing in complexity up to [[differential geometry]] and [[partial differential equation]]s. [[Probability theory]] was used in [[statistical mechanics]]. Geometric intuition played a strong role in the first two and, accordingly, [[Relativity physics|theories of relativity]] were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called [[classical physics]], and in particular within the same mathematical structures. The most sophisticated example of this is the [[Sommerfeld–Wilson–Ishiwara quantization]] rule, which was formulated entirely on the classical [[phase space]].

== History of the formalism ==

=== The "old quantum theory" and the need for new mathematics ===

{{Main|Old quantum theory}}
In the 1890s, [[Max Planck|Planck]] was able to derive the [[blackbody spectrum]], which was later used to avoid the classical [[ultraviolet catastrophe]] by making the unorthodox assumption that, in the interaction of [[electromagnetic radiation]] with [[matter]], energy could only be exchanged in discrete units which he called [[quantum|quanta]]. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, {{math|''h''}}, is now called the [[Planck constant]] in his honor.

In 1905, [[Albert Einstein|Einstein]] explained certain features of the [[photoelectric effect]] by assuming that Planck's energy quanta were actual particles, which were later dubbed [[photons]].
[[File:Bohr atom model (mul).svg|right|300px | light at the right frequency]]

All of these developments were [[Phenomenology (particle physics)|phenomenological]] and challenged the theoretical physics of the time. [[Old quantum theory|Bohr and Sommerfeld]] went on to modify [[classical mechanics]] in an attempt to deduce the [[Bohr model]] from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of the Planck constant were actually allowed. The most sophisticated version of this formalism was the so-called [[Sommerfeld–Wilson–Ishiwara quantization]]. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable [[3-body problem]]) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

In 1923, [[Louis de Broglie|de Broglie]] proposed that [[wave–particle duality]] applied not only to photons but to electrons and every other physical system.

The situation changed rapidly in the years 1925–1930, when working mathematical foundations were found through the groundbreaking work of [[Erwin Schrödinger]], [[Werner Heisenberg]], [[Max Born]], [[Pascual Jordan]], and the foundational work of [[John von Neumann]], [[Hermann Weyl]] and [[Paul Dirac]], and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after [[Werner Heisenberg]] discovered the uncertainty relations and [[Niels Bohr]] introduced the idea of [[Complementarity (physics)|complementarity]].

=== The "new quantum theory" ===

Werner Heisenberg's [[matrix mechanics]] was the first successful attempt at replicating the observed quantization of [[atomic spectra]]. Later in the same year, Schrödinger created his [[Schrödinger equation|wave mechanics]]. Schrödinger's formalism was considered easier to understand, visualize and calculate as it led to [[differential equations]], which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.

Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the [[Absolute value#Complex numbers|absolute square]] of the wave function of an [[electron]] should be interpreted as the [[charge density]] of an object smeared out over an extended, possibly infinite, volume of space. It was [[Max Born]] who introduced the interpretation of the [[Absolute value#Complex numbers|absolute square]] of the wave function as the probability distribution of the position of a [[Point particle|''pointlike'' object]]. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the [[Copenhagen interpretation]] of quantum mechanics. Schrödinger's [[wave function]] can be seen to be closely related to the classical [[Hamilton–Jacobi equation]]. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, [[Paul Dirac]]{{sfn|Dirac|1925}} discovered that the equation for the operators in the [[Heisenberg picture|Heisenberg representation]], as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through [[Poisson bracket]]s, a procedure now known as [[canonical quantization]].

Already before Schrödinger, the young postdoctoral fellow Werner Heisenberg invented his [[matrix mechanics]], which was the first correct quantum mechanics – the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, [[linear algebra]] was not generally popular with physicists in its present form.

Although Schrödinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic ''[[The Principles of Quantum Mechanics]]''. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the [[bra–ket notation]], together with an abstract formulation in terms of the [[Hilbert space]] used in functional analysis; he showed that Schrödinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in many types of generalizations of the field.

The first complete mathematical formulation of this approach, known as the [[Dirac–von Neumann axioms]], is generally credited to [[John von Neumann]]'s 1932 book ''[[Mathematical Foundations of Quantum Mechanics]]'', although [[Hermann Weyl]] had already referred to Hilbert spaces (which he called ''unitary spaces'') in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical [[spectral theory]] based on linear operators rather than the [[quadratic form]]s that were [[David Hilbert]]'s approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about [[Interpretation of quantum mechanics|''interpretation'' of the theory]], and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.

=== Later developments ===

The application of the new quantum theory to electromagnetism resulted in [[quantum field theory]], which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases. 
* [[Path integral formulation]]
* [[Phase-space formulation]] of quantum mechanics & [[geometric quantization]]
* [[quantum field theory in curved spacetime]]
* [[Wightman axioms|axiomatic]], [[local quantum physics|algebraic]] and [[constructive quantum field theory]]
* [[C*-algebra]] [[Formalism (mathematics)|formalism]]
* [[POVM|Generalized statistical model of quantum mechanics]]

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called [[classical limit of quantum mechanics]]. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, [[Quantization (physics)|quantization]], namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.

Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called [[hidden-variable theory|hidden-variable theories]]. The issue of hidden variables has become in part an experimental issue with the help of [[quantum optics]].

== Postulates of quantum mechanics ==
A physical system is generally described by three basic ingredients: [[Quantum state|states]]; [[observable]]s; and [[Dynamics (mechanics)|dynamics]] (or law of [[time evolution]]) or, more generally, a [[Gauge invariance|group of physical symmetries]]. A classical description can be given in a fairly direct way by a phase space [[model (abstract)|model]] of mechanics: states are points in a phase space formulated by [[symplectic manifold]], observables are real-valued functions on it, time evolution is given by a one-parameter [[group (mathematics)|group]] of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a [[Hilbert space]] of states, observables are [[self-adjoint operator]]s on the space of states, time evolution is given by a [[Stone's theorem on one-parameter unitary groups|one-parameter group]] of unitary transformations on the Hilbert space of states, and physical symmetries are realized by [[unitary transformation]]s. (It is possible, to map this Hilbert-space picture to a [[phase space formulation]], invertibly. See below.)

