Hilbert space Definition and 236 Threads

Hello, I have some troubles understanding Hilbert representations for the standard free quantum particle On the one hand, we can represent Heisenberg algebra [Xi,Pj]= i delta ij on the space of square integrable functions on, say, R^3, with the X operator represented as multiplication and P...

Hello, I am having a question regarding how eigenfunctions are orthogonal in Hilbert space, or what does that even mean (other than the inner product is zero). I mean, I know in ##\mathbb {R^{3}}##, vectors are orthogonal when they are right angles to each other. However, how can functions be...

This is one of those "existential doubts" that most likely have a trivial solution which I can't see. Veltman says in the Diagrammatica book: Although the reasoning makes perfect sense for a Hilbert space spanned by momentum states, intuitively it doesn't make sense to me, because a...

When we talk about a two-state quantum system being a two-dimensional complex Hilbert space are we implicitly considering the "existence of time"? Why is all this additional structure (of a two-dimensional complex Hilbert space) necessary if, even with a full quantum mechanical perspective...

Hello, Maybe it's a silly question, but why the space ##L^2[a,b]## has always to have bounded limits? Why can't we define the space of functions ##f(x)## where ##x \in \mathbb{R}## and ##\int_{-\infty}^\infty |f(x)|^2 dx \le M## for some ## M \in \mathbb{R^+}##? As far as I know the sum of two...

Hi, I am reading the paper http://arxiv.org/abs/quant-ph/0502053 listed in the reference of Wikipedia Rigged Hilbert Space. I have a question about the relation, Φ ⊂ H ⊂ Φ', where H is Hilbert space, Φ is its subspace and Φ' is dual space of Φ. Φ⊂H and Φ⊂Φ' are obvious. How can we say H ⊂...

According to Wikipedia:http://en.wikipedia.org/wiki/Hilbert_space the inner product \langle x | y \rangle is linear in the first argument and anti-linear in the second argument. That is: \langle \lambda_1 x_1 + \lambda_2 x_2 | y \rangle = \lambda_1 \langle x_1 | y \rangle + \lambda_2 \langle...

I've started few days ago to study quantum physics, and there's a thing which isn't clear to me. I know that a quantum state is represented by a ray in a Hilbert space (so that ##k \left| X \right\rangle## is the same state of ##\left| X \right\rangle##). Suppose now to have these three states...

Can anyone please explain to me what is the Ising model, Hilbert space, and Hamiltonian ? However, please explain it as simple as possible because I am a freshman. I have looked up all three things. I've tried my best to make some sense of it, but I am, honestly, still confused on what any of...

Hi All, AFAIK, the key property that separates/differentiates a Hilbert Space H from your generic normed inner-product vector space is that , in H, the norm is generated by an inner-product, i.e., for every vector ##v##, we have## ||v||_H= <v,v>_H^{1/2} ##, and a generalized version of the...

Let \text{ } H \text{ }be \text{ }a \text{ } Hilbert \text{ } space, \text{ }K \text{ }be \text{ }a \text{ }closed\text{ }convex\text{ }subset \text{ } of \text{ }H \text{ }and \text{ }x_{0}\in K. \text{ }Then \\N_{K}(x_{0}) =\{y\in K:\langle y,x-x_{0}\rangle \leq 0,\forall x\in K\} .\text{...

I have started reading formal definitions of Hilbert Spaces. I don't understand the requirement of separability postulate. I have proved that it leads to count ability of basis but again why is that required at first place.

Hello all! I have the following question with regards to quantum mechanics. If ##H## is a Hilbert space with a countably-infinite orthonormal basis ##\{ \left | n \right \rangle \}_{n \ \in \ \mathbb{N} }##, and two operators ##R## and ##L## on ##H## are defined by their action on the basis...

So lately I've been thinking about whether or not it'd be possible to have the commutation relation [x,p]=i \hbar in a Hilbert Space of finite dimension d. Initially, I was trying to construct a lattice universe and a translation operator that takes a particle from one lattice point to the...

Maybe someone here can explain me something I never understood in QM: The wave function lives in the Hilbert space spanned by the measurement operator. Is there any mathematical relation of those spaces with each other?

Hi, if ket is 2+3i , than its bra is 2-3i , my question is 2+3i is in Hilbert space than 2-3i can be represented in same hilbert space, but in books it is written we need dual Hilbert space for bra?

