What is Hilbert space: Definition and 229 Discussions

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is a vector space equipped with an inner product, an operation that allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used.
Hilbert spaces arise naturally and frequently in mathematics and physics, typically as infinite-dimensional function spaces. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications. The success of Hilbert space methods ushered in a very fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, and Hardy spaces of holomorphic functions.
Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace (the analog of "dropping the altitude" of a triangle) plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes (an orthonormal basis), in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, the Hilbert space can also be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is often in the older literature referred to as the Hilbert space. Linear operators on a Hilbert space are likewise fairly concrete objects: in good cases, they are simply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectrum.

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  1. cianfa72

    I Representation of Spin 1/2 quantum state

    Hi, I'm aware of the wave function ##\Psi## of a quantum system represents basically the "continuous components" of a quantum state (a point/vector in the infinite-dimension Hilbert space) in a basis. If we take the ##\delta(x - \bar x)## eigenfunctions as basis on Hilbert space then the wave...
  2. nomadreid

    I Basic question about Bra-Ket notation

    This is so basic as to be embarrassing, but I haven't figured out my misunderstanding of some basic notation. [1] If v and w are two vectors in a Hilbert space, then <v|w> is interpreted as the probability amplitude of w collapsing into v. [2] However, if P is a projection, then <v|Pv>...
  3. non_physicist

    What can AI do and not do in physics currently?

    What can AI do, and not do, in physics currently? Can it navigate hilbert space (I don't know what this is, just coming from an HPS undergrad background)? Can it design atomic bombs? Has it solved any problems? I suppose much research uses AI. In what forms?
  4. ARoyC

    I Tensor Product of Two Hilbert Spaces

    How to prove that the tensor product of two same-dimensional Hilbert spaces is also a Hilbert space? I understand that I need to prove the Cauchy Completeness of the new Hilbert space. I am stuck in the middle.
  5. lsdragon

    A How to derive the quantum detailed balance condition?

    In the "On The detailed balance conditions for non-Hamiltonian systems", I learned that for a Markov open quantum system to satisfying the master equation with the Liouvillian superoperators, the detailed balance condition will be > Definition 2: The open quantum Markovian system...
  6. Euge

    POTW Weak Topology on an Infinite-Dimensional Hilbert Space

    Prove that the weak topology on an infinite-dimensional Hilbert space is non-metrizable.
  7. S

    I Operators in finite dimension Hilbert space

    I have a question about operators in finite dimension Hilbert space. I will describe the context before asking the question. Assume we have two quantum states | \Psi_{1} \rangle and | \Psi_{2} \rangle . Both of the quantum states are elements of the Hilbert space, thus | \Psi_{1} \rangle , |...
  8. A

    A An ab initio Hilbert space formulation of Lagrangian mechanics

    I want to share my recent results on the foundation of classical mechanics. Te abstract readWe construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, several concepts from analytical mechanics will be rediscovered...
  9. P

    A About the rigged Hilbert space in QM

    In Quantum Mechanics, how can you justify the use of distributions like the delta functional without introducing a rigged Hilbert space? I see that some texts do not make any reference to this subject.
  10. einheit

    I Are all (pure) states physically realizable?

    To elaborate that summary a bit, suppose ##\mathcal H## is the Hilbert space of the particle, with ##\mathcal{H}_2\subseteq\mathcal{H}## its two-dimensional spin subspace. Now consider any ##|x\rangle\in\mathcal{H}## such that ##|x\rangle\perp\mathcal{H}_2##, i.e., ##\forall ~...
  11. D

    A Fidelity for quantum state at t=0

    fidelity for pure state with respect to t=0 is 1. My teacher told me this. But I am not getting this. This is my detailed question the initial state(t=0)##|\psi\rangle=|\alpha\rangle|0\rangle## the final state (t) ##|\chi\rangle= |i\alpha\sin(t)\rangle|cos(t)\alpha\rangle## Fidelity between the...
  12. Giulio Prisco

