A Hilbert space is a vector space equipped with an inner product and complete with respect to the norm induced by that inner product.^[1] It provides the natural mathematical setting for quantum mechanics, where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.^[2] The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension.

Quantum Mathematics: Foundations of Quantum Theory Visualized

Definition

A Hilbert space $ℋ$ is a vector space over $ℝ$ or $ℂ$ together with an inner product

$⟨ ϕ ∣ ψ ⟩$

such that the induced norm

$‖ ψ ‖ = \sqrt{⟨ ψ ∣ ψ ⟩}$

makes $ℋ$ a complete metric space.^[3]

In quantum theory, Hilbert spaces are usually taken to be complex. A normalized state vector satisfies

$⟨ ψ ∣ ψ ⟩ = 1 .$

Geometric interpretation

Hilbert space extends the geometry of ordinary three-dimensional space to possibly infinitely many dimensions. The inner product determines:

the length of a vector through its norm;
the angle between vectors through their overlap;
the notion of orthogonality, when $⟨ ϕ ∣ ψ ⟩ = 0$ ;
the projection of one vector onto another or onto a subspace.

These ideas are central in quantum mechanics.^[4]

Examples

Euclidean space

The finite-dimensional space $ℝ^{n}$ with the standard dot product is a simple example of a Hilbert space. Likewise, $ℂ^{n}$ with

$⟨ ϕ ∣ ψ ⟩ = \sum_{i = 1}^{n} ϕ_{i}^{*} ψ_{i}$

is the standard Hilbert space of finite-dimensional quantum systems.^[5]

Function spaces

An important infinite-dimensional example is the space $L^{2}$ of square-integrable functions. The inner product is

$⟨ ϕ ∣ ψ ⟩ = \int ϕ^{*} (x) ψ (x) d x .$

Wavefunctions in nonrelativistic quantum mechanics are elements of such spaces.^[6]

Sequence spaces

Another standard example is the space $ℓ^{2}$ of square-summable sequences.

Basis and expansion

A Hilbert space may have an orthonormal basis ${e_{n}}$ , meaning

$⟨ e_{m} ∣ e_{n} ⟩ = δ_{m n} .$

Any vector can be expanded as

$ψ = \sum_{n} c_{n} e_{n} .$

These expansions generalize Fourier series.^[7]

Hilbert space in quantum mechanics

Hilbert space provides the formal setting for quantum states.^[8]

The probability amplitude between two states is

$⟨ ϕ ∣ ψ ⟩$

and probabilities are given by its squared magnitude.

Operators and observables

Physical quantities are represented by operators acting on Hilbert space.^[9]

Observables correspond to self-adjoint operators with real eigenvalues.^[10]

The expectation value is

$⟨ \hat{A} ⟩ = ⟨ ψ ∣ \hat{A} ∣ ψ ⟩ .$

Commutators

The commutator

$[\hat{x}, \hat{p}] = i ℏ$

leads to the uncertainty principle.^[11]

Spectral theorem

The spectral theorem decomposes self-adjoint operators into projection operators.^[12]

$\hat{A} = \sum_{n} a_{n} P_{n}$

This provides the mathematical basis for quantum measurement.

Density matrices

A general quantum state is described by a density operator $ρ$ .^[13]

$⟨ A ⟩ = T r (ρ A)$

Pure states satisfy $ρ^{2} = ρ$ , while mixed states satisfy $T r (ρ^{2}) < 1$ .

Table of contents (139 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

Matter (by scale)

Matter (by scale)

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

Methods and tools

Methods and tools

Full contents

1. Foundations (7)↑

2. Conceptual and interpretations (10)↑

3. Mathematical structure and systems (11)↑

4. Atomic and spectroscopy (11)↑

Physics:Quantum Atomic structure and spectroscopy
Physics:Quantum Hydrogen atom
Physics:Quantum Multi-electron atoms
Physics:Quantum Fine structure
Physics:Quantum Hyperfine structure
Physics:Quantum Isotopic shift
Physics:Quantum Zeeman effect
Physics:Quantum Stark effect
Physics:Quantum Spectral lines and series
Physics:Quantum Selection rules
Physics:Quantum Fermi's golden rule

5. Wavefunctions and modes (8)↑

Physics:Quantum Wavefunction
Physics:Quantum Superposition principle
Physics:Quantum Eigenstates and eigenvalues
Physics:Quantum Boundary conditions and quantization
Physics:Quantum Standing waves and modes
Physics:Quantum Normal modes and field quantization
Physics:Number of independent spatial modes in a spherical volume
Physics:Quantum Density of states

6. Quantum dynamics and evolution (9)↑

7. Measurement and information (6)↑

8. Quantum information and computing (6)↑

9. Quantum optics and experiments (4)↑

Physics:Quantum Nonlinear King plot anomaly in calcium isotope spectroscopy
Physics:Quantum optics beam splitter experiments
Physics:Quantum Ultra fast lasers
Physics:Quantum Experimental quantum physics

Template:Quantum optics operators

10. Open quantum systems (7)↑

11. Quantum field theory (18)↑

12. Statistical mechanics and kinetic theory (10)↑

13. Condensed matter and solid-state physics (7)↑

14. Plasma and fusion physics (10)↑

Physics:Quantum Fusion
Physics:Quantum Fusion reactions and Lawson criterion
Physics:Quantum Plasma (fusion context)
Physics:Quantum Magnetic confinement fusion
Physics:Quantum Inertial confinement fusion
Physics:Quantum Plasma instabilities and turbulence
Physics:Quantum Tokamak
Physics:Quantum Tokamak core plasma
Physics:Quantum Tokamak edge physics and recycling asymmetries
Physics:Quantum Stellarator

15. Timeline (8)↑

16. Advanced and frontier topics (6)↑

References

↑ Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press. ISBN 9780125850506.
↑ Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press. https://archive.org/details/principlesofquan0000dira.
↑ Hall, Brian C. (2013). Quantum Theory for Mathematicians. Springer. https://link.springer.com/book/10.1007/978-1-4614-7116-5.
↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. https://doi.org/10.1017/CBO9780511976667.
↑ Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.
↑ Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics. Academic Press.
↑ von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press. https://archive.org/details/mathematicalfoun0000vonn.
↑ Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press.
↑ Hall, Brian C. (2013). Quantum Theory for Mathematicians. Springer.
↑ Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.
↑ Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics. Academic Press.
↑ Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

Author: Harold Foppele

5.00

(one vote)

[1] Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis. Academic Press. ISBN 9780125850506.

[2] Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press. https://archive.org/details/principlesofquan0000dira.

[3] Hall, Brian C. (2013). Quantum Theory for Mathematicians. Springer. https://link.springer.com/book/10.1007/978-1-4614-7116-5.

[4] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.

[5] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press. https://doi.org/10.1017/CBO9780511976667.

[6] Sakurai, J. J. (1994). Modern Quantum Mechanics. Addison-Wesley.

[7] Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics. Academic Press.

[8] von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press. https://archive.org/details/mathematicalfoun0000vonn.

[9] Dirac, P. A. M. (1930). The Principles of Quantum Mechanics. Oxford University Press.

[10] Hall, Brian C. (2013). Quantum Theory for Mathematicians. Springer.

[11] Griffiths, David J. (2018). Introduction to Quantum Mechanics. Pearson.

[12] Reed, Michael; Simon, Barry (1980). Methods of Modern Mathematical Physics. Academic Press.

[13] Nielsen, Michael A.; Chuang, Isaac L. (2010). Quantum Computation and Quantum Information. Cambridge University Press.

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