Physics:Quantum Postulates

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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]

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.[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⟩∈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:

|ψk⟩∼|ψl⟩⇔|ψk⟩=eiα|ψl⟩,α∈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 tr(ρ)=1.[7]

A separable bipartite mixed state can be written as

ρ=∑kpkρ1k⊗ρ2k,∑kpk=1.

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

ρ=ρ1⊗ρ2.

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(an)=|⟨an|ψ⟩|2.

For a discrete degenerate spectrum,

ℙ(an)=∑ign|⟨ani|ψ⟩|2.

For a continuous nondegenerate spectrum,

dℙ(α)=|⟨α|ψ⟩|2dα.

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

⟨A⟩=tr(Aρ),

and the probability of obtaining eigenvalue an is

ℙ(an)=tr(Pnρ),

where Pn is the projection operator onto the eigensubspace associated with an.

State update after measurement

In an ideal projective measurement, once the result an 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 an, then the state immediately after the measurement is the normalized projection of |ψ⟩ onto the eigensubspace associated with an.

|ψ⟩⟹anPn|ψ⟩⟨ψ|Pn|ψ⟩.

For a mixed state, the corresponding update rule is

ρ′=PnρPn†tr(PnρPn†).

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

iℏddt|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;t0)|ψ(t0)⟩.

For a mixed state,

ρ(t)=U(t;t0)ρ(t0)U†(t;t0).

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=−Sℏ,−(S−1)ℏ,…,0,…,(S−1)ℏ,Sℏ.

A single-particle state of spin S is therefore represented by a (2S+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(…,𝐫i,σi,…,𝐫j,σj,…)=(−1)2Sψ(…,𝐫j,σj,…,𝐫i,σi,…)

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

See also

Table of contents (185 articles)

Index

Full contents

9. Quantum optics and experiments (5) ↑ Back to index
14. Plasma and fusion physics (8) ↑ Back to index
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.
Conceptual illustration of plasma physics in a fusion context, showing magnetically confined ionized gas in a tokamak and the collective behavior governed by electromagnetic fields and transport processes.

References

  1. ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. ISBN 978-3-527-82272-0. 
  2. ↑ Weyl, Hermann (1950). The Theory of Groups and Quantum Mechanics. Dover. Bibcode: 1950tgqm.book.....W. 
  3. ↑ Bäuerle, Gerard G. A.; de Kerf, Eddy A. (1990). Lie Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics. Studies in Mathematical Physics. Amsterdam: North Holland. ISBN 0-444-88776-8. 
  4. ↑ Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics 23 (2): 185–195. doi:10.1007/BF01883623. Bibcode: 1993FoPh...23..185S. 
  5. ↑ Bäuerle, Gerard G. A.; de Kerf, Eddy A. (1990). Lie Algebras, Part 1: Finite and Infinite Dimensional Lie Algebras and Applications in Physics. Studies in Mathematical Physics. Amsterdam: North Holland. ISBN 0-444-88776-8. 
  6. ↑ Jauch, J. M.; Wigner, E. P.; Yanase, M. M. (1997). "Some Comments Concerning Measurements in Quantum Mechanics". Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 475–482. doi:10.1007/978-3-662-09203-3_52. ISBN 978-3-642-08179-8. https://archive-ouverte.unige.ch/unige:162146. 
  7. ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. ISBN 978-3-527-82272-0. 
  8. ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. ISBN 978-3-527-82272-0. 
  9. ↑ Jauch, J. M.; Wigner, E. P.; Yanase, M. M. (1997). "Some Comments Concerning Measurements in Quantum Mechanics". Part I: Particles and Fields. Part II: Foundations of Quantum Mechanics. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 475–482. doi:10.1007/978-3-662-09203-3_52. ISBN 978-3-642-08179-8. https://archive-ouverte.unige.ch/unige:162146. 
  10. ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. ISBN 978-3-527-82272-0. 
  11. ↑ Weyl, Hermann (1950). The Theory of Groups and Quantum Mechanics. Dover. Bibcode: 1950tgqm.book.....W. 
  12. ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. ISBN 978-3-527-82272-0. 
  13. ↑ Sakurai, Jun John; Napolitano, Jim (2021). Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. ISBN 978-1-108-47322-4. 
  14. ↑ Sakurai, Jun John; Napolitano, Jim (2021). Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. ISBN 978-1-108-47322-4. 
  15. ↑ Sakurai, Jun John; Napolitano, Jim (2021). Modern quantum mechanics (3rd ed.). Cambridge: Cambridge University Press. ISBN 978-1-108-47322-4. 
  16. ↑ Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (2020). Quantum mechanics. Volume 2: Angular momentum, spin, and approximation methods. Weinheim: Wiley-VCH Verlag GmbH & Co. KGaA. ISBN 978-3-527-82272-0. 
Author: Harold Foppele




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