Hilbert Definition and 293 Threads

I think this is right, but could someone confirm (or deny) this for me? While a particle like an electron - or a finite set of particles for that matter - is represented by a single normed vector in Hilbert space which is acted on by operators such as ones for energy, position and momentum...

Hi, everyone: I have been curious for a while about the similarity between the correlation function and an inner-product: Both take a pair of objects and spit out a number between -1 and 1, so it seems we could define a notion of orthogonality in a space of random variables, so...

What is the relation between Hilbert Space and space-time? Are the two disjoint or is there something relating the two?

Homework Statement Prove that for a linear set M a subset of Hilbert space, that the set perpendicular to the set perpendicular to M is equal to M iff M is closed. The Attempt at a Solution I already have my proof -- but what is an example of a linear subset of H that is not closed? I think...

Thant helped. thank you!

1. Problem description Let (e_n)_{n=1}^{\infty} be an orthonormal(ON) basis for H (Hilbert Space). Assume that (f_n)_{n=1}^{\infty} is an ON-sequence in H that satisfies \sum_{n=1}^{\infty} ||e_n-f_n|| < 1 . Show that (f_n)_{n=1}^{\infty} is an ON-basis for H. Homework Equations...

So I was recently learned that for some square integrable position wave-functions in Hilbert Space have the momentum function is not square integrable. Thus the momentum function are not in hilbert space. However, due to "Fourier's Trick" Dirac discovered for quantum mechanics, the momentum...

Homework Statement Show that \int {{f^*}(x)g(x) \cdot dx} is an inner product on the set of square-integrable complex functions. Homework Equations Schwarz inequality: \left| {\int {{f^*}(x)g(x) \cdot dx} } \right| \le \sqrt {\int {{{\left| {f(x)} \right|}^2} \cdot dx} \int {{{\left|...

I have a Hilbert space H; given a closed subspace U of H let PU denote the orthogonal projection onto U. I also have a lattice L of closed subspaces of H, such that for all U and U' in L, PU and PU' commute. The problem is to find an orthonormal basis B of H, such that for every element b of B...

Hello, I hope I am asking this in the right area of the forums. My teacher wrote the following formula down at our last meeting, and I was wondering if it was true ( \mathcal{H} is the infinite dimension separable Hilbert space): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H}...

Hello, I hope I am asking this in the right area of the forums. I wanted to ask if the following formula is true (assuming H is infinite dimensional and separable): \mathcal{K} (\mathcal{H}) \approx \mathcal{K} (\mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H} \oplus \mathcal{H})\approx...

Hi, I asked this question in the quantum physics forum https://www.physicsforums.com/showthread.php?t=406171 but (afaics) we could not figure out a proof. Let me start with a description of the problem in quantum mechanical terms and then try to translate it into a more rigorous mathematical...

Hilbert Space... Can somebody tell me the difference between H1 & H2 hilbert spaces?

Hi, I discussed this with some friends but we could not figure out a proof. Usually when considering bound states of the Schrödinger equation of a given potential V(x) one assumes that the wave function converges to zero for large x. One could argue that this is due to the requirement...

Homework Statement three distiguishable spin 1/2 particles interact via H = \lamda ( S_1 \cdot S_2 + S_2 \cdot S_3 + S_3 \cdot S_1 ) a) What is the demension of the hilbert space? b) Express H in terms of J^2 where J = S_1 + S_2 + S_3 c) I then need to find the energy and...

no idea what could be on the final list when it's released. I wonder what people here think: http://www.physorg.com/news190538259.html

this question is in reference to eq 3.9 and footnote 6 in griffith's intro to quantum mechanics consider a function f(x). the inner product <f|f> = int [ |f(x)|^2 dx] which is zero only* when f(x) = 0 only points to footnote 6, where Griffith points out: "what about a function that is...

Let H be a Hilbert space and let S be the set of linear operators on H. Is there a proper subset of S that is dense in S?

\begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} \ldots & \ldots & \frac{1}{n}\\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \ldots & \ldots & \frac{1}{n + 1}\\ \frac{1}{3} & \frac{1}{4} & \right \frac{1}{5}\ldots & \ldots & \frac{1}{n + 2}\\ \vdots & \vdots & \vdots & \ddots &...