The following summary of the mathematical framework of quantum mechanics can be partly traced back to the [[Dirac–von Neumann axioms]].{{sfn | Cohen-Tannoudji | Diu | Laloë | 2020 | p=}}

=== Description of the state of a system ===
Each isolated physical system is associated with a (topologically) [[Separable space|separable]] [[complex number|complex]] [[Hilbert space]] {{math|''H''}} with [[inner product]] {{math|⟨''φ''{{!}}''ψ''⟩}}.

{{Quote box
| title = Postulate I
| quote = The state of an isolated physical system is represented, at a fixed time {{nowrap|<math>t</math>,}} by a state vector <math>|\psi \rangle</math> belonging to a Hilbert space <math>\mathcal{H}</math> called the ''state space''.
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[[Hilbert space#Separable spaces|Separability]] is a mathematically convenient hypothesis, with the physical interpretation that the state is uniquely determined by countably many observations. Quantum states can be identified with [[equivalence class]]es in {{math|''H''}}, where two vectors (of length 1) represent the same state if they differ only by a [[phase factor]]:{{sfn|Bäuerle|de Kerf|1990|page=330}}{{sfn | Solem | Biedenharn | 1993 }}
<math display="block">|\psi_k \rangle \sim |\psi_l\rangle \;\; \Leftrightarrow \;\; |\psi_k \rangle = e^{i\alpha} |\psi_l\rangle, \quad\ \alpha\in\mathbb{R}.</math>
As such, a quantum state forms a [[Wigner's theorem#Rays and ray space|'''ray''']] in [[projective Hilbert space]], not a ''vector''.{{sfn|Bäuerle|de Kerf|1990|page=341}}

Accompanying Postulate I is the composite system postulate:{{sfn | Jauch | Wigner | Yanase | 1997 }}

{{Quote box
| title = Composite system postulate
| quote = The Hilbert space of a composite system is the Hilbert space [[tensor product]] of the state spaces associated with the component systems. For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.
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In the presence of [[quantum entanglement]], the quantum state of the composite system cannot be factored as a tensor product of states of its local constituents; Instead, it is expressed as a sum, or [[Quantum superposition|superposition]], of tensor products of states of component subsystems. A subsystem in an entangled composite system generally cannot be described by a state vector (or a ray), but instead is described by a [[density matrix|density operator]]; Such quantum state is known as a [[Mixed state (physics)|mixed state]]. The [[density operator]] of a mixed state is a [[trace class]], nonnegative ([[Positive semi-definite matrix|positive semi-definite]]) [[Self-adjoint operator|self-adjoint]] operator {{math|''ρ''}} normalized to be of [[Trace (linear algebra)|trace]] 1. In turn, any [[density operator]] of a mixed state can be represented as a subsystem of a larger composite system in a pure state (see [[Schrödinger–HJW theorem|purification theorem]]).

In the absence of quantum entanglement, the quantum state of the composite system is called a [[separable state]]. The density matrix of a bipartite system in a separable state can be expressed as <math>
\rho=\sum_k p_k \rho_1^k \otimes \rho_2^k
</math>, where <math>\;
\sum_k p_k = 1
</math>. If there is only a single non-zero <math>p_k</math>, then the state can be expressed just as <math display="inline"> \rho = \rho_1 \otimes \rho_2 , </math> and is called simply separable or product state.

=== Measurement on a system <span class="anchor" id="Measurement"></span> ===

==== Description of physical quantities ====
Physical observables are represented by [[Hermitian matrix|Hermitian]] matrices on {{math|''H''}}. Since these operators are Hermitian, their [[eigenvalues]] are always real, and represent the possible outcomes/results from measuring the corresponding observable. If the spectrum of the observable is [[Discrete spectrum|discrete]], then the possible results are ''quantized''.

{{Quote box
| title = Postulate II.a
| quote = Every measurable physical quantity <math>\mathcal{A}</math> is described by a Hermitian operator <math>A</math> acting in the state space <math>\mathcal{H}</math>. This operator is an observable, meaning that its [[eigenvectors]] form a basis for <math>\mathcal{H}</math>. The result of measuring a physical quantity <math>\mathcal{A}</math> must be one of the eigenvalues of the corresponding observable <math>A</math>.
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==== Results of measurement ====
By spectral theory, we can associate a [[probability measure]] to the values of {{math|''A''}} in any state {{math|''ψ''}}. We can also show that the possible values of the observable {{math|''A''}} in any state must belong to the [[spectrum of an operator|spectrum]] of {{math|''A''}}. The [[expected value|expectation value]] (in the sense of probability theory) of the observable {{math|''A''}} for the system in state represented by the unit vector {{math|''ψ''}} ∈ ''H'' is <math>\langle\psi|A|\psi\rangle</math>. If we represent the state {{math|''ψ''}} in the basis formed by the eigenvectors of {{math|''A''}}, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.

{{Quote box
| title = Postulate II.b
| quote = When the physical quantity <math>\mathcal{A}</math> is measured on a system in a normalized state <math>|\psi\rangle</math>, the probability of obtaining an eigenvalue (denoted <math>a_n</math> for discrete spectra and <math>\alpha</math> for continuous spectra) of the corresponding observable <math>A</math> is given by the ''amplitude squared'' of the appropriate wave function (projection onto corresponding eigenvector).

