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{{Quantum book backlink|Foundations}}
A '''Hilbert space''' is a [[Wikipedia:Vector space|vector space]] equipped with an [[Wikipedia:Inner product space|inner product]] and complete with respect to the norm induced by that inner product.<ref>{{cite book
|last=Reed
|first=Michael
|last2=Simon
|first2=Barry
|title=Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis
|publisher=Academic Press
|year=1980
|isbn=9780125850506
}}</ref> It provides the natural mathematical setting for [[Wikipedia:Quantum mechanics|quantum mechanics]], where physical states are represented by vectors or, more precisely, by rays in a complex Hilbert space.<ref>{{cite book |last=Dirac |first=P. A. M. |title=The Principles of Quantum Mechanics |publisher=Oxford University Press |year=1930 |url=https://archive.org/details/principlesofquan0000dira}}</ref> The geometry of Hilbert space generalizes the familiar Euclidean notions of length, angle, orthogonality, and projection to spaces of finite or infinite dimension.
'''[[File:Foundations of Quantum Theory-1.jpg|thumb|350px|Quantum Mathematics: Foundations of Quantum Theory Visualized]]'''

== Definition ==

A Hilbert space <math>\mathcal{H}</math> is a vector space over <math>\mathbb{R}</math> or <math>\mathbb{C}</math> together with an inner product

<math>
\langle \phi \mid \psi \rangle
</math>

such that the induced norm

<math>
\|\psi\| = \sqrt{\langle \psi \mid \psi \rangle}
</math>

makes <math>\mathcal{H}</math> a complete metric space.<ref>{{cite book |last=Hall |first=Brian C. |title=Quantum Theory for Mathematicians |publisher=Springer |year=2013 |url=https://link.springer.com/book/10.1007/978-1-4614-7116-5}}</ref>

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

<math>
\langle \psi \mid \psi \rangle = 1.
</math>

== 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 <math>\langle \phi \mid \psi \rangle = 0</math>;
* the ''projection'' of one vector onto another or onto a subspace.

These ideas are central in quantum mechanics.<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018}}</ref>

== Examples ==

=== Euclidean space ===

The finite-dimensional space <math>\mathbb{R}^n</math> with the standard dot product is a simple example of a Hilbert space. Likewise, <math>\mathbb{C}^n</math> with

<math>
\langle \phi \mid \psi \rangle = \sum_{i=1}^{n} \phi_i^* \psi_i
</math>

is the standard Hilbert space of finite-dimensional quantum systems.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010 |url=https://doi.org/10.1017/CBO9780511976667}}</ref>

=== Function spaces ===

An important infinite-dimensional example is the space <math>L^2</math> of square-integrable functions. The inner product is

<math>
\langle \phi \mid \psi \rangle = \int \phi^*(x)\psi(x)\,dx.
</math>

Wavefunctions in nonrelativistic quantum mechanics are elements of such spaces.<ref>{{cite book |last=Sakurai |first=J. J. |title=Modern Quantum Mechanics |publisher=Addison-Wesley |year=1994}}</ref>

=== Sequence spaces ===

Another standard example is the space <math>\ell^2</math> of square-summable sequences.

== Basis and expansion ==

A Hilbert space may have an orthonormal basis <math>\{e_n\}</math>, meaning

<math>
\langle e_m \mid e_n \rangle = \delta_{mn}.
</math>

Any vector can be expanded as

<math>
\psi = \sum_n c_n e_n.
</math>

These expansions generalize Fourier series.<ref>{{cite book |last=Reed |first=Michael |last2=Simon |first2=Barry |title=Methods of Modern Mathematical Physics |publisher=Academic Press |year=1980}}</ref>

[[File:Fourier series and transform.gif|thumb|Orthonormal expansions in Hilbert space.]]

== Hilbert space in quantum mechanics ==

Hilbert space provides the formal setting for quantum states.<ref>{{cite book |last=von Neumann |first=John |title=Mathematical Foundations of Quantum Mechanics |publisher=Princeton University Press |year=1955 |url=https://archive.org/details/mathematicalfoun0000vonn}}</ref>

The probability amplitude between two states is

<math>
\langle \phi \mid \psi \rangle
</math>

and probabilities are given by its squared magnitude.

=Operators and observables=

Physical quantities are represented by operators acting on Hilbert space.<ref>{{cite book |last=Dirac |first=P. A. M. |title=The Principles of Quantum Mechanics |publisher=Oxford University Press |year=1930}}</ref>

Observables correspond to self-adjoint operators with real eigenvalues.<ref>{{cite book |last=Hall |first=Brian C. |title=Quantum Theory for Mathematicians |publisher=Springer |year=2013}}</ref>

The expectation value is

<math>
\langle \hat{A} \rangle = \langle \psi \mid \hat{A} \mid \psi \rangle.
</math>

== Commutators ==

The commutator

<math>
[\hat{x},\hat{p}] = i\hbar
</math>

leads to the [[Wikipedia:Uncertainty principle|uncertainty principle]].<ref>{{cite book |last=Griffiths |first=David J. |title=Introduction to Quantum Mechanics |publisher=Pearson |year=2018}}</ref>

=Spectral theorem=

The spectral theorem decomposes self-adjoint operators into projection operators.<ref>{{cite book |last=Reed |first=Michael |last2=Simon |first2=Barry |title=Methods of Modern Mathematical Physics |publisher=Academic Press |year=1980}}</ref>

<math>
\hat{A} = \sum_n a_n P_n
</math>

This provides the mathematical basis for quantum measurement.

=Density matrices=

A general quantum state is described by a density operator <math>\rho</math>.<ref>{{cite book |last=Nielsen |first=Michael A. |last2=Chuang |first2=Isaac L. |title=Quantum Computation and Quantum Information |publisher=Cambridge University Press |year=2010}}</ref>

<math>
\langle A \rangle = \mathrm{Tr}(\rho A)
</math>

Pure states satisfy <math>\rho^2 = \rho</math>, while mixed states satisfy <math>\mathrm{Tr}(\rho^2) < 1</math>.
=See also=
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}

=References=
{{reflist|3}}

{{Author|Harold Foppele}}

{{Sourceattribution|Quantum Mathematical Foundations of Quantum Theory|1}}

Physics:Quantum Mathematical Foundations of Quantum Theory - Revision history

imported>WikiHarold: Repair Quantum Collection B backlink template

imported>WikiHarold: Repair Quantum Collection B backlink template