I was wondering what is Hilbert Space exactly? I read the Wikipedia page, but its one of those situations u understand what your reading but don't full grasp the concept. I was just hoping someone could explain it to me.

In article #34 of a recent thread about Haag's theorem, i.e., https://www.physicsforums.com/showthread.php?t=334424&page=3 a point of view was mentioned which I'd like to discuss further. Here's the context: I think I see a flaw in the argument above. Suppose I want to know the...

This will sound like a very amateur question but please read: I have been puzzled for a while about the *precise* mathematical meaning of "quantum fluctuation". I know what a classical fluctuation is (as found in classical statistical dynamics). I also know what a superposition is. These seem...

Homework Statement Let f(x) be the discontinuous function f(x)=e^{-x},\text{for }x>0 f(x)=x,\text{for }x\leq 0 Construct explicitly a sequence of functions f_n(x), such that ||f_n(x)-f(x)||<\frac{1}{n}, and f_n(x) is a continuous function of x, for any finite n. Here ||\;|| represents...

Homework Statement Prove that for q>=p and any f which is continuous in [a,b] then || f ||_p<=c* || f ||_q, for some positive constant c. Homework Equations The norm is defined as: ||f||_p=(\int_{a}^{b} f^p)^\frac{1}{p}. The Attempt at a Solution Well, I think that because f is...

Hi, I was wondering how the state vector for a particle in a 1-D box can be expanded as a linear combination of the discrete energy eigenkets as well as a linear combination of the continuous position eigenkets. It seems to me that this is a contradiction because one basis is countable whereas...

Homework Statement for what range of n is the function f(x) = x^n in Hilbert space, on the interval (0,1)? assume n is real. Homework Equations functions in Hilbert space are square integrable from -inf to inf The Attempt at a Solution I am having trouble with the language of the...

I've been reading about them (briefly), and can't see any large difference between them and metric spaces or even euclidean spaces for that matter. What am I missing? I read a Hilbert Space is a complete inner product space. But a metric space is a complete space as well with the only...

can somebody explain eigenvalues inside hilbert spaces??

Why is the Hilbert Action and the matter actionof the Einstein's Gravitation theory diffeomorphism invariant, as Wald said in his textbook General Relativity on Page 456 and Sean Carroll said in his Spacetime and Geometry on Page 435. In other words，why do we have to set \delta S_{M} to be...

Hi there, In 3-dimensional real linear space, the simplest bases can be taken as the canonical bases \hat{x} = \left(\begin{matrix}1 \\ 0 \\ 0\end{matrix}\right), \qquad \hat{y} = \left(\begin{matrix}0 \\ 1 \\0\end{matrix}\right), \qquad \hat{z} = \left(\begin{matrix}0 \\ 0 \\...

In real linear space, we can use the rotation matrix in terms of Euler angle to rotate any vector in that space. I know in hilbert space, the corresponding rotation matrix is so-called unitary operator. I wonder how do I construct such matrix to rotate a complex vector in hilbert space? Can I...

When quantizing a classical field theory, e.g. Klein-Gordon theory, the Hilbert space of one-particle states is taken to be a set of equivalence classes of (Lebesgue) measurable and square integrable solutions of the classical field equation, but how do you use the classical theory to construct...

What exactly is the Hilbert space of a massive spin 0 particle in non-relativistic QM? The following construction defines a Hilbert space H, but is it the right one? We could e.g use some subspace of H instead. And what if we use Riemann integrals instead of Lebesgue integrals? Let G be the set...

Homework Statement http://img523.imageshack.us/img523/4456/56166304yr3.png Homework Equations http://img356.imageshack.us/img356/2793/40249940is8.png The Attempt at a Solution I defined K:[a,b] --> [a,b] with k(s,t) = \frac{(t-s)^{n-1}}{(n-1)!} I found for the norm: \int_a^b \int_a^b...

Hi there! Repeating the question on the title: Are Hilbert spaces uniquely defined for a given system? I started to think about this when I was reading about Schrödinger and Heisenberg pictures/formulations. From my understanding, you can describe a system analyzing the time dependent...

