1. ### 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...
2. ### A How are incompatible ##\hat p\hat q\neq\hat q\hat p## related to ##p\wedge q## ?

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...
3. ### 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...
4. ### 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...
5. ### 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...
6. ### A Is there any theory that can be modeled in any type of space?

Is there any theory in physics that can be modelled 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?
7. ### 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...
8. ### 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)?
9. ### 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...
10. ### 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...
11. ### 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: > $$\psi_1=(\phi_1+2\phi_2)/\sqrt{5}$$ >...
12. ### 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}}=...
13. ### 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...
14. ### 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...
15. ### 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.
16. ### 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.

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...
18. ### 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...
19. ### 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...
20. S

### I Eigenvectors and inner product

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...
21. S

### A Operator mapping in Hilbert space

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...
22. S

### I How to check if a matrix is Hilbert space and unitary?

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}...
23. S

### I Norm of a Functional and wavefunction analysis

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...
24. ### The addition of 3 spins

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...
25. ### A Topological Quantum Field Theory: Help reading a paper

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...
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### I Hilbert space in Everettian QM

Is it assumed that Hilbert space is an infinite manifold that the non-collapsing wave function occupies in Everettian QM? Thank you.
27. ### I Phase Space and two dimensional Hilbert Space

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 ?
28. ### I Why are Hilbert spaces used in quantum mechanics?

In classical mechanics we use a 6n-dimensional phase space, itself a vector space, to describe the state of a given system at any one 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...
29. ### I What is the outer product of a tensor product of vectors?

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...
30. ### I What does "completeness" mean in completeness relations

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...