Hilbert space Definition and 232 Threads

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...

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...

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...

Hi, what is the physical meaning, or also the geometrical meaning of the inner product of two eigenvectors of a matrix? I learned from the previous topics that a vectors space is NOT Hilbert space, however an inner product forms a Hilbert space if it is complete. Can two eigenvectors which...

Hi, I have an operator given by the expression: L = (d/dx +ia) where a is some constant. Applying this on x, gives a result in the subspace C and R. Can I safely conclude that the operator L can be given as: \begin{equation} L: \mathcal{H} \rightarrow \mathcal{H} \end{equation} where H is...

I have a matrix, [ a, ib; -1 1] where a and b are constants. I have to represent and analyse this matrix in a Hilbert space: I take the space C^2 of this matrix is Hilbert space. Is it sufficient to generate the inner product: <x,y> = a*ib -1 and obtain the norm by: \begin{equation}...

Hi, I am working on a home-task to analyse the properties of a ODE and its solution in a Hilbert space, and in this context I have: 1. Generated a matrix form of the ODE, and analysed its phase-portrait, eigenvalues and eigenvectors, the limits of the solution and the condition number of the...

Hello, I have derived the matrix form of one ODE, and found a complex matrix, whose phase portrait is a spiral source. The matrix indicates further that the ODE has diffeomorphic flow and requires stringent initial conditions. I have thought about including limits for the matrix, however the...

Homework Statement Given 3 spins, #1 and #3 are spin-1/2 and #2 is spin-1. The particles have spin operators ## \vec{S}_i, i=1,2,3 ##. The particles are fixed in space. Let ## \vec{S} = \vec{S}_1 + \vec{S}_2 + \vec{S}_3 ## be the total spin operator for the particles. (ii) Find the eigenvalues...

https://www.ma.utexas.edu/users/dafr/OldTQFTLectures.pdf I'm reading the paper linked above (page 10) and have a simple question about notation and another that's more of a sanity check. Given a space ##Y## and a spacetime ##X## the author talks about the associated Quantum Hilbert Spaces...

Hello, I Have a particle with wavefunction Psi(x) = e^ix and would like to find its Hilbert space representation for a period of 0-2pi. Which steps should I follow? Thanks!

Is it assumed that Hilbert space is an infinite manifold that the non-collapsing wave function occupies in Everettian QM? Thank you.

A fictitious system having three degenerate angular momentum states with ##\ell=1## is described by the Hamiltonian \hat H=\alpha (\hat L^2_++\hat L^2_-) where ##\alpha## is some positive constant. How to find the energy eigenvalues of ##\hat H##?

I'm looking for a rigorous mathematical description of the quantum mechanical space state of, for instance, a particle with no internal states. At university we were told that it the Hilbert state of wave functions. They gave us no particular restrictions on these functions, such as continuity...

Some posts in another thread lead me to a search which ended when I read the following "kets such as ##|\psi\rangle## are elements of abstract Hilbert Space". That lead me to this paper. http://www.phy.ohiou.edu/~elster/lectures/qm1_1p2.pdf "The abstract Hilbert space ##l^2## is given by a...

I always had this doubt,but i guess i never asked someone. What's the main difference between the Classical phase space, and the two dimensional Hilbert Space ?

This is basically just a comprehension question, but what makes elements of the Hilbert space exist in infinite dimensions? I understand that the number of base vectors to write out an element, like a wavefunction, are infinite: \begin{equation*} \psi(x) = \int c_s u_s (x) ds = \sum_k^{\infty}...

When you cut an object with a knife.. say a sausage. Does it's Hilbert Space or Quantum State split into two too? Or is it like in a holographic film.. in which even after cutting it, all the original image is in each of the cut portion?

As the title says, why does the set of hydrogen bound states form an orthonormal basis? This is clearly not true in general since some potentials (such as the finite square well and reversed gaussian) only admit a finite number of bound states.

Is it possible to approximately calculate the dynamics of a "phi-fourth" interacting Klein-Gordon field by using a finite dimensional Hilbert state space where the possible values of momentum are limited to a discrete set ##-p_{max},-\frac{N-1}{N}p_{max},-\frac{N-2}{N}p_{max}...

Why cannot we represent mixed states with a ray in a Hilbert space like a Pure state. I know Mixed states corresponds to statistical mixture of pure states, If we are able to represent Pure state as a ray in Hilbert space, why we can't represent mixed states as ray or superposition of rays in...

In classical mechanics we use a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at anyone point in time, with the evolution of the state of a system being described in terms of a trajectory through the corresponding phase space. However, in quantum...

If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...

From my humble (physicist) mathematics training, I have a vague notion of what a Hilbert space actually is mathematically, i.e. an inner product space that is complete, with completeness in this sense heuristically meaning that all possible sequences of elements within this space have a...

