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{{Short description|Learning project introducing the quantum concept and the fundamental principles of quantum mechanics, from quantization to quantum technologies}}

{{Quantum book backlink|Foundations}}

'''Quantum postulates''' are the core assumptions of [[Physics:Quantum mechanics|quantum mechanics]]. They specify how physical systems are represented, how observables are defined, how measurements produce probabilities and state updates, and how states evolve in time. In the standard formulation, a system is described by a complex [[Physics:Quantum Hilbert space|Hilbert space]], observables are represented by self-adjoint operators, measurement statistics are given by the [[Physics:Born rule|Born rule]], and time evolution is governed by the [[Physics:Quantum Schrödinger equation|Schrödinger equation]].<ref>{{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}}</ref>

[[File:Quantum postulates1.jpg|thumb|400px|Basic structure of the quantum postulates: states, observables, measurement, and time evolution.]]

=Physical system=
A physical system is described by states, observables, and dynamics. In classical mechanics, states are points in phase space and observables are real-valued functions on that space. In quantum mechanics, by contrast, states are rays or density operators on a Hilbert space, and observables are self-adjoint operators acting on that space.<ref>{{cite book | last=Weyl | first=Hermann | title=The Theory of Groups and Quantum Mechanics | title-link=Gruppentheorie und Quantenmechanik | publisher=Dover | translator-first=H. P. | translator-last=Robertson | year=1950 | orig-year=1931 | bibcode=1950tgqm.book.....W }}</ref>

== State space and quantum states ==

Each isolated physical system is associated with a separable complex [[Physics:Quantum Hilbert space|Hilbert space]] <math>\mathcal{H}</math> with inner product. At a fixed time, the physical state is represented by a normalized vector <math>|\psi\rangle \in \mathcal{H}</math>, up to an overall phase.<ref>{{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}}</ref><ref>{{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 }}</ref>

{{Quote box
| title = Postulate I
| quote = The state of an isolated physical system is represented, at a fixed time <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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Two normalized vectors represent the same physical state if they differ only by a phase factor:

<math display="block">|\psi_k \rangle \sim |\psi_l\rangle \;\Leftrightarrow\; |\psi_k \rangle = e^{i\alpha} |\psi_l\rangle,\qquad \alpha\in\mathbb{R}.</math>

Thus the physical state is properly a ray in projective Hilbert space rather than a single vector.<ref>{{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}}</ref>

=== Composite systems ===

For a composite system, the total state space is the tensor product of the state spaces of the component subsystems.<ref>{{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 }}</ref>

{{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.
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If a composite state cannot be factored into subsystem states, the system is [[Physics:Quantum entanglement|entangled]]. In that case, subsystems are generally described not by state vectors but by [[Physics:Quantum Density matrix|density operators]] <math>\rho</math>, which are positive self-adjoint trace-class operators normalized by <math>\mathrm{tr}(\rho)=1</math>.<ref>{{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}}</ref>

A separable bipartite mixed state can be written as

<math display="block">\rho=\sum_k p_k\,\rho_1^k\otimes\rho_2^k,\qquad \sum_k p_k=1.</math>

If only one term is present, the state is a product state:

<math display="block">\rho=\rho_1\otimes\rho_2.</math>

== Observables and measurement ==

A measurable physical quantity is represented by a self-adjoint operator on <math>\mathcal{H}</math>. Its eigenvalues are the possible outcomes of measurement.

{{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>. The result of measuring <math>\mathcal{A}</math> must be one of the eigenvalues of <math>A</math>.
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Since self-adjoint operators have real spectra, measurement results are real numbers. For discrete spectra, the outcomes are quantized.

=== Born rule ===

The probabilities of measurement outcomes are determined by the projection of the state onto the eigenspaces of the observable.<ref>{{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}}</ref>

{{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 of the corresponding observable <math>A</math> is given by the amplitude squared of the projection onto the corresponding eigenspace.
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For a discrete nondegenerate spectrum,

<math display="block">\mathbb{P}(a_n)=|\langle a_n|\psi\rangle|^2.</math>

For a discrete degenerate spectrum,

<math display="block">\mathbb{P}(a_n)=\sum_i^{g_n}|\langle a_n^i|\psi\rangle|^2.</math>

For a continuous nondegenerate spectrum,

<math display="block">d\mathbb{P}(\alpha)=|\langle \alpha|\psi\rangle|^2\,d\alpha.</math>

For a mixed state <math>\rho</math>, the expectation value of an observable <math>A</math> is

<math display="block">\langle A\rangle=\mathrm{tr}(A\rho),</math>

and the probability of obtaining eigenvalue <math>a_n</math> is

<math display="block">\mathbb{P}(a_n)=\mathrm{tr}(P_n\rho),</math>

where <math>P_n</math> is the projection operator onto the eigensubspace associated with <math>a_n</math>.

=== State update after measurement ===

In an ideal projective measurement, once the result <math>a_n</math> is obtained, the state updates to the normalized projection onto the associated eigensubspace.

