Math 2.0

Quantum theory Definition

Hamiltonian operator

Builds on ObservableStone's theorem on one-parameter unitary groupsUnitary operatorSchrödinger operatorSchrödinger equationMomentum operatorPosition operator

Definition

Let ℏ>0\hbar > 0, and let ℋ\mathcal{H} be the complex Hilbert space of a quantum system. The Hamiltonian of the system is a self-adjoint operator HH on ℋ\mathcal{H}, the observable of total energy, that generates the time evolution. A state vector ψ\psi at time 00 becomes ψ(t)=e−itH/ℏ ψ\psi(t) = e^{-itH/\hbar}\, \psi at time t∈ℝt \in \R.

Remarks

By Stone's theorem, U(t)=e−itH/ℏU(t) = e^{-itH/\hbar} is a strongly continuous one-parameter group of unitary operators, and every such group arises from exactly one self-adjoint HH. For ψ∈D(H)\psi \in D(H), the curve ψ(t)\psi(t) is differentiable and solves the abstract Schrödinger equation iℏ ψ′(t)=Hψ(t)i\hbar\, \psi'(t) = H\psi(t).

Consider a particle of mass m>0m > 0 in ℝn\R^n with a real potential VV. Its Hamiltonian is built from the momentum operators and the position operators: H=12m∑j=1nPj2+V(X)=−ℏ22mΔ+V.H = \frac{1}{2m}\sum_{j=1}^n P_j^2 + V(X) = -\frac{\hbar^2}{2m}\Delta + V. It is the Schrödinger operator of the Schrödinger equation. One must check that it is self-adjoint on a suitable domain; for a bounded measurable VV this holds on the domain of the Laplacian, the Sobolev space H2(ℝn)H^2(\R^n).

Example: for a free particle on ℝ\R, H=P2/2mH = P^2/2m. The Fourier transform turns it into multiplication by ℏ2k2/2m\hbar^2 k^2/2m, a real function, so HH is self-adjoint with spectrum [0,∞)[0, \infty).

Non-example: the operator iIiI on ℂ2\C^2 is not self-adjoint. It would give e−it(iI)/ℏ=et/ℏIe^{-it(iI)/\hbar} = e^{t/\hbar} I, which is not unitary and makes the norm of a state vector grow.

Reading path

To understand this page from the ground up, read these 27 pages in order. The foundations come first.

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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…conceptHilbert space, concept, no evidence levelHilbert spaceconceptHilbert space adjoint, concept, no evidence levelHilbert space adjointconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionMomentum operator, concept, no evidence levelMomentum operatorconceptObservable, concept, no evidence levelObservableconceptPosition operator, concept, no evidence levelPosition operatorconceptProjection-valued measure, concept, no evidence levelProjection-valued mea…conceptSchrödinger equation, concept, no evidence levelSchrödinger equationconceptSchrödinger operator, concept, no evidence levelSchrödinger operatorconceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptSpectrum of an operator, concept, no evidence levelSpectrum of an operat…conceptUnbounded operator, concept, no evidence levelUnbounded operatorconceptUnitary operator, concept, no evidence levelUnitary 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 …statementStone's theorem on one-parameter unitary groups, statement, no evidence levelStone's theorem on on…statementHamiltonian operator, definition, no evidence levelHamiltonian operatordefinition
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. Hilbert space concept
  6. Hilbert space adjoint concept
  7. Identity matrix object
  8. Matrix definition
  9. Momentum operator concept
  10. Observable concept
  11. Position operator concept
  12. Projection-valued measure concept
  13. Schrödinger equation concept
  14. Schrödinger operator concept
  15. Self-adjoint operator concept
  16. Spectrum of an operator concept
  17. Unbounded operator concept
  18. Unitary 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. Stone's theorem on one-parameter unitary groups statement
  27. Hamiltonian operator definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
About this page
TagF5252H0MFD
URIm2://F5252H0MFD
Kinddefinition
WritingW0 stub
MSC81Q10, 47B25, 81Q05
DefinesHamiltonian operator
UsesObservable, Stone's theorem on one-parameter unitary groups, Unitary operator, Schrödinger operator, Schrödinger equation, Momentum operator, Position operator
Changed3 November 2026

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