The quantum-mechanics tag has no wiki summary.

**2**

votes

**2**answers

240 views

### Existence of a projection operator onto a classical set of density matrices

I have a Hilbert space of quantum density matrices written in the Glauber-Sudarshan P representation - ie. we have coherent states $|\alpha \rangle$ and we write density matrices as
$$ \rho = \int ...

**13**

votes

**1**answer

314 views

### Reconciling two notions of geometric quantization.

Let $(M,\omega)$ be a compact symplectic manifold and $(L,\nabla)$ a prequantum line bundle. There are two schemes to quantize this data:
Choose a polarization $P$ of $M$ and define the quantum ...

**1**

vote

**0**answers

321 views

### How to estimate the quantum fidelity between two given states

There is a well-known theorem, firstly obtained by Denes Petz, in quantum information theory, which is described as follows:
$\mathbf{Theorem.}$ Let $\rho$ and $\sigma$ be two states on $\mathcal H$, ...

**3**

votes

**1**answer

133 views

### Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials).
Is ...

**0**

votes

**0**answers

238 views

### Functional Analysis and Differential Manifold incompatibility

From the mathematical point of view, where is the idea of the incompatibility between functional analysis, at the basis of quantum mechanics, and differential geometry, at the basis of general ...

**2**

votes

**2**answers

306 views

### Translation of an article

I need to read this article
"On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups"
authors:G.Zhislin, A. ...

**2**

votes

**0**answers

104 views

### Three body problem with two fermions and a different particle

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?

**3**

votes

**1**answer

252 views

### Solvable models in quantum mechanics

Is there anyone who studied on the book "Solvable Models In Quantum Mechanics" by Albeverio? I don't succed in understanding the proof of page 116 about the eigenvalues of the Hamiltonian with point ...

**8**

votes

**0**answers

269 views

### Uncertainty principle in Entropy terms

Math Questions:
Consider Hilbert space $L_2(\mathbb{R})$ with a standard norm
$
||\psi|| = ( \int_{\mathbb{R}}{ |\psi(t)|^2 dt } )^{1/2},
$
and Fourier transform
$
(F\psi)(\xi) =
...

**6**

votes

**7**answers

689 views

### Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this ons is slightly different, and more restrictive. I know (some)
quantum mechanics, ...

**12**

votes

**3**answers

2k views

### Mathematical “proof” of the stability of atoms?

I am trying to find proofs of the stability of an atom, says, for simplicity, the hydrogen atom. There are positive answers and negative answers in various atom models.
The naive "solar system" model ...

**7**

votes

**1**answer

433 views

### Intuitive meaning of Double Commutant Theorem

Is there any intuitive explanation of the Double Commutant Theorem for Von Neumann Algebras? By intuitive I mean in terms of Quantum Mechanics. For example, duality of states and observables in the ...

**1**

vote

**1**answer

464 views

### Beginner question on constraints of a wave function in quantum mechanics

I am working through Griffiths’ Introduction to Quantum Mechanics. In chapter 1, he attempts to impose a condition such that
$$\frac{d}{dt}\int_{-\infty}^\infty\left|\psi(x,t)\right|^2dx=0$$
so that ...

**17**

votes

**8**answers

1k views

### A novice question on Quantum Mechanics

I'm currently working through Dirac's book The Principles of Quantum Mechanics. In it, he describes the nature of superpositions and at one point states:
"... if the ket vector corresponding to a ...

**4**

votes

**1**answer

133 views

### Interpretation of a parameter in forming a pseudodifferential operator

In Zworski's Semiclassical Analysis, he defines the following method of quantization: for a symbol $a = a(x,\xi) \in \mathscr{S}(\mathbb{R}^{2n})$ and $u \in \mathscr{S}(\mathbb{R}^n)$,
$$ ...

**3**

votes

**2**answers

651 views

### Infinite dimensional manifold

In Hamiltonian mechanics, one essentially work with $\mathbb{R}^{2n}$. However, this is only a local description of our configuration manifold $M$. More precisely, the mechanical system is regarded as ...

**3**

votes

**0**answers

208 views

### is there any connection between the consistent histories interpretation of quantum mechanics and kripke semantics?

Kripke semantics interpret intuitionistic logic by a partially ordered set of worlds/situations. Consistent histories interpretation of QM elaborates the copenhagen interpretation where a consistent ...

**-1**

votes

**2**answers

540 views

### Problem of quantization: state of the art

The "problem of quantization":
Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate ...