<math>\begin{align}\mathbb{P}(a_n)&= |\langle a_n|\psi \rangle|^2 & \text{(Discrete, nondegenerate spectrum)} \\ \mathbb{P}(a_n) &= \sum_i^{g_n} |\langle a_n^i|\psi \rangle|^2 & \text{(Discrete, degenerate spectrum)} \\ d\mathbb{P}(\alpha) &= |\langle \alpha | \psi \rangle |^2 d\alpha & \text{(Continuous, nondegenerate spectrum)} \end{align}</math>
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For a mixed state {{math|''ρ''}}, the expected value of {{math|''A''}} in the state {{math|''ρ''}} is <math> \operatorname{tr}(A\rho)</math>, and the probability of obtaining an eigenvalue <math> a_n </math> in a discrete, nondegenerate spectrum of the corresponding observable <math> A </math> is given by <math> \mathbb P(a_n)=\operatorname{tr}(|a_n\rangle\langle a_n|\rho)=\langle a_n|\rho|a_n\rangle </math>.

If the eigenvalue <math> a_n </math> has [[Degenerate_energy_levels|degenerate]], orthonormal eigenvectors <math> \{|a_{n1}\rangle,|a_{n2}\rangle, \dots , |a_{nm}\rangle\} </math>, then the [[Projection (linear algebra)|projection operator]] onto the eigensubspace can be defined as the identity operator in the eigensubspace:
<math display="block"> P_n=|a_{n1}\rangle\langle a_{n1}|+|a_{n2}\rangle\langle a_{n2}| + \dots + |a_{nm}\rangle\langle a_{nm}|, </math>
and then <math> \mathbb P(a_n)=\operatorname{tr}(P_n\rho) </math>.

Postulates II.a and II.b are collectively known as the [[Born rule]] of quantum mechanics.

==== Effect of measurement on the state ====

When a measurement is performed, only one result is obtained (according to some [[interpretations of quantum mechanics]]). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore, the state vector must change as a result of measurement, and ''collapse'' onto the eigensubspace associated with the eigenvalue measured. {{Quote box
| title = Postulate II.c
| quote = If the measurement of the physical quantity <math>\mathcal{A}</math> on the system in the state <math>|\psi\rangle </math> gives the result <math>a_n </math>, then the state of the system immediately after the measurement is the normalized projection of <math> |\psi\rangle</math> onto the eigensubspace associated with <math>a_n </math><br/><br/> <math>|\psi\rangle \quad \overset{a_n}{\Longrightarrow}\quad \frac{P_n|\psi\rangle}{\sqrt{\langle \psi | P_n | \psi \rangle}}</math>
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For a mixed state {{math|''ρ''}}, after obtaining an eigenvalue <math> a_n </math> in a discrete, nondegenerate spectrum of the corresponding observable <math> A </math>, the updated state is given by <math display="inline"> \rho'=\frac{P_n\rho P_n^\dagger}{\operatorname{tr}(P_n\rho P_n^\dagger)} </math>. If the eigenvalue <math> a_n </math> has degenerate, orthonormal eigenvectors <math> \{|a_{n1}\rangle,|a_{n2}\rangle, \dots ,|a_{nm}\rangle\} </math>, then the [[Projection (linear algebra)|projection operator]] onto the eigensubspace is <math> P_n=|a_{n1}\rangle\langle a_{n1}|+|a_{n2}\rangle\langle a_{n2}| + \dots + |a_{nm}\rangle\langle a_{nm}| </math>.

Postulates II.c is sometimes called the "state update rule" or "collapse rule"; Together with the Born rule (Postulates II.a and II.b), they form a complete representation of [[Measurement in quantum mechanics|measurements]], and are sometimes collectively called the measurement postulate(s).

Note that the [[projection-valued measure]]s (PVM) described in the measurement postulate(s) can be generalized to [[POVM|positive operator-valued measures]] (POVM), which is the most general kind of measurement in quantum mechanics. A POVM can be understood as the effect on a component subsystem when a PVM is performed on a larger, composite system (see [[Naimark's dilation theorem]]).

=== Time evolution of a system ===
Though it is possible to derive the Schrödinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the [[De Broglie Hypothesis|de Broglie hypothesis]] or [[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics|path integrals]].

{{Quote box
| title = Postulate III
| quote = The time evolution of the state vector <math>|\psi(t)\rangle</math> is governed by the Schrödinger equation, where <math>H(t)</math> is the observable associated with the total energy of the system (called the [[Hamiltonian (quantum mechanics)|Hamiltonian]])<br/><br/><math>i\hbar \frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle </math>
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Equivalently, the time evolution postulate can be stated as:

{{Quote box
| title = Postulate III
| quote = The time evolution of a [[Closed system#In quantum physics|closed system]] is described by a [[Unitary transformation (quantum mechanics)|unitary transformation]] on the initial state.<br/><br/><math>|\psi(t)\rangle = U(t;t_0)|\psi(t_0)\rangle </math>
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For a closed system in a mixed state {{math|''ρ''}}, the time evolution is <math>\rho(t)=U(t;t_0)\rho(t_0) U^\dagger(t;t_0)</math>.

The evolution of an [[open quantum system]] can be described by [[quantum operation]]s (in an [[Quantum operation#Statement of the theorem|operator sum]] formalism) and [[quantum instrument]]s, and generally does not have to be unitary.

=== Other implications of the postulates ===
* Physical symmetries act on the Hilbert space of quantum states [[unitary operator|unitarily]] or [[antiunitary|antiunitarily]] due to [[Wigner's theorem]] ([[supersymmetry]] is another matter entirely).
* Density operators are those that are in the closure of the [[convex hull]] of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are [[extreme point]]s of the set of density operators. Physicists also call one-dimensional orthogonal projectors ''pure states'' and other density operators ''mixed states''.
* One can in this formalism state Heisenberg's [[uncertainty principle]] and [[mathematical proof|prove]] it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.
* Recent research has shown{{sfn | Carcassi | Maccone | Aidala | 2021}} that the composite system postulate (tensor product postulate) can be derived from the state postulate (Postulate I) and the measurement postulates (Postulates II); Moreover, it has also been shown{{sfn | Masanes | Galley | Müller | 2019}} that the measurement postulates (Postulates II) can be derived from "unitary quantum mechanics", which includes only the state postulate (Postulate I), the composite system postulate (tensor product postulate) and the unitary evolution postulate (Postulate III).