Hello all, I'm working through the following paper on topological quantum computing. http://www.qip2010.ethz.ch/tutorialprogram/JiannisPachosLecture In particular I'm trying to derive and solve the pentagon equation in order to evaluate the F matrices for ising anyons. One thing I'm trying...

My book says that "the countability of the ONS in a hilbert space H entails that H can be represented as closure of the span of countably many elements". I must admit my english is probably not that good. At least the above quote does not make sense to me. What is it trying to say? Previously...

Let H be a Hilbert space. Let F be a subset of H. F is dense in H if: <f,h>=0 for all f in F => h=0 Now take an orthonormal set (ek) (this is a countable sequence indexed by k) in H. My book says that obviously: \bigcupspan(ek) is dense in H (the union runs over all k) => g=Ʃ<g,ek>ek Now...

Homework Statement Find the spectrum of the Momentum operator in the Hilbert Space defined by L^2([-L,L]), consisting of all square integrable functions ψ(x) in the range -L, to L Homework Equations We can get the resolvent set containting all λ in ℂ such that you can always find a...

Suppose that H, K are Hilbert spaces, and A : H -> K is a bounded linear operator and an isomorphism. If X is a dense set in H, then is A(X) a dense set in K? Any references to texts would also be helpful.

For example, I have a 3D particle that experiences a harmonic oscillator potential only in the X,Y plane for all Z ie. a free particle in the Z direction. This seems like cylindrical coordinates but I'm not sure how to express the Hilbert space if I want to be able to describe states and...

Homework Statement Consider the states with the quantum numbers n = l = 1 and s = 1/2 Let J = L + S What is the dimension of the Hilbert space to describe all states with these quantum numbers? Homework Equations The Attempt at a Solution I believe the dimension of the Hilbert...

(All that follows assumes we are talking about a self-adjoint operator A on a Hilbert space \mathscr H.) The first volume of Reed-Simon defines \mathscr H_{\rm pp} = \left\{ \psi \in \mathscr H: \mu_\psi \text{ is pure point} \right\}. The book seems to take for granted that \mathscr H_{\rm...

One of the most important results of functional analysis is that for every bounded linear functional f: H → ℂ on a Hilbert space H, there exists a fixed |v> in H such that f(|u>) is equal to the inner product of |v> with |u> for all |u> in H. This justifies the labeling of f as <v| in the...

also see http://planetmath.org/exampleofnonseparablehilbertspace the main difficulty is about the completeness, which is hard to prove, the author's hint seems don't work here, for you can not use the monotone convergence theorem directly , f(x)χ[-N,N]/sqrt[N] is not monotone

Hi, Let H = \{(x_n)_n \subseteq \mathbb{R} | \sum_{n=1}^{\infty} x_n < \infty \} and for $(x_n)_n \in H$ define $$\|(x_n)_n\|_H = \sup_{n} \left|\sum_{k=0}^{n} x_k \right|$$ Prove that $H$ is complete. Is $H$ a Hilbert space? What is the best way to prove $H$ is complete? To prove it's a...

in quantum mechanics we have something called hilbert space. What does the dimensions of this space represent for that system? also is ψ(x) same as |ψ> in the dirac notation?

Hi all, I have a question about the concept of complete set when I apply the perturbation theory in two situations -Finite Hilbert Space and Infinite Hilbert Space. Consider a Hamiltonian H=H0+H', where H0 is the unperturbed Hamiltonian and H' is the perturbed Hamiltonian. Let ψ_n be the...

I am wondering, what is the dimension of a ray in a Hilbert space? For example here (page 2, bottom of page) I have read: I understand why a state is represented by all multiples of a vector, not just the vector. But is the ray really one-dimensional? It would be one-dimensional if we multiply...

Homework Statement let \ell^{2} denote the space of sequences of real numbers \left\{a_{n}\right\}^{\infty}_{1} such that \sum_{1 \leq n < \infty } a_{n}^{2} < \infty a) Verify that \left\langle \left\{a_{n}\right\}^{\infty}_{1}, \left\{b_{n}\right\}^{\infty}_{1} \right\rangle =...

I've been taught (in the context of Sturm-Liouville problems) that Fourier series can be explained using inner products and the idea of projection onto eigenfunctions in a Hilbert space. In those cases, the eigenvalues are infinite, but discrete. I'm now taking a quantum mechanics course, and...

Is something wrong in my assertions below? Suppose we have two quantum systems N and X. Let N is described by discrete observable \hat{n} (bounded s.a. operator with discrete infinite spectrum) with eigenvectors |n\rangle. Let X is described by continuous observable \hat{x} (unbounded s.a...