    A Fundamental reality: Hilbert space

    What do you guys think of this soberly elegant proposal by Sean Carroll? Reality as a Vector in Hilbert Space Fundamental reality lives in Hilbert space and everything else (space, fields, particles...) is emergent. Seems to me a step in the right conceptual direction.
  13. U

    Questions on field operator in QFT and interpretations

    For a real scalar field, I have the following expression for the field operator in momentum space. $$\tilde{\phi}(t,\vec{k})=\frac{1}{\sqrt{2\omega}}\left(a_{\vec{k}}e^{-i\omega t}+a^{\dagger}_{-\vec{k}}e^{i\omega t}\right)$$ Why is it that I can discard the phase factors to produce the time...
  14. J

    A Do we really need the Hilbert space for Quantum Mechanics?

    Let's play this game, let's assume the infinite Hilbert Space, the operators and all the modern machinery introduced by Von Neuman were not allowed. How would be the formalism? Thanks
  15. snypehype46

    I Functor between the category of Hilbert Space and the category of sets

    I have a question that is related to categories and physics. I was reading this paper by John Baez in which he describes a TQFT as a functor from the category nCob (n-dimensional cobordisms) to Vector spaces. https://arxiv.org/pdf/quant-ph/0404040.pdf. At the beginning of the paper @john baez...
  16. J

    B Are subspaces of Hilbert space real?

    When orthogonal states of a quantum system is projected into subspaces A and B are A and B real spaces?
  17. Decimal

    I Completeness relations in a tensor product Hilbert space

    Hello, Throughout my undergrad I have gotten maybe too comfortable with using Dirac notation without much second thought, and I am feeling that now in grad school I am seeing some holes in my knowledge. The specific context where I am encountering this issue currently is in scattering theory...
  18. L

    A Integrability along a Hilbert space?

    Suppose we have an infinite dimensional Hilbert-like space but that is incomplete, such as if a subspace isomorphic to ##\mathbb{R}## had countably many discontinuities and we extended it to an isomorphism of ##\mathbb{R}^{\infty}##. Is there a measure of integrating along any closed subset of...
  19. thaiqi

    I Dynamical System & Hilbert Space: Analyzing the Relationship

    Is there any relation between dynamical system and Hilbert space(functional analysis)?
  20. forkosh

    A Is the "op" lattice ##\mathscr{L_H}^\perp## also atomistic....?

    Let ##\mathscr{L_H}## be the usual lattice of subspaces of Hilbert space ##\mathscr{H}##, where for ##p,q\in\mathscr{H}## we write ##p\leq q## iff ##p## is a subspace of ##q##. Then, as discussed by, e.g., Beltrametti&Cassinelli https://books.google.com/books?id=yWoq_MRKAgcC&pg=PA98, this...
  21. nomadreid

    I Quantum logic based on closed Hilbert space subspaces

    One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps...
  22. forkosh

    A Exploring Basis Vector Relationships in Incompatible Propositions

    If propositions ##p,q\in{\mathscr L}_{\mathcal H}## (i.e., the lattice of subspaces of ##\mathcal H##) are incompatible, then ##\hat p\hat q\neq\hat q\hat p##. But since it's a lattice, there exists a unique glb ##p\wedge q=q\wedge p##. How are they mathematically related? In particular, I...
  23. J

    A Looking Under the Hood of Feynman Diagrams

    I'm currently working my way through Griffith's Elementary Particles text, and I'm looking to understand what's going on with the underlying Hilbert space of a system described using a Feynman diagram. I'm fairly well acquainted with non relativistic QM, but not much with QFT. In particular, I'd...
  24. TTT

    I Rigged Hilbert Space X: Eq (1) and (2)

    X=e+or-kx (1) <X(x)|Φ(x)>=∫-∞∞X*(x)Φ(x)dx (2) where Φ(x) satisfies the following. ∫-∞∞|Φ(x)|2(1+|x|)ndx is finte if n=0, 1, 2,...
  25. C

    I Question regarding a Free particle and Hilbert space (QM)

    In quantum mechanics, the Eigenfunction resulting from the Hamiltonian of a free particle in 1D system is $$ \phi = \frac{e^{ikx} }{\sqrt{2\pi} } $$ We know that a function $$ f(x) $$ belongs to Hilbert space if it satisfies $$ \int_{-\infty}^{+\infty} |f(x)|^2 dx < \infty $$ But since the...
  26. nomadreid

    I Quantum states: only vectors?