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of [[Spin (physics)|spin]] and Pauli's [[Pauli exclusion principle|exclusion principle]], see below.

=== Spin ===

In addition to their other properties, all particles possess a quantity called [[Spin (physics)|spin]], an ''intrinsic angular momentum''. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position {{math|'''r'''}} and time {{math|''t''}} as continuous variables, {{math|1=''ψ'' = ''ψ''('''r''', ''t'')}}. For spin wavefunctions the spin is an additional discrete variable: {{math|1=''ψ'' = ''ψ''('''r''', ''t'', ''σ'')}}, where {{math|''σ''}} takes the values;
<math display="block">\sigma = -S \hbar , -(S-1) \hbar , \dots, 0, \dots ,+(S-1) \hbar ,+S \hbar \,.</math>

That is, the state of a single particle with spin {{math|''S''}} is represented by a {{math|(2''S'' + 1)}}-component [[spinor]] of complex-valued wave functions.

Two classes of particles with ''very different'' behaviour are [[boson]]s which have integer spin ({{math|1=''S'' = 0, 1, 2, ...}}), and [[fermion]]s possessing half-integer spin ({{math|1=''S'' = {{frac|1|2}}, {{frac|3|2}}, {{frac|5|2}}, ...}}).

=== Symmetrization postulate ===
{{Main|Identical particles}}
In quantum mechanics, two particles can be distinguished from one another using two methods. By performing a measurement of intrinsic properties of each particle, particles of different types can be distinguished. Otherwise, if the particles are identical, their trajectories can be tracked which distinguishes the particles based on the locality of each particle. While the second method is permitted in classical mechanics, (i.e. all classical particles are treated with distinguishability), the same cannot be said for quantum mechanical particles since the process is infeasible due to the fundamental uncertainty principles that govern small scales. Hence the requirement of indistinguishability of quantum particles is presented by the symmetrization postulate. The postulate is applicable to a system of bosons or fermions, for example, in predicting the spectra of [[helium atom]]. The postulate, explained in the following sections, can be stated as follows:

{{Quote box
| title = Symmetrization postulate{{sfn|Sakurai|Napolitano|2021|p=443}}
| quote = The wavefunction of a system of ''N'' identical particles (in 3D) is either totally symmetric (Bosons) or totally antisymmetric (Fermions) under interchange of any pair of particles.
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Exceptions can occur when the particles are constrained to two spatial dimensions where existence of particles known as [[anyon]]s are possible which are said to have a continuum of statistical properties spanning the range between fermions and bosons.{{sfn|Sakurai|Napolitano|2021|p=443}} The connection between behaviour of identical particles and their spin is given by [[Spin–statistics theorem|spin statistics theorem]].

It can be shown that two particles localized in different regions of space can still be represented using a symmetrized/antisymmetrized wavefunction and that independent treatment of these wavefunctions gives the same result.{{sfn|Sakurai|Napolitano|2021|p=434-437}} Hence the symmetrization postulate is applicable in the general case of a system of identical particles.

==== Exchange Degeneracy ====
In a system of identical particles, let ''P'' be known as exchange operator that acts on the wavefunction as:
: <math>P \bigg(\cdots|\psi\rang |\phi\rang \cdots\bigg) \equiv \cdots |\phi\rang |\psi\rang \cdots </math>

If a physical system of identical particles is given, wavefunction of all particles can be well known from observation but these cannot be labelled to each particle. Thus, the above exchanged wavefunction represents the same physical state as the original state which implies that the wavefunction is not unique. This is known as exchange degeneracy.{{sfn | Cohen-Tannoudji | Diu | Laloë | 2020 | p=1375–1377}}

More generally, consider a linear combination of such states, <math>|\Psi\rangle </math>. For the best representation of the physical system, we expect this to be an eigenvector of ''P'' since exchange operator is not excepted to give completely different vectors in projective Hilbert space. Since <math>P^2 = 1</math>, the possible eigenvalues of ''P'' are +1 and −1. The <math>|\Psi\rangle </math> states for identical particle system are represented as symmetric for +1 eigenvalue or antisymmetric for -1 eigenvalue as follows:
: <math>P|\cdots n_i,n_j \cdots; S\rang = + |\cdots n_i,n_j \cdots; S\rang  </math>
: <math>P|\cdots n_i, n_j \cdots; A\rang = - |\cdots n_i, n_j \cdots; A\rang</math>

The explicit symmetric/antisymmetric form of <math>|\Psi\rangle </math> is [[Identical particles#Wavefunction representation|constructed]] using a symmetrizer or [[antisymmetrizer]] operator. Particles that form symmetric states are called [[boson]]s and those that form antisymmetric states are called as fermions. The relation of spin with this classification is given from [[Spin–statistics theorem|spin statistics theorem]] which shows that integer spin particles are bosons and half integer spin particles are fermions.

==== Pauli exclusion principle ====
The property of spin relates to another basic property concerning systems of {{math|''N''}} identical particles: the [[Pauli exclusion principle]], which is a consequence of the following permutation behaviour of an {{math|''N''}}-particle wave function; again in the position representation one must postulate that for the transposition of any two of the {{math|''N''}} particles one always should have

{{Equation box 1
|indent=:
|title='''Pauli principle'''
|equation=<math>\psi (\dots, \,\mathbf r_i,\sigma_i, \, \dots, \,\mathbf r_j,\sigma_j, \,\dots) = (-1)^{2S}\cdot \psi ( \dots, \,\mathbf r_j,\sigma_j, \, \dots, \mathbf r_i,\sigma_i,\, \dots)</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4
}}

i.e., on [[Transposition (mathematics)|transposition]] of the arguments of any two particles the wavefunction should ''reproduce'', apart from a prefactor {{math|(−1)<sup>2''S''</sup>}} which is {{math|+1}} for bosons, but ({{math|−1}}) for [[fermions]].
Electrons are fermions with {{math|1=''S'' = 1/2}}; quanta of light are bosons with {{math|1=''S'' = 1}}.

