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Quantum theory Definition

Mathematical formulation of quantum mechanics

Builds on Hilbert spaceSelf-adjoint operatorUnbounded operatorProjection-valued measureUnitary operatorTrace class operatorTensor product of Hilbert spaces

Definition

In the mathematical formulation of quantum mechanics, a physical system is described by the following data and rules.

  1. States: the system has a complex separable Hilbert space HH. A pure state is a unit vector ψ∈H\psi \in H. The vectors ψ\psi and eiαψe^{i\alpha}\psi with α∈ℝ\alpha \in \R give the same state.
  2. Observables: each measurable quantity is a self-adjoint operator AA on HH, possibly unbounded, with projection-valued measure PAP^A from the spectral theorem, so that A=∫ℝλ dPA(λ)A = \int_\R \lambda \, dP^A(\lambda).
  3. Measurement (Born rule): measuring AA in the state ψ\psi gives a result in a Borel set E⊆ℝE \subseteq \R with probability ⟨PA(E)ψ,ψ⟩\langle P^A(E)\psi, \psi \rangle. So the possible results lie in the spectrum of AA. For ψ\psi in the domain of AA, the expected value is ⟨Aψ,ψ⟩\langle A\psi, \psi \rangle.
  4. Dynamics: there is a self-adjoint operator H^\hat H, the Hamiltonian. A state evolves by ψ(t)=e−itH^/ℏψ(0)\psi(t) = e^{-it\hat H/\hbar}\psi(0), where ℏ\hbar is the reduced Planck constant. The operators e−itH^/ℏe^{-it\hat H/\hbar} form a strongly continuous one-parameter group of unitary operators.
  5. Composite systems: the Hilbert space of a system made of two parts with spaces H1H_1 and H2H_2 is the tensor product H1⊗H2H_1 \otimes H_2.

Mixed states are density operators: positive trace class operators ρ\rho with tr⁡ρ=1\operatorname{tr}\rho = 1; the probability in clause 3 is then tr⁡(ρPA(E))\operatorname{tr}(\rho P^A(E)).

Remarks

Example: a spin-12\tfrac12 system has H=ℂ2H = \C^2 and spin observable σz=diag⁡(1,−1)\sigma_z = \operatorname{diag}(1, -1), whose spectral projections are P({1})=diag⁡(1,0)P(\set{1}) = \operatorname{diag}(1, 0) and P({−1})=diag⁡(0,1)P(\set{-1}) = \operatorname{diag}(0, 1). For ψ=(1,1)/2\psi = (1, 1)/\sqrt 2, the two results each have probability 12\tfrac12, and the expected value is ⟨σzψ,ψ⟩=0\langle \sigma_z \psi, \psi \rangle = 0.

Non-example: the matrix N=(0100)N = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} is not self-adjoint, so clause 2 does not make it an observable. It has no spectral decomposition into orthogonal projections: its only eigenvalue is 00, but N≠0N \ne 0.

When ψ\psi is a unit vector, the pure state ψ\psi corresponds to the density operator ρ=⟨⋅,ψ⟩ψ\rho = \langle \cdot, \psi \rangle \psi, the projection onto ℂψ\C\psi. By Stone's theorem, every strongly continuous one-parameter unitary group has the form of clause 4 for a unique self-adjoint generator.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Hilbert space, concept, no evidence levelHilbert spaceconceptProjection-valued measure, concept, no evidence levelProjection-valued mea…conceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hil…conceptTrace class operator, concept, no evidence levelTrace class operatorconceptUnbounded operator, concept, no evidence levelUnbounded operatorconceptUnitary operator, concept, no evidence levelUnitary operatorconceptMathematical formulation of quantum mechanics, definition, no evidence levelMathematical formulat…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Hilbert space concept
  2. Projection-valued measure concept
  3. Self-adjoint operator concept
  4. Tensor product of Hilbert spaces concept
  5. Trace class operator concept
  6. Unbounded operator concept
  7. Unitary operator concept
  8. Mathematical formulation of quantum mechanics definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics, Postulates of quantum mechanics
About this page
TagF8R88XFMQK
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Kinddefinition
WritingW0 stub
MSC81P05, 46N50, 81Q10
DefinesMathematical formulation of quantum mechanics
UsesHilbert space, Self-adjoint operator, Unbounded operator, Projection-valued measure, Unitary operator, Trace class operator, Tensor product of Hilbert spaces
Changed2 November 2026

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