Physics:Quantum Hilbert space

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Hilbert space is a real or complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of finite-dimensional Euclidean space to possibly infinite-dimensional settings. The inner product extends the familiar dot product, making it possible to define length, angle, and orthogonality, while completeness guarantees that limits of Cauchy sequences remain inside the space.

The state of a vibrating string can be modeled as a point in a Hilbert space. Its decomposition into distinct overtones corresponds to orthogonal projection onto coordinate directions.

Development

Hilbert spaces were developed in the early 20th century through the work of David Hilbert, Erhard Schmidt, and Frigyes Riesz, and later placed in an abstract setting by Biography:John von Neumann. They became central in functional analysis, Fourier analysis, the theory of partial differential equations, and the mathematical formulation of quantum mechanics.

A Hilbert space is in particular a Banach space, but with additional geometric structure coming from the inner product. This makes possible such notions as orthogonal projection, orthonormal bases, and Fourier expansion, all of which extend familiar geometric ideas from finite-dimensional vector spaces to infinite-dimensional ones.

Definition

Let $H$ be a complex vector space. An inner product on $H$ is a function assigning to each pair $x,y\in H$ a complex number $⟨x,y⟩$ such that:

1. $⟨y,x⟩=\overline{⟨x,y⟩}$ (conjugate symmetry),
2. $⟨a{x}_1+b{x}_2,y⟩=a⟨x_1,y⟩+b⟨x_2,y⟩$ (linearity in the first argument),
3. $⟨x,x⟩\ge 0$, with equality if and only if $x=0$ (positive definiteness).

It follows that the inner product is conjugate-linear in the second argument: $$⟨x,a{y}_1+b{y}_2⟩=\bar{a}⟨x,y_1⟩+\bar{b}⟨x,y_2⟩.$$

Inner product

The inner product induces a norm $$\Vert x\Vert =\sqrt{⟨x,x⟩}$$ and hence a distance function $$d(x,y)=\Vert x-y\Vert .$$ This turns $H$ into a metric space.

A Hilbert space is an inner product space that is complete with respect to this norm, meaning that every Cauchy sequence converges to an element of the space.

Basic geometric properties

The norm and inner product satisfy the Cauchy–Schwarz inequality $$|⟨x,y⟩|\le \Vert x\Vert \Vert y\Vert ,$$

If $u$ and $v$ are orthogonal, then the Pythagorean theorem holds: $$\Vert u+v{\Vert }^2=\Vert u{\Vert }^2+\Vert v{\Vert }^2.$$ More generally, Hilbert spaces satisfy the parallelogram law: $$\Vert u+v{\Vert }^2+\Vert u-v{\Vert }^2=2(\Vert u{\Vert }^2+\Vert v{\Vert }^2).$$

Important

One of the most important consequences of completeness is the existence of orthogonal projection onto closed subspaces: if $V$ is a closed linear subspace of a Hilbert space $H$, then every element $x\in H$ can be written uniquely as $$x=v+w$$ with $v\in V$ and $w\in V^{\perp }$.

Standard examples

A basic finite-dimensional example is ${\mathbf{𝐑}}^3$ with the usual dot product $$\left(\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right)\cdot \left(\begin{array}{c}{y}_1\\ y_2\\ y_3\end{array}\right)=x_1{y}_1+x_2{y}_2+x_3{y}_3.$$

Every finite-dimensional inner product space is automatically complete, and hence is a Hilbert space.

A fundamental infinite-dimensional example is the sequence space ${\ell }^2$, consisting of all sequences $(z_1,z_2,\dots )$ such that $$\sum _{n=1}^{\mathrm{\infty }}|z_n{|}^2<\mathrm{\infty }.$$ Its inner product is $$⟨z,w⟩=\sum _{n=1}^{\mathrm{\infty }}{z}_n\overline{w_n}.$$

Major class

Another major class is formed by Lebesgue spaces $L^2(X,\mu )$, where $$\underset{X}{\int }|f{|}^2\mathrm{d}\mu <\mathrm{\infty }.$$ The inner product is $$⟨f,g⟩=\underset{X}{\int }f(t)\overline{g(t)}\mathrm{d}\mu (t).$$

Role in analysis and physics

Hilbert spaces provide the natural setting for Fourier series and Fourier transforms, since orthonormal bases allow functions to be expanded into convergent series of coefficients.

They are equally central in the weak formulation of partial differential equations, especially through Sobolev spaces and the Lax–Milgram theorem.

In modern quantum mechanics, pure states are represented by unit vectors in a complex Hilbert space, and observables by self-adjoint operators acting on that space.

History

The theory emerged from work on integral equations and orthogonal expansions. David Hilbert and Erhard Schmidt studied integral operators and eigenfunction expansions, while Frigyes Riesz and Ernst Otto Fischer proved the completeness of $L^2$ spaces, establishing one of the first genuine infinite-dimensional Hilbert spaces.

The abstract concept was clarified by Biography:John von Neumann, who introduced the term Hilbert space and made it central to operator theory and quantum theory.

References

Author: Harold Foppele