I hate to be 'that guy', but I've heard so-called "Hilbert Space" referenced many times. I can imagine that it's derived from physicist David Hilbert. I'd guess that you'd learn about it in an Undergrad course.

Or is it only theoretical.

Introduction If Quantum Mechanics is more fundamental than General Relativity as most Physicists believe, and Quantum Mechanics is described using Hilbert Spaces wouldn't finding a compatible version of General Relativity that operates within the confines of a Hilbert Space be of utmost...

I understand this question is rather marginal, but still think I might get some help here. I previously asked a question regarding the so-called computable Universe hypothesis which, roughly speaking, states that a universe, such as ours, may be (JUST IN PRINCIPLE) simulated on a large enough...

Recently I've been studying Angular Momentum in Quantum Mechanics and I have a doubt about the eigenstates of orbital angular momentum in the position representation and the relation to the spherical harmonics. First of all, we consider the angular momentum operators L^2 and L_z. We know that...

The equivalent of a dot product in Hilbert space is: \langle f | g \rangle = \int f(x) g(x) dx And you can find the angle between functions/vectors f and g via: \theta = arccos\left( \frac{\langle f | g \rangle}{\sqrt{\langle f|f \rangle \langle g|g \rangle}} \right) So is it possible to...

If Hilbert space is just a mathematical tool like a column for an accountant and doesn't have factual existence. How about the quantum vacuum. Isn't it quantum vacuum is just another tool? Is it like Hilbert space or does the quantum vacuum have more factual existence? If the quantum vacuum is...

Does Hilbert Space contain the fine structure constant or store the values of other constants of nature or their information or does it only contain the position, momentum basis information of particles?

Homework Statement [/B] Let M, N be a subset of a Hilbert space and be two closed linear subspaces. Assume that (u,v)=0, for all u in M and v in N. Prove that M+N is closed. Homework Equations I believe that (u,v)=0 is an inner product space The Attempt at a Solution This is a problem from...

The text does it thusly: imgur link: http://i.imgur.com/Xj2z1Cr.jpg But, before I got to here, I attempted it in a different way and want to know if it is still valid. Check that f^{*}f is finite, by checking that it converges. f^{*}f = a_0^2 + a_1^2 cos^2x + b_1^2sin^2x + a_2^2cos^22x +...

Homework Statement Given a orthonormal basis of the hilbert space of qutrit states: H = span (|0>, |1>, |2>) write in abstract notation and also a chosen consistent matrix representation, the states a) An equiprobable quantum superposition of the three elements of the basis b) An...

Depending on interpretation of QM, can hilbert space be considered just as real as space time? In MWI the wave function is real, but still lies in hilbert space, so would hilbert space be considered a real space according to this interpretation?

What is the dimensionality, N, of the Hilbert space (i.e., how many basis vectors does it need)? To be honest I am entirely lost on this question. I've heard of Hilbert space being both finite and infinite so I'm not sure as to a solid answer for this question. Does the Hilbert space need 4...

I see I am not the only one finds Hilbert confusing - because all it's properties seem so familiar. I have gathered together what I could find, please comment? A Hilbert space is a vector space that: Has an inner product: • Inner product of a pair of elements in the space must be equal to...

Just reviewing some QM again and I think I'm forgetting something basic. Just consider a qubit with basis {0, 1}. On the one hand I thought 0 and -0 are NOT the same state as demonstrated in interference experiments, but on the other hand the literature seems to say the state space is...

The state, ##| S\rangle##, say, of a system is represented as a vector in a Hilbert space. ##\psi (x, t)## is the representation of the state vector in the position eigenbasis; ##\psi (p, t)## in the momentum eigenbasis et cetera. That is, ##\psi (x, t) = \langle x|S\rangle##; ##\psi (p, t) =...

Given x,y elements of a hilbert space H, how do we conclude that x = y? Since there is an inner product, we can say that x = y only if (x,z) = (y,z) for all z in H. But is there a definition of equality which does not depend on the inner product? A hilbert space is a special instance of...

In quantum mechanics, a Hilbert space always means (in mathematical terms) a Hilbert space together with a distinguished irreducible unitary representation of a given Lie algebra of preferred observables on a common dense domain. Two Hilbert spaces are considered (physically) different if this...

State-space trajectories in classical mechanics can be used to nicely represent the time evolution of a given system. In the case of the harmonic oscillator, for instance, we get ellipses. How does this situation carry over to quantum mechanics? Can the time evolution of, say, the quantum...

Hello, could you please give me an insight on how to get through this proof involving operators? Proof: Given an eigenvalue-eigenvector equation, suppose that the vectorstate depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t...

Homework Statement (a) For what range of ##\nu## is the function ##f(x) = x^{\nu}## in Hilbert space, on the interval ##(0,1)##. Assume ##\nu## is real, but not necessarily positive. (b) For the specific case ##\nu = \frac{1}{2}##, is ##f(x)## in Hilbert space? What about ##xf(x)##? What...

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...