Due to the form of anti-symmetrized wavefunction:
: <math>\Psi^{(A)}_{n_1 \cdots n_N} (x_1, \ldots, x_N) =
  \frac{1}{\sqrt{N!}} \left|
  \begin{matrix}
    \psi_{n_1}(x_1) & \psi_{n_1}(x_2) & \cdots & \psi_{n_1}(x_N) \\
    \psi_{n_2}(x_1) & \psi_{n_2}(x_2) & \cdots & \psi_{n_2}(x_N) \\
             \vdots &          \vdots & \ddots &          \vdots \\
    \psi_{n_N}(x_1) & \psi_{n_N}(x_2) & \cdots & \psi_{n_N}(x_N) \\
  \end{matrix}
  \right|</math>
if the wavefunction of each particle is completely determined by a set of quantum number, then two fermions cannot share the same set of quantum numbers since the resulting function cannot be anti-symmetrized (i.e. above formula gives zero). The same cannot be said of Bosons since their wavefunction is:
: <math>|x_1 x_2 \cdots x_N; S \rangle = \frac{\prod_j n_j!}{N!} \sum_p \left|x_{p(1)}\right\rangle \left|x_{p(2)}\right\rangle \cdots \left|x_{p(N)}\right\rangle       </math>
where <math>n_j </math> is the number of particles with same wavefunction.

==== Exceptions for symmetrization postulate ====
In nonrelativistic quantum mechanics all particles are either bosons or [[fermions]]; in relativistic quantum theories also [[Supersymmetry|"supersymmetric"]] theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension {{math|1=''d'' = 2}} can one construct entities where {{math|(−1)<sup>2''S''</sup>}} is replaced by an arbitrary complex number with magnitude 1, called [[anyon]]s. In relativistic quantum mechanics, spin statistic theorem can prove that under certain set of assumptions that the integer spins particles are classified as bosons and half spin particles are classified as [[fermion]]s. Anyons which form neither symmetric nor antisymmetric states are said to have fractional spin.

Although ''spin'' and the ''Pauli principle'' can only be derived from relativistic generalizations of quantum mechanics, the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the [[periodic system]] of chemistry, are consequences of the two properties.

== Mathematical structure of quantum mechanics ==

=== Pictures of dynamics ===
{{Main article|Dynamical pictures}}
{{unordered list
| 1 = In the so-called [[Schrödinger picture]] of quantum mechanics, the dynamics is given as follows:

The [[time evolution]] of the state is given by a differentiable function from the real numbers {{math|'''R'''}}, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows:
If {{math|{{braket|ket|''ψ''(''t'')}}}} denotes the state of the system at any one time {{math|''t''}}, the following Schrödinger equation holds:

{{Equation box 1
|indent=:
|title='''Schrödinger equation ''' ''(general)''
|equation=<math> i\hbar\frac{d}{d t}\left|\psi(t)\right\rangle=H\left|\psi(t)\right\rangle</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
where {{math|''H''}} is a densely defined self-adjoint operator, called the system [[Hamiltonian (quantum mechanics)|Hamiltonian]], {{math|''i''}} is the [[complex number|imaginary unit]] and {{math|''ħ''}} is the [[reduced Planck constant]]. As an observable, {{math|''H''}} corresponds to the total [[energy]] of the system.

Alternatively, by [[Stone's theorem on one-parameter unitary groups|Stone's theorem]] one can state that there is a strongly continuous one-parameter unitary map {{math|''U''(''t'')}}: {{math|''H'' → ''H''}} such that
<math display="block">\left|\psi(t+s)\right\rangle=U(t)\left|\psi(s)\right\rangle</math>
for all times {{math|''s'', ''t''}}. The existence of a self-adjoint Hamiltonian {{math|''H''}} such that
<math display="block">U(t)=e^{-(i/\hbar)t H}</math>
is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that {{math|''H''}} does not depend on time and that the perturbation starts at {{math|1=''t''<sub>0</sub> = 0}}; otherwise one must use the [[Dyson series]], formally written as
<math display="block">U(t)=\mathcal{T}\left[\exp\left(-\frac{i}{\hbar} \int_{t_0}^t dt'\, H(t')\right)\right],</math>
where <math>{\mathcal{T}}</math> is Dyson's [[time-ordering]] symbol.

(This symbol permutes a product of noncommuting operators of the form
<math display="block"> B_1(t_1)\cdot  B_2(t_2)\cdot\dots \cdot B_n(t_n)</math>
into the uniquely determined re-ordered expression
<math display="block">B_{i_1}(t_{i_1})\cdot  B_{i_2}(t_{i_2})\cdot\dots \cdot B_{i_n}(t_{i_n})</math> with <math>t_{i_1}\ge t_{i_2}\ge\dots\ge t_{i_n}\,.</math>

The result is a causal chain, the primary ''cause'' in the past on the utmost r.h.s., and finally the present ''effect'' on the utmost l.h.s.&nbsp;.)

| 2 = The [[Heisenberg picture]] of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus:
<math display="block">\left|\psi\right\rangle = \left|\psi(0)\right\rangle</math>
<math display="block">A(t) = U(-t)AU(t). </math>
It is then easily checked that the expected values of all observables are the same in both pictures
<math display="block">\langle\psi\mid A(t)\mid\psi\rangle=\langle\psi(t)\mid A\mid\psi(t)\rangle</math>
and that the time-dependent Heisenberg operators satisfy
{{Equation box 1
|indent =:
|title='''Heisenberg picture''' ''(general)''
|equation =
<math>\frac{d}{dt}A(t)=\frac{i}{\hbar}[H,A(t)]+\frac{\partial A(t)}{\partial t},</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}
which is true for time-dependent {{math|1=''A'' = ''A''(''t'')}}. Notice the commutator expression is purely formal when one of the operators is [[Unbounded operator|unbounded]]. One would specify a representation for the expression to make sense of it.