    Elementary question: Is there ever a case where the solutions for a wave equation turn out not to be a vector (in Hilbert space of infinite complex-valued dimensions, or a restriction to a subspace thereof) , but something else -- say, (higher-order) tensors or bivectors, or some such? My...
  27. A

    A Recent paper on QED using finite-dimensional Hilbert space - validity?

    I've been struggling with a somewhat-recent paper by Charles Francis, "A construction of full QED using finite dimensional Hilbert space," available here: https://arxiv.org/pdf/gr-qc/0605127.pdf But also published in...
  28. S

    A Is there any theory that can be modeled in any type of space?

    Is there any theory in physics that can be modeled in any type of space (Hilbert space, Euclidean, Non-Euclidean...etc)? And if yes, could that theory also contain/be compatible with all types of (physical) symmetries?
  29. W

    I Is quantum mechanics formulated from 1st principles?

    I was surprised recently to learn that one of the reasons both Newton and Einstein were so revolutionary was that they performed a neat mathematical trick. For Newton, it was the mathematical derivation of Kepler's laws from Newton's laws of gravitation and motion. For Einstein, it was the...
  30. Jd_duarte

    I Orthonormal Basis of Wavefunctions in Hilbert Space

    Hello, I've a fundamental question that seems to keep myself confused about the mathematics of quantum mechanics. For simplicity sake I'll approach this in the discrete fashion. Consider the countable set of functions of Hilbert space, labeled by i\in \mathbb{N} . This set \left...
  31. LarryC

    Simultaneous Diagonalization for Two Self-Adjoint Operators

    (a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?
  32. Rabindranath

    A Lagrange multipliers on Banach spaces (in Dirac notation)

    I want to prove Cauchy–Schwarz' inequality, in Dirac notation, ##\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2##, using the Lagrange multiplier method, minimizing ##\left|\left<\psi\middle|\phi\right>\right|^2## subject to the...
  33. Decimal

    I Inner products on a Hilbert space

    Hello, I am taking a quantum mechanics course using the Griffiths textbook and encountering some confusion on the definition of inner products on eigenfunctions of hermitian operators. In chapter 3 the definition of inner products is explained as follows: $$ \langle f(x)| g(x) \rangle = \int...
  34. cromata

    I Complete sets and complete spaces

    -Definition of complete space: if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in converges in M. (and from this definition we can define Hilbert Space) -Definition of Hilbert space: A Hilbert space is a vector space with an...
  35. K

    B Infinite dimensional Hilbert Space

    Could someone tell me in what sense the following photo of Hilbert is a infinite dimensional Hilbert Space? It's shown in a pdf I'm reading. Perhaps I'm putting the chariot in front of the horses as one would say here in our country, by considering infinite as infinite dimensional?
  36. A

    Quantum state of system after measurement

    > Operator $$\hat{A}$$ has two normalized eigenstates $$\psi_1,\psi_2$$ with > eigenvalues $$\alpha_1,\alpha_2$$. Operator $$\hat{B}$$ has also two > normalized eigenstates $$\phi_1,\phi_2$$ with eigenvalues > $$\beta_1,\beta_2$$. Eigenstates satisfy: > $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$ >...
  37. M

    A Functions in a Hilbert space

    Hi PF! Given a function ##B## defined as $$B[f(x)]\equiv f''(x) + f(x) + const.$$ Evidently in order for this function to be in the real Hilbert space ##H## we know $$const. = -\frac{1}{x_1-x_0}\int_{x_0}^{x_1} (f''(x) + f(x))\,dx.$$ Can someone please explain why? I can elaborate further if...
  38. G

