Hilbert space

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.

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 $T r (ρ^{2}) < 1$ .

↑ 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

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[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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Physics:Quantum Mathematical Foundations of Quantum Theory

Hilbert space

Definition

Geometric interpretation

Examples

Euclidean space

Function spaces

Sequence spaces

Basis and expansion

Hilbert space in quantum mechanics

Operators and observables

Commutators

Spectral theorem

Density matrices

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