| 3 = The so-called [[Dirac picture]] or [[interaction picture]] has time-dependent ''states'' and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the observables can be solved exactly, confining any complications to the evolution of the states. For this reason, the Hamiltonian for the observables is called "free Hamiltonian" and the Hamiltonian for the states is called "interaction Hamiltonian". In symbols:

{{Equation box 1
|indent =:
|title='''Dirac picture'''
|equation =
<math> i\hbar\frac{d}{dt}\left|\psi(t)\right\rangle = {H}_{\rm int}(t) \left|\psi(t)\right\rangle</math>

<math>i\hbar \frac{d}{dt}A(t) = [A(t),H_{0}].</math>
|cellpadding= 6
|border
|border colour = #0073CF
|background colour=#F9FFF7}}

The interaction picture does not always exist, though. In interacting quantum field theories, [[Haag's theorem]] states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a [[superselection sector]]. Moreover, even if in the Schrödinger picture the Hamiltonian does not depend on time, e.g. {{math|1=''H'' = ''H''<sub>0</sub> + ''V''}}, in the interaction picture it does, at least, if {{math|''V''}} does not commute with {{math|''H''<sub>0</sub>}}, since
<math display="block">H_{\rm int}(t)\equiv e^{{(i/\hbar})tH_0}\,V\,e^{{(-i/\hbar})tH_0}.</math>

So the above-mentioned Dyson-series has to be used anyhow.

The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical [[Poisson bracket]]s); but this is already rather "high-browed", and the Schrödinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in [[perturbation theory]], and is specially associated to [[quantum field theory]] and [[many-body theory|many-body physics]].

Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated with the symmetry (for instance, angular or linear momentum).
}}
'''Summary''':
{{Pictures in quantum mechanics}}

=== Representations ===

The original form of the Schrödinger equation depends on choosing a particular representation of Heisenberg's [[canonical commutation relations]]. The [[Stone–von Neumann theorem]] dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the [[phase space formulation]] of quantum mechanics, which works in full [[phase space]] instead of [[Hilbert space]], so then with a more intuitive link to the [[classical limit]] thereof. This picture also simplifies considerations
of [[Quantization (physics)|quantization]], the deformation extension from classical to quantum mechanics.

The [[quantum harmonic oscillator]] is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrödinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the [[Oscillator representation|Segal–Bargmann (Fock-space or coherent state) representation]] (named after [[Irving Segal]] and [[Valentine Bargmann]]). All four are unitarily equivalent.

=== Time as an operator ===

The framework presented so far singles out time as ''the'' parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter {{math|''s''}}, and in that case the time ''t'' becomes an additional generalized coordinate of the physical system. At the quantum level, translations in {{math|''s''}} would be generated by a "Hamiltonian" {{math|''H'' − ''E''}}, where ''E'' is the energy operator and {{math|''H''}} is the "ordinary" Hamiltonian. However, since ''s'' is an unphysical parameter, ''physical'' states must be left invariant by "''s''-evolution", and so the physical state space is the kernel of {{math|''H'' − ''E''}} (this requires the use of a [[rigged Hilbert space]] and a renormalization of the norm).

This is related to the [[Dirac bracket|quantization of constrained systems]] and [[quantization of gauge theories]]. It
is also possible to formulate a quantum theory of "events" where time becomes an observable.{{sfn | Edwards | 1979}}

== Problem of measurement ==
{{Main|Measurement in quantum mechanics}}
The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of [[measurement]].{{sfn | Greenstein | Zajonc | 2006 | p=215}} The von Neumann description of quantum measurement of an observable {{math|''A''}}, when the system is prepared in a pure state {{math|''ψ''}} is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time – more specifically the [[Compton scattering|Compton–Simon experiment]]; it is not applicable to most present-day measurements within the quantum domain):
* Let {{math|''A''}} have spectral resolution <math display="block"> A = \int \lambda \, d \operatorname{E}_A(\lambda),</math> where {{math|E<sub>''A''</sub>}} is the [[resolution of the identity]] (also called [[projection-valued measure]]) associated with {{math|''A''}}. Then the probability of the measurement outcome lying in an interval {{math|''B''}} of {{math|'''R'''}} is {{math|{{!}}E<sub>''A''</sub>(''B'') ''ψ''{{!}}<sup>2</sup>}}. In other words, the probability is obtained by integrating the characteristic function of {{math|''B''}} against the countably additive measure <math display="block"> \langle \psi \mid \operatorname{E}_A \psi \rangle. </math>
* If the measured value is contained in {{math|''B''}}, then immediately after the measurement, the system will be in the (generally non-normalized) state {{math|E<sub>''A''</sub>(''B'')''ψ''}}. If the measured value does not lie in {{math|''B''}}, replace {{math|''B''}} by its complement for the above state.

For example, suppose the state space is the {{math|''n''}}-dimensional complex Hilbert space {{math|'''C'''<sup>''n''</sup>}} and {{math|''A''}} is a Hermitian matrix with eigenvalues {{math|''λ''<sub>''i''</sub>}}, with corresponding eigenvectors {{math|''ψ<sub>i</sub>''}}. The projection-valued measure associated with {{math|''A''}}, {{math|E<sub>''A''</sub>}}, is then
<math display="block"> \operatorname{E}_A (B) = | \psi_i\rangle \langle \psi_i|, </math>
where {{math|''B''}} is a Borel set containing only the single eigenvalue {{math|''λ<sub>i</sub>''}}. If the system is prepared in state
<math display="block">| \psi \rangle </math>
Then the probability of a measurement returning the value {{math|''λ<sub>i</sub>''}} can be calculated by integrating the spectral measure
<math display="block"> \langle \psi \mid \operatorname{E}_A \psi \rangle </math>
over {{math|''B<sub>i</sub>''}}. This gives trivially
<math display="block"> \langle \psi| \psi_i\rangle \langle \psi_i \mid \psi \rangle = | \langle \psi \mid \psi_i\rangle | ^2. </math>

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the ''projection postulate''.