Suppose that we have rigged Gilbert space Ω\subsetH\subsetΩ\times (H is infinite-dimensional and separable). Is the Ω a separable space? Is the Ω\times a separable space? Consider the complete set of commuting observables (CSCO) which contain both bounded and unbounded operators...

The expansion theorem in quantum mechanics states that a general state of a system can be represented by a unique linear combination of the eigenstates of any Hermitian operator. If that's the case then that would imply we would be able to represent the spin state of a particle in terms of...

Hello, I'm reading Gaussian measures on Hilbert spaces by S. Maniglia (available via google) and I have the following issue, regarding the proof. He states in Lemma 1.1.4: Let μ be a finite Borel measure on H. Then the following assertions are equivalent: (1) \int_H |x|^2 \mu(dx) < \infty (2)...

Homework Statement unitary operators on hilbert space Homework Equations is there a unitary operator on a (finite or infinite) Hilbert space so that cU(x)=y, for some constant (real or complex), where x and y are fixed non-zero elements in H ? The Attempt at a Solution I know the...

In Griffith's intro to QM it says on page 95 (in footnote 6) : "In Hilbert space two functions that have the same square integral are considered equivalent. Technically, vectors in Hilbert space represent equivalence classes of functions." But that means that if we take for example...

Square integrable functions -- Hilbert space and light on Dirac Notation I started off with Zettilis Quantum Mechanics ... after being half way through D.Griffiths ... Now Zettilis precisely defines what a Hibert space is and it includes the Cauchy sequence and convergence of the same ... is...

Homework Statement Let H be an \infty-dimensional Hilbert space and T:H\to{H} be an operator. Show that if T is compact, bounded and has closed range, then T has finite rank. Do not use the open-mapping theorem. Let B(H) denote the space of all bounded operators mapping H\to{H}, K(H) denote...

Note: I am NOT talking about the classical limit of quantum mechanics, where in the limit of numbers that are large compared to h the average values approach the classical values, nor am I talking about Lagrangin/Hamiltonian mechanics in phase space; I am talking about using vectors with...

Definitions of a rigged Hilbert space typically talk about the "dual space" of a certain dense subspace of a given Hilbert space H. Do they mean the algebraic or the continuous dual space (continuous wrt the norm topology on H)?

What is the difference between Rigid Hilbert Space and Hilbert Space?

If we assume the inner product is linear in the second argument, the polarization identity reads (x,y) = \frac 14 \| x + y \|^2 - \frac 14 \| x - y \|^2 - \frac i4 \|x + iy\|^2 + \frac i4 \| x - iy \|^2. But there is another identity that I've seen referred to in some texts as the...

So one of the postulate of quantum mechanics is that observables have complete eigenfunctions. Can someone let me know if I am understanding this properly: Basically you postulate for example, position kets |x> such that any state can be represented by a linear combination of these states...

Homework Statement Let S be a linearly independent subset of a Hilbert space. Prove that span(S) is a subspace, that is a linear manifold and a closed set, if and only if S is finite. Homework Equations The Attempt at a Solution Assuming S is finite means that S is a closed set...

Hi everyone. Many texts when describing QFT start immediately discussing about free field theories, Fock spaces etc.. I want to understand general properties of the Hilbert space, and how to find a basis of it, and how to find a particle interpretation. I know there are very mathematical...

I never thought about this stuff much before, but I am getting confused by a couple of things. For example, would the state ∣E,l,m⟩ be the tensor product of ∣E⟩, ∣l⟩, ∣m⟩, ie. ∣E⟩∣l⟩∣m⟩? I always just looked at this as a way to keep track of operators that had simultaneous eigenfunctions in a...

I never thought about this stuff much before, but I am getting confused by a couple of things. For example, would the state ∣E,l,m⟩ be the tensor product of ∣E⟩, ∣l⟩, ∣m⟩, ie. ∣E⟩∣l⟩∣m⟩? I always just looked at this as a way to keep track of operators that had simultaneous eigenfunctions in a...

Homework Statement P_{2} \subset L_{2} is the set of all polynomials of degree n \leq 2. Complete the following approximation. In other words find the polynomial of degree 2 that minimises the following expression: \int \left|cos(\frac{\pi t}{2}) - p(t)\right|^{2}dt = min with -1 <= t >= 1...