{{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 immediately after the measurement is the normalized projection of <math>|\psi\rangle</math> onto the eigensubspace associated with <math>a_n</math>.
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<math display="block">|\psi\rangle \quad \overset{a_n}{\Longrightarrow}\quad \frac{P_n|\psi\rangle}{\sqrt{\langle \psi | P_n | \psi \rangle}}.</math>

For a mixed state, the corresponding update rule is

<math display="block">\rho'=\frac{P_n\rho P_n^\dagger}{\mathrm{tr}(P_n\rho P_n^\dagger)}.</math>

The Born rule together with this state-update rule gives the standard projective measurement scheme. More general quantum measurements are described by [[Physics:Quantum POVM|positive operator-valued measures]].<ref>{{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 }}</ref>

== Time evolution ==

The time evolution of a closed quantum system is governed by the [[Physics:Quantum Schrödinger equation|Schrödinger equation]].

{{Quote box
| title = Postulate III
| quote = The time evolution of the state vector <math>|\psi(t)\rangle</math> is governed by the Schrödinger equation
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<math display="block">i\hbar \frac{d}{dt}|\psi(t)\rangle = H(t)|\psi(t)\rangle,</math>

where <math>H(t)</math> is the [[Physics:Quantum Hamiltonian|Hamiltonian]] of the system.

Equivalently, time evolution may be expressed by a unitary operator:

<math display="block">|\psi(t)\rangle = U(t;t_0)|\psi(t_0)\rangle.</math>

For a mixed state,

<math display="block">\rho(t)=U(t;t_0)\rho(t_0)U^\dagger(t;t_0).</math>

Open systems generally evolve nonunitarily and are instead described by [[Physics:Quantum operation|quantum operations]], quantum instruments, or master-equation formalisms.<ref>{{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}}</ref>

== Further implications ==

Several important consequences follow from the postulates.

* Physical symmetries act on the Hilbert space by unitary or antiunitary transformations, as stated by [[Physics:Quantum Wigner's theorem|Wigner's theorem]].<ref>{{cite book | last=Weyl | first=Hermann | title=The Theory of Groups and Quantum Mechanics | title-link=Gruppentheorie und Quantenmechanik | publisher=Dover | translator-first=H. P. | translator-last=Robertson | year=1950 | orig-year=1931 | bibcode=1950tgqm.book.....W }}</ref>
* Pure states correspond to one-dimensional orthogonal projectors, while general density operators describe mixed states.
* The [[Physics:Quantum Uncertainty principle|uncertainty principle]] can be derived as a theorem of the operator formalism.

These show that the postulates are not merely interpretive statements but the basis of the full mathematical structure of quantum theory.

== Spin ==

All particles possess intrinsic angular momentum called [[Physics:Quantum spin|spin]]. Unlike classical rotation, quantum spin is an intrinsic property with no direct classical analogue. For a particle of spin <math>S</math>, the spin degree of freedom introduces the discrete values

<math display="block">\sigma=-S\hbar,\;-(S-1)\hbar,\;\dots,\;0,\;\dots,\;(S-1)\hbar,\;S\hbar.</math>

A single-particle state of spin <math>S</math> is therefore represented by a <math>(2S+1)</math>-component spinor. Integer-spin particles are [[Physics:Quantum boson|boson]]s, while half-integer-spin particles are [[Physics:Quantum fermion|fermion]]s.<ref>{{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}}</ref>

== Symmetrization postulate ==

For a system of identical particles, the total wavefunction must be either symmetric or antisymmetric under exchange of any pair of particles.<ref>{{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}}</ref>

{{Quote box
| title = Symmetrization postulate
| quote = The wavefunction of a system of <math>N</math> identical particles in three dimensions is either totally symmetric (bosons) or totally antisymmetric (fermions) under interchange of any pair of particles.
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This requirement underlies the distinction between bosons and fermions and is closely related to the [[Physics:Quantum spin-statistics theorem|spin-statistics theorem]]. In two spatial dimensions, more general exchange behavior can occur, leading to [[Physics:Quantum anyon|anyon]]s.<ref>{{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}}</ref>

=== Pauli exclusion principle ===

For fermions, antisymmetry of the wavefunction implies the [[Physics:Quantum Pauli exclusion principle|Pauli exclusion principle]]: no two identical fermions can occupy the same one-particle quantum state.

{{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}\psi(\dots,\mathbf r_j,\sigma_j,\dots,\mathbf r_i,\sigma_i,\dots)</math>
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For bosons the prefactor is <math>+1</math>; for fermions it is <math>-1</math>. This distinction underlies atomic shell structure and many properties of matter.<ref>{{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}}</ref><ref>{{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}}</ref>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}
{{Author|Harold Foppele}}

[[Category:Quantum mechanics]]

{{Sourceattribution|Quantum Postulates|1}}

Physics:Quantum Postulates - Revision history

imported>WikiHarold at 00:00, 11 May 2026

imported>WikiHarold at 00:00, 11 May 2026