A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections
<math display="block"> | \psi_i\rangle \langle \psi_i| </math>
by a finite set of positive operators
<math display="block"> F_i F_i^* </math>
whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {{math|{''λ''<sub>1</sub> ... ''λ<sub>n</sub>''} }} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is {{math|''λ<sub>i</sub>''}}. Instead of collapsing to the (unnormalized) state
<math display="block"> | \psi_i\rangle \langle \psi_i |\psi\rangle </math>
after the measurement, the system now will be in the state
<math display="block"> F_i |\psi\rangle. </math>

Since the {{math|''F<sub>i</sub> F<sub>i</sub>*''}} operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general [[mixed state (physics)|mixed state]]s.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other [[quantum operation]]s, which are described by [[completely positive map]]s which do not increase the trace.

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).

== List of mathematical tools ==

Part of the folklore of the subject concerns the [[mathematical physics]] textbook [[Methods of Mathematical Physics]] put together by [[Richard Courant]] from [[David Hilbert]]'s [[Göttingen University]] courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.

The main tools include:
* [[linear algebra]]: [[complex numbers]], [[eigenvector]]s, [[eigenvalue]]s
* [[functional analysis]]: [[Hilbert space]]s, [[linear operator]]s, [[spectral theory]]
* [[differential equations]]: [[partial differential equations]], [[separation of variables]], [[ordinary differential equations]], [[Sturm–Liouville theory]], [[eigenfunction]]s
* [[harmonic analysis]]: [[Fourier transform]]s

== See also ==
* [[List of mathematical topics in quantum theory]]
* [[Symmetry in quantum mechanics]]

== Notes ==
{{reflist}}

== References ==
* {{cite book | last1=Bäuerle | first1=Gerard G. A. | last2=de Kerf | first2=Eddy A. | title=Lie Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics |series= Studies in Mathematical Physics| publisher=North Holland | publication-place=Amsterdam | date=1990 | isbn=0-444-88776-8}}
* {{cite book | last1=Byron | first1=Frederick W. | last2=Fuller | first2=Robert W. | title=Mathematics of Classical and Quantum Physics | publisher=Courier Corporation | publication-place=New York | year=1992 | isbn=978-0-486-67164-2}}
* {{cite journal | last1=Carcassi | first1=Gabriele | last2=Maccone | first2=Lorenzo | last3=Aidala | first3=Christine A. | title=Four Postulates of Quantum Mechanics Are Three | journal=Physical Review Letters | publisher=American Physical Society (APS) | volume=126 | issue=11 | date=2021-03-16 | page=110402 | issn=0031-9007 | doi=10.1103/physrevlett.126.110402| pmid=33798366 | arxiv=2003.11007 | bibcode=2021PhRvL.126k0402C | s2cid=214623241 }}
* {{cite book | last1=Cohen-Tannoudji | first1=Claude | last2=Diu | first2=Bernard | last3=Laloë | first3=Franck | title=Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods | publisher=Wiley-VCH Verlag GmbH & Co. KGaA | publication-place=Weinheim | date=2020 | isbn=978-3-527-82272-0}}
* {{cite journal | last1 = Dirac | first1 = P. A. M. | title = The Fundamental Equations of Quantum Mechanics | doi = 10.1098/rspa.1925.0150 | journal = Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume = 109 | issue = 752 | pages = 642–653 | year = 1925 |bibcode = 1925RSPSA.109..642D | doi-access = free }}
* {{cite journal | last=Edwards | first=David A. | title=The mathematical foundations of quantum mechanics | journal=Synthese | publisher=Springer Science and Business Media LLC | volume=42 | issue=1 | year=1979 | issn=0039-7857 | doi=10.1007/bf00413704 | pages=1–70| s2cid=46969028 }}
* {{cite book | last1=Greenstein | first1=George | last2=Zajonc | first2=Arthur | title=The Quantum Challenge | publisher=Jones & Bartlett Learning | publication-place=Sudbury, Mass. | date=2006 | isbn=978-0-7637-2470-2}}
* {{cite book | last1=Jauch | first1=J. M. | last2=Wigner | first2=E. P. | last3=Yanase | first3=M. M. | title=Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics | chapter=Some Comments Concerning Measurements in Quantum Mechanics | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | year=1997 | pages=475–482 | isbn=978-3-642-08179-8 | doi=10.1007/978-3-662-09203-3_52 | chapter-url=https://archive-ouverte.unige.ch/unige:162146 }}
* {{cite journal | last1=Masanes | first1=Lluís | last2=Galley | first2=Thomas D. | last3=Müller | first3=Markus P. | title=The measurement postulates of quantum mechanics are operationally redundant | journal=Nature Communications | publisher=Springer Science and Business Media LLC | volume=10 | issue=1 | date=2019 | page=1361 | issn=2041-1723 | doi=10.1038/s41467-019-09348-x| pmid=30911009 | pmc=6434053 | arxiv=1811.11060 | bibcode=2019NatCo..10.1361M | doi-access=free }}
* {{cite journal |last1=Solem |first1=J. C. |last2=Biedenharn |first2=L. C. |date=1993 |title=Understanding geometrical phases in quantum mechanics: An elementary example |journal=Foundations of Physics |volume=23 |issue=2 |pages=185–195 |doi=10.1007/BF01883623|bibcode=1993FoPh...23..185S |s2cid=121930907 }}
* {{cite book | last1=Streater | first1=Raymond Frederick | last2=Wightman | first2=Arthur Strong | title=PCT, Spin and Statistics, and All that | publisher=Princeton University Press | publication-place=Princeton, NJ | date=2000 | isbn=978-0-691-07062-9}}
* {{cite book | last1=Sakurai | first1=Jun John | last2=Napolitano | first2=Jim | title=Modern quantum mechanics | publisher=Cambridge University Press | isbn=978-1-108-47322-4|edition=3rd|location=Cambridge|year=2021}}

