Math 2.0

Quantum theory Definition

Wave function collapse

Builds on Hilbert spaceObservableOrthogonal projectionQuantum stateDensity matrixBorn rule

Definition

Let ℋ\mathcal{H} be a complex Hilbert space and let AA be an observable on ℋ\mathcal{H} with discrete spectrum. For each eigenvalue aa of AA, let PaP_a be the orthogonal projection onto the eigenspace of aa. Let the system be in the pure state given by a unit vector ψ∈ℋ\psi \in \mathcal{H}. Let a measurement of AA give the value aa, which by the Born rule happens with probability pa=∥Paψ∥2p_a = \norm{P_a \psi}^2, and assume pa>0p_a > 0. The collapse, or projection postulate, replaces ψ\psi by the post-measurement state ψa=Paψ∥Paψ∥.\psi_a = \frac{P_a \psi}{\norm{P_a \psi}}. For a density operator ρ\rho with pa=tr⁡(Paρ)>0p_a = \tr(P_a \rho) > 0, the post-measurement state is ρa=PaρPatr⁡(Paρ).\rho_a = \frac{P_a \rho P_a}{\tr(P_a \rho)}.

Remarks

Example: on ℂ2\C^2 let A=(100−1)A = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} and ψ=(1,1)/2\psi = (1, 1)/\sqrt{2}. Then P1=(1000)P_1 = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}, p1=1/2p_1 = 1/2 and ψ1=(1,0)\psi_1 = (1, 0), a unit eigenvector of AA for the eigenvalue 11. Non-example: the vector P1ψ=(1/2,0)P_1 \psi = (1/\sqrt{2}, 0) is not the post-measurement state, since its norm is not 11. The update is idempotent: measuring AA again in the state ψa\psi_a gives aa with probability 11. The density operator form is also called the Lüders rule.

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Adjoint of an unbounded operator, concept, no evidence levelAdjoint of an unbound…conceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptContinuous functional calculus, concept, no evidence levelContinuous functional…conceptDensely defined operator, concept, no evidence levelDensely defined opera…conceptDensity matrix, concept, no evidence levelDensity matrixconceptExpectation value, concept, no evidence levelExpectation valueconceptHilbert space, concept, no evidence levelHilbert spaceconceptHilbert space adjoint, concept, no evidence levelHilbert space adjointconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionObservable, concept, no evidence levelObservableconceptOrthogonal projection, concept, no evidence levelOrthogonal projectionconceptProbability measure, concept, no evidence levelProbability measureconceptProjection-valued measure, concept, no evidence levelProjection-valued mea…conceptQuantum state, concept, no evidence levelQuantum stateconceptSpectrum of an operator, concept, no evidence levelSpectrum of an operat…conceptTrace class operator, concept, no evidence levelTrace class operatorconceptUnbounded operator, concept, no evidence levelUnbounded operatorconceptTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionNormal matrix, definition, no evidence levelNormal matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionSpectral theorem for normal matrices, statement, no evidence levelSpectral theorem for …statementSpectral theorem for bounded normal operators, statement, no evidence levelSpectral theorem for …statementSpectral theorem for unbounded self-adjoint operators, statement, no evidence levelSpectral theorem for …statementBorn rule, statement, no evidence levelBorn rulestatementWave function collapse, definition, no evidence levelWave function collapsedefinition
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.
  • no evidence level
  1. Adjoint of an unbounded operator concept
  2. Borel sigma-algebra concept
  3. Continuous functional calculus concept
  4. Densely defined operator concept
  5. Density matrix concept
  6. Expectation value concept
  7. Hilbert space concept
  8. Hilbert space adjoint concept
  9. Identity matrix object
  10. Matrix definition
  11. Observable concept
  12. Orthogonal projection concept
  13. Probability measure concept
  14. Projection-valued measure concept
  15. Quantum state concept
  16. Spectrum of an operator concept
  17. Trace class operator concept
  18. Unbounded operator concept
  19. Transpose definition
  20. Conjugate transpose definition
  21. Normal matrix definition
  22. Unitary matrix definition
  23. Spectral theorem for normal matrices statement
  24. Spectral theorem for bounded normal operators statement
  25. Spectral theorem for unbounded self-adjoint operators statement
  26. Born rule statement
  27. Wave function collapse definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wave_function_collapse
About this page
TagQPCBY8SZJW
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Kinddefinition
WritingW0 stub
MSC81P15, 81P05
DefinesWave function collapse
UsesHilbert space, Observable, Orthogonal projection, Quantum state, Density matrix, Born rule
Changed3 November 2026

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