I know that all Hilbert spaces are Banach spaces, and that the converse is not true, but I've been unable to come up with a (hopefully simple!) example of a Banach space that is not also a Hilbert space. Any help would be appreciated!

We know that the Hilbert space of wavefunctions can be spanned by the |x> basis which is a non-countable set of infinite basis kets. Now consider the case of a particle in a box. We say that the space can be spanned by the energy eigenkets of the hamiltonian (each eigenket corresponds to an...

In QM we require that an operator acting on a state vector gives the corresponding observable multiplied by the vector. Spin up can be represented by the state vector \left( \begin{array}{c} 1 \\ 0 \end{array} \right) , while spin down can be represented by \left( \begin{array}{c} 0 \\ 1...

Homework Statement Let H be a Hilbert space. Prove that if \left\{ x _{n} \right\} is a sequence such that lim_{n\rightarrow\infty}\left\langle x_{n},y\right\rangle exists for all y\in H, then there exists x\in H such that lim_{n\rightarrow\infty} \left\langle x_{n},y\right\rangle =...

Hi there, Can anyone give me an hint/idea of how to prove Hilbert-Schmidt operators are compact? More specifically, if X is a seperable Hilbert space and T:X->X is a linear operator such that there exists an orthonormal basis (e_{n}) such that \sum_{n} ||T(e_{n})||^{2}<\infty then show that T...

Can you define a space that is locally homeomorphic to an infinite dimensional Hilbert space analogously to how you define a (smooth) manifold by an atlas defining local homeomorphisms to R^n? So the charts would just map to the Hilbert space rather than R^n. Could the rest of the definition...

A question arose to me while reading the first chapter of Sakurai's Modern Quantum Mechanics. Given a Hilbert space, is the outer product \mathcal{H}\times \mathcal{H}^\ast \to End(\mathcal{H}); (| \alpha\rangle,\langle \beta|)\mapsto | \alpha\rangle\langle \beta| a surjection? Ie, can any...

[SOLVED] Little bit of convex analysis on a Hilbert space Homework Statement Let H be a Hilbert space over R and f:H-->R a function that is bounded below, convex and lower semi continuous (i.e., f(x) \leq \liminf_{y\rightarrow x}f(y) for all x in H). (a) For all x in H and lambda>0, show that...

[SOLVED] Hilbert space &amp; orthogonal projection Homework Statement Let H be a real Hilbert space, C a closed convex non void subset of H, and a: H x H-->R be a continuous coercive bilinear form (i.e. (i) a is linear in both arguments (ii) There exists M \geq 0 such that |a(x,y)| \leq...

Homework Statement Is it true/possible to show that in a Hilbert space, if z_n is a sequence (not known to converge a priori) such that (z_n,y)-->0 for all y, then z_n-->0 ? The Attempt at a Solution I've shown that if z_n converges, then it must be to 0. But does it converge?

Homework Statement Is there a way to prove that E={sin(nx), cos(nx): n in N u {0}} is a maximal orthonormal basis for the Hilbert space L²([0,2pi], R) of square integrable functions (actually the equivalence classes "modulo equal almost everywhere" of the square integrable functions)? I...

Homework Statement The theorem about the closest point property says: If A is a convex, closed subspace of a hilbert space H, then \forall x \in H\,\, \exists y \in A:\,\,\,\, \| x-y\| = \inf_{a\in A}\|x-a\| I have to show that it is enough to show this theorem for x = 0 only, by...

[SOLVED] Hilbert Space Homework Statement For What Values of \psi(x)=\frac{1}{x^{\alpha}} belong in a Hilbert Sapce?Homework Equations \int x^{a}=\frac{1}{a+1} x^{a+1} The Attempt at a Solution I tried to use the condition that function in Hilbert space should satisfy: \int\psi^{2}=A but it...

I've now encountered two different definitions for a projection. Let X be a Banach space. An operator P on it is a projection if P^2=P. Let H be a Hilbert space. An operator P on it is a projection if P^2=P and if P is self-adjoint. But the Hilbert space is also a Banach space, and there's...

the NCG blog has various interesting stuff one thing was this link to a 4 minute talk by David Hilbert http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.mp3 wide audience, nontechnical here's NCG blog http://noncommutativegeometry.blogspot.com/ and the brief post about...