== Further reading ==
* {{cite book | last=Auyang | first=Sunny Y. | title=How is Quantum Field Theory Possible? | publisher=Oxford University Press on Demand | publication-place=New York, NY | date=1995 | isbn=978-0-19-509344-5}}
* {{cite book | last=Emch | first=Gérard G. | title=Algebraic Methods in Statistical Mechanics and Quantum Field Theory | publisher=John Wiley & Sons | publication-place=New York | date=1972 | isbn=0-471-23900-3}}
* {{cite book | last1=Giachetta | first1=Giovanni | last2=Mangiarotti | first2=Luigi | last3=Sardanashvily | first3=Gennadi | title=Geometric and Algebraic Topological Methods in Quantum Mechanics | publisher=WORLD SCIENTIFIC | year=2005 | isbn=978-981-256-129-9 | doi=10.1142/5731| arxiv=math-ph/0410040 }}
* {{cite journal | last=Gleason | first=Andrew M. | title=Measures on the Closed Subspaces of a Hilbert Space | journal=Journal of Mathematics and Mechanics | publisher=Indiana University Mathematics Department | volume=6 | issue=6 | year=1957 | jstor=24900629 | pages=885–893 | url=http://www.jstor.org/stable/24900629}}
* {{cite book | last=Hall | first=Brian C. | title=Quantum Theory for Mathematicians | series=Graduate Texts in Mathematics | publisher=Springer New York | publication-place=New York, NY | year=2013 | volume=267 | isbn=978-1-4614-7115-8 | issn=0072-5285 | doi=10.1007/978-1-4614-7116-5| bibcode=2013qtm..book.....H | s2cid=117837329 }}
* {{cite book | last=Jauch | first=Josef Maria | title=Foundations of Quantum Mechanics | publisher=Addison-Wesley | publication-place=Reading, Mass. | date=1968 | isbn=0-201-03298-8}}
* {{cite book | last=Jost | first=R. | title=The General Theory of Quantized Fields | publisher=American Mathematical Society | series=Lectures in applied mathematics | year=1965}}
* {{cite book | last=Kuhn | first=Thomas S. | title=Black-Body Theory and the Quantum Discontinuity, 1894-1912 | publisher=University of Chicago Press | publication-place=Chicago | date=1987| isbn=978-0-226-45800-7}}
* {{cite book | last=Landsman | first=Klaas | title=Foundations of Quantum Theory | series=Fundamental Theories of Physics | publisher=Springer International Publishing | publication-place=Cham | year=2017 | volume=188 | isbn=978-3-319-51776-6 | issn=0168-1222 | doi=10.1007/978-3-319-51777-3}}
* {{cite book | last=Mackey | first=George W. | title=Mathematical Foundations of Quantum Mechanics | publisher=Courier Corporation | publication-place=Mineola, N.Y | year=2004 | isbn=978-0-486-43517-6}}
* {{cite book | last=McMahon | first=David | title=Quantum Mechanics Demystified, 2nd Edition | publisher=McGraw-Hill Prof Med/Tech | publication-place=New York, NY | date=2013 |url=https://theswissbay.ch/pdf/Gentoomen%20Library/Misc/Demystified%20Series/McGraw-Hill%20-%20Quantum%20Mechanics%20Demystified%20%282006%29.pdf | isbn=978-0-07-176563-3}}
* {{cite book | last=Moretti | first=Valter | title=Spectral Theory and Quantum Mechanics | series=Unitext | publisher=Springer International Publishing | publication-place=Cham | year=2017 | volume=110 | isbn=978-3-319-70705-1 | issn=2038-5714 | doi=10.1007/978-3-319-70706-8| s2cid=125121522 }}
* {{cite book | last=Moretti | first=Valter | title=Fundamental Mathematical Structures of Quantum Theory | publisher=Springer International Publishing | publication-place=Cham | year=2019 | isbn=978-3-030-18345-5 | doi=10.1007/978-3-030-18346-2| s2cid=197485828 }}
* {{cite book | last=Prugovecki | first=Eduard | title=Quantum Mechanics in Hilbert Space | publisher=Courier Dover Publications | publication-place=Mineola, NY | date=2006 | isbn=978-0-486-45327-9}}
* {{cite book | last1=Reed | first1=Michael | last2=Simon | first2=Barry | title=Methods of Modern Mathematical Physics | publisher=Academic Press | publication-place=New York | date=1972 | isbn=978-0-12-585001-8}}
* {{cite book | last=Shankar | first=R. | title=Principles of Quantum Mechanics | publisher=Springer | date=2013 | url=https://www.fisica.net/mecanica-quantica/Shankar%20-%20Principles%20of%20quantum%20mechanics.pdf | isbn=978-1-4615-7675-4}}
* {{cite book | last=Teschl | first=Gerald | title=Mathematical Methods in Quantum Mechanics | publisher=American Mathematical Soc. | publication-place=Providence, R.I |url= https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/schroe.pdf| date=2009 | isbn=978-0-8218-4660-5}}
* {{cite book | last= von Neumann | first=John | title=Mathematical Foundations of Quantum Mechanics | publisher=Princeton University Press | publication-place=Princeton Oxford | year=2018 | isbn=978-0-691-17856-1}}
* {{cite book | last=Weaver | first=Nik | title=Mathematical Quantization | publisher=Chapman and Hall/CRC | date=2001 | isbn=978-0-429-07514-8 | doi=10.1201/9781420036237}}
* {{cite book | last=Weyl | first=Hermann | title=The Theory of Groups and Quantum Mechanics | publisher=Courier Corporation | publication-place=Mineola, NY | year=1950 | isbn=978-0-486-60269-1}}

{{Quantum mechanics topics}}
{{Functional analysis}}

[[Category:Quantum mechanics| ]]
[[Category:Mathematical physics]]
[[Category:History of physics]]