    2D subspace of a Hilbert space

    Homework Statement Have to read a paper and somewhere along the line it claims that for any distinct ## \ket{\phi_{0}}## and ##\ket{\phi_{1}}## we can choose a basis s.t. ## \ket{\phi_{0}}= \cos\frac{\theta}{2}\ket{0} + \sin\frac{\theta}{2}\ket{1}, \hspace{0.5cm} \ket{\phi_{1}}=...
  39. M

    A How to determine constant to be in Hilbert space

    Hi PF! I'm trying to solve an ODE through the Ritz method, which is to say approximate the solution through a series $$\Phi = \sum_{i=1}^N a_if_i,\\ f_i = \phi_i-d_i.$$ Here ##a_i## are constants to be determined and ##f_i## are prescribed functions, where ##\phi_i## is a function and ##d_i##...
  40. A

    I State Vectors vs. Wavefunctions

    Hi physicsforums, I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc. I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...
  41. A

    Mathematics behind Signal and Systems

    I am looking for a signal processing textbook that uses real, complex, and functional analysis with measure theory. In other words, mathematically rigorous signal processing. Specifically, I prefer the kind that takes time to review all the topics from mathematical analysis before jumping into...
  42. fresh_42

    Insights Hilbert Spaces And Their Relatives - Part II - Comments

    Greg Bernhardt submitted a new PF Insights post Hilbert Spaces And Their Relatives - Part II Continue reading the Original PF Insights Post.
  43. M

    I Why do these functions form complete orthogonal systems in the Hilbert space?

    Hi PF! A text states that the following two functions $$ \psi^o_k = \sin(\pi(k-1/2)x)\cosh(\pi(k-1/2)(z+h)): k\in\mathbb{N},\\ \psi^e_k = \cos(\pi kx)\cosh(\pi k(z+h)): k\in\mathbb{N} $$ each form complete orthogonal systems in two mutually orthogonal subspaces, which compose the Hilbert...
  44. Robert Shaw

    I Do we need a reference frame in Quantum Hilbert space?

    Entangled states are only separable relative to certain basis states. So does that mean that reference frames have importance beyond those in spacetime?
  45. fresh_42

    Insights Hilbert Spaces and Their Relatives - Comments

    Greg Bernhardt submitted a new PF Insights post Hilbert Spaces and Their Relatives Continue reading the Original PF Insights Post.
  46. M

    I Understanding Spin States in Hilbert Space

    Hello In our Quantum Mechanics lecture we have been discussing a simplified model of the Stern-Gerlach experiment. Let ##|+>## and ##|->## denote an electron that is "spin up" and "spin down" (with respect to ##\hat{z}##), respectively. Our professor then asserted that ##|+>## and ##|->## acted...
  47. SemM

    A Hilbert-adjoint operator vs self-adjoint operator

    Hi, while reading a comment by Dr Du, I looked up the definition of Hilbert adjoint operator, and it appears as the same as Hermitian operator: https://en.wikipedia.org/wiki/Hermitian_adjoint This is ok, as it implies that ##T^{*}T=TT^{*}##, however, it appears that self-adjointness is...
  48. SemM

    A What separates Hilbert space from other spaces?

    Hi, I have the impression that the special thing about Hilbert space for Quantum Mechanics is that it is simply an infinite space, which allows for infinitively integration and derivation of its elements, f(x), g(x), their linear combination, or any other complex function, given that the main...
  49. SemM

    I Paper on Hilbert spaces in QM

    Hi, I found this article very interesting, given the loads of question I have posted in this regard in the last months. I cannot recall where I got the link from, and if it came from Bill Hobba in some discussion, thanks Bill! If not, thanks anyway for your answers and contributions. Here is...
  50. amjad-sh

    I Hilbert space and conjugate of a wave function

    Take a wavefunction ##\psi## and let this wavefunction be a solution of Schroedinger equation,such that: ##i \hbar \frac{\partial \psi}{\partial t}=H\psi## The complex conjugate of this wavefunction will satisfy the "wrong-sign Schrodinger equation" and not the schrodinger equation,such that ##i...
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