Quantum postulates are the core assumptions of 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 Hilbert space, observables are represented by self-adjoint operators, measurement statistics are given by the Born rule, and time evolution is governed by the Schrödinger equation.^[1]

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.^[2]

State space and quantum states

Each isolated physical system is associated with a separable complex Hilbert space $ℋ$ with inner product. At a fixed time, the physical state is represented by a normalized vector $| ψ ⟩ \in ℋ$ , up to an overall phase.^[3]^[4]

Postulate I

The state of an isolated physical system is represented, at a fixed time

t

, by a state vector

| ψ ⟩

belonging to a Hilbert space

ℋ

called the state space.

Two normalized vectors represent the same physical state if they differ only by a phase factor:

$| ψ_{k} ⟩ \sim | ψ_{l} ⟩ \Leftrightarrow | ψ_{k} ⟩ = e^{i α} | ψ_{l} ⟩, α \in ℝ .$

Thus the physical state is properly a ray in projective Hilbert space rather than a single vector.^[5]

Composite systems

For a composite system, the total state space is the tensor product of the state spaces of the component subsystems.^[6]

Composite system postulate

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems.

If a composite state cannot be factored into subsystem states, the system is entangled. In that case, subsystems are generally described not by state vectors but by density operators $ρ$ , which are positive self-adjoint trace-class operators normalized by $t r (ρ) = 1$ .^[7]

A separable bipartite mixed state can be written as

$ρ = \sum_{k} p_{k} ρ_{1}^{k} \otimes ρ_{2}^{k}, \sum_{k} p_{k} = 1 .$

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

$ρ = ρ_{1} \otimes ρ_{2} .$

Observables and measurement

A measurable physical quantity is represented by a self-adjoint operator on $ℋ$ . Its eigenvalues are the possible outcomes of measurement.

Postulate II.a

Every measurable physical quantity

𝒜

is described by a Hermitian operator

A

acting in the state space

ℋ

. The result of measuring

𝒜

must be one of the eigenvalues of

A

.

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.^[8]

Postulate II.b

When the physical quantity

𝒜

is measured on a system in a normalized state

| ψ ⟩

, the probability of obtaining an eigenvalue of the corresponding observable

A

is given by the amplitude squared of the projection onto the corresponding eigenspace.

For a discrete nondegenerate spectrum,

$ℙ (a_{n}) = | ⟨ a_{n} | ψ ⟩ |^{2} .$

For a discrete degenerate spectrum,

$ℙ (a_{n}) = \sum_{i}^{g_{n}} | ⟨ a_{n}^{i} | ψ ⟩ |^{2} .$

For a continuous nondegenerate spectrum,

$d ℙ (α) = | ⟨ α | ψ ⟩ |^{2} d α .$

For a mixed state $ρ$ , the expectation value of an observable $A$ is

$⟨ A ⟩ = t r (A ρ),$

and the probability of obtaining eigenvalue $a_{n}$ is

$ℙ (a_{n}) = t r (P_{n} ρ),$

where $P_{n}$ is the projection operator onto the eigensubspace associated with $a_{n}$ .

State update after measurement

In an ideal projective measurement, once the result $a_{n}$ is obtained, the state updates to the normalized projection onto the associated eigensubspace.

Postulate II.c

If the measurement of the physical quantity

𝒜

on the system in the state

| ψ ⟩

gives the result

a_{n}

, then the state immediately after the measurement is the normalized projection of

| ψ ⟩

onto the eigensubspace associated with

a_{n}

.

$| ψ ⟩ \overset{a_{n}}{⟹} \frac{P_{n} | ψ ⟩}{\sqrt{⟨ ψ | P_{n} | ψ ⟩}} .$

For a mixed state, the corresponding update rule is

$ρ^{'} = \frac{P_{n} ρ P_{n}^{†}}{t r (P_{n} ρ P_{n}^{†})} .$

The Born rule together with this state-update rule gives the standard projective measurement scheme. More general quantum measurements are described by positive operator-valued measures.^[9]

Time evolution

The time evolution of a closed quantum system is governed by the Schrödinger equation.

Postulate III

The time evolution of the state vector

| ψ (t) ⟩

is governed by the Schrödinger equation

$i ℏ \frac{d}{d t} | ψ (t) ⟩ = H (t) | ψ (t) ⟩,$

where $H (t)$ is the Hamiltonian of the system.

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

$| ψ (t) ⟩ = U (t; t_{0}) | ψ (t_{0}) ⟩ .$

For a mixed state,

$ρ (t) = U (t; t_{0}) ρ (t_{0}) U^{†} (t; t_{0}) .$

Open systems generally evolve nonunitarily and are instead described by quantum operations, quantum instruments, or master-equation formalisms.^[10]

Further implications

Several important consequences follow from the postulates.

Physical symmetries act on the Hilbert space by unitary or antiunitary transformations, as stated by Wigner's theorem.^[11]
Pure states correspond to one-dimensional orthogonal projectors, while general density operators describe mixed states.
The 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 spin. Unlike classical rotation, quantum spin is an intrinsic property with no direct classical analogue. For a particle of spin $S$ , the spin degree of freedom introduces the discrete values

$σ = - S ℏ, - (S - 1) ℏ, \dots, 0, \dots, (S - 1) ℏ, S ℏ .$

A single-particle state of spin $S$ is therefore represented by a $(2 S + 1)$ -component spinor. Integer-spin particles are bosons, while half-integer-spin particles are fermions.^[12]

Symmetrization postulate

For a system of identical particles, the total wavefunction must be either symmetric or antisymmetric under exchange of any pair of particles.^[13]

Symmetrization postulate

The wavefunction of a system of

N

identical particles in three dimensions is either totally symmetric (bosons) or totally antisymmetric (fermions) under interchange of any pair of particles.

This requirement underlies the distinction between bosons and fermions and is closely related to the spin-statistics theorem. In two spatial dimensions, more general exchange behavior can occur, leading to anyons.^[14]

Pauli exclusion principle

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

Pauli principle

$ψ (\dots, 𝐫_{i}, σ_{i}, \dots, 𝐫_{j}, σ_{j}, \dots) = (- 1)^{2 S} ψ (\dots, 𝐫_{j}, σ_{j}, \dots, 𝐫_{i}, σ_{i}, \dots)$

For bosons the prefactor is $+ 1$ ; for fermions it is $- 1$ . This distinction underlies atomic shell structure and many properties of matter.^[15]^[16]

Table of contents (137 articles)

Index

Core theory

Foundations
Conceptual and interpretations
Mathematical structure and systems
Atomic and spectroscopy
Wavefunctions and modes
Quantum dynamics and evolution
Measurement and information
Quantum information and computing

Applications and extensions

Quantum optics and experiments
Open quantum systems
Quantum field theory
Statistical mechanics and kinetic theory
Condensed matter and solid-state physics
Plasma and fusion physics
Timeline
Advanced and frontier topics

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Physics:Quantum Postulates

Physical system

State space and quantum states

Composite systems

Observables and measurement

Born rule

State update after measurement

Time evolution

Further implications

Spin

Symmetrization postulate

Pauli exclusion principle

See also

Table of contents (137 articles)

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