**2**

votes

**2**answers

549 views

### Can one derive Wigner’s theorem in the complex case from the real case?

Wigner's theorem states that every symmetry of complex Hilbert space is either unitary or anti-unitary, up to multiplication by a unit scalar function. Here $f:\mathcal{H} \rightarrow \mathcal{H}$ is ...

**2**

votes

**1**answer

228 views

### Tracking down locality assumption in CHSH inequality

CHSH inequality requires both locality and realism. I will equate here realism with counterfactual definiteness.
Now counterfactual definiteness tells us that given two different measurements on the ...

**6**

votes

**2**answers

702 views

### Quantum mechanics basics [closed]

Hello. I'm thinking about where does the basic quantum mechanics things comes from. I mean the forms of operators and a Shroedinger equation. The more intuitive explanation is better.
To get forms of ...

**3**

votes

**1**answer

149 views

### Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian

We are considering the instantaneous eigenstates of an analytically time-dependent hamiltonian and I would like to know how legitimate it is to extend them to the complex plane.
Specifically, our ...

**5**

votes

**2**answers

2k views

### Classical Limit of Feynman Path Integral

I understand that in the limit that h_bar goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation ...

**11**

votes

**2**answers

2k views

### Classical Limit of Quantum Mechanics

There is a well-known principle that one can recover classical mechanics from quantum mechanics in the limit as $\hbar$ goes to zero. I am looking for the strongest statement one can make concerning ...

**6**

votes

**0**answers

215 views

### Lovász function of the Möbius ladder

Quantum motivation
Noncontextuality inequalities (and in particular Bell inequalities) can be mapped into graphs, in such a way that its relevant properties can be calculated via some simple ...

**2**

votes

**3**answers

393 views

### Cardinality of a certain set of distinct subsets of $\mathbb{N}$

A recent question here has convinced me that folks here have a warm heart for the foundations of quantum mechanics, so I decided to ask a question that has been bothering me for a while.
Quantum ...

**30**

votes

**1**answer

5k views

### “psi-epistemic theories” in 3 or more dimensions

In their recent paper The Quantum State Can Be Interpreted Statistically, Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for ...

**12**

votes

**2**answers

639 views

### Is zero a hydrogen eigenvalue?

This question has been bugging me for some time.
Take the hamiltonian for the hydrogen atom: $$\hat{H}=-\frac{1}{2}\nabla^2-\frac{1}{r},$$ acting on (a domain contained in) $L^2(\mathbb{R}^3)$. It is ...

**2**

votes

**1**answer

918 views

### Square root of a stochastic process

I am currently working on the understanding of the stochastic nature of the Schroedinger equation. This has a notable history dating back to Nelson's works and relative criticisms. But one can take a ...

**2**

votes

**0**answers

319 views

### Quantum sheaves

Are the following definitions known?
Consider H a Hilbert space. A "quantum topology" on H is a set Sigma of closed subspaces satisfying the following conditions:
(a) {0} and H lie in Sigma
(b) If ...

**21**

votes

**3**answers

2k views

### On the periods in the periodic table (or Why is a noble gas stable?)

Added 22, November:
I've succeeded in making the question entirely unintelligible with all my additions.
So I thought I would summarize it in the simplest form I could manage and add it to the title. ...

**2**

votes

**1**answer

640 views

### Bounding the von Neumann entropy of a density matrix with the Hilbert-Schmidt norm

Question
Suppose I have a $D$-dimensional density matrix $\rho_0$
$\rho_0^\dagger = \rho_0 \quad, \quad \mathrm{Tr} \rho_0 = 1 \quad, \quad \rho_0 > 0,$
with a known spectrum $\{\lambda_i^0\}$ ...

**33**

votes

**6**answers

4k views

### Is the Mendeleev table explained in quantum mechanics?

Does anybody know if there exists a mathematical explanation of Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the authors ...

**8**

votes

**4**answers

2k views

### Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...

**3**

votes

**1**answer

415 views

### Are all quantum cellular automata invertible & representable?

A little background: As far as I know there is no standard definition of a quantum cellular automaton in the literature. Different authors use different definitions. Here I propose my own definition ...

**2**

votes

**2**answers

554 views

### Continuous Measurement in Quantum Mechanics

Let $\mathcal{P}(S^{\infty})$ denote the set of probability measures on the unit sphere $S^{\infty} \subset \mathcal{H}$ in the Hilbert space of states of a quantum mechanical system. Measurement of ...

**5**

votes

**2**answers

1k views

### Space with 720° / not 2$\pi$ rotational symmetry?

Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)?
The reason I am asking is because in quantum ...

**6**

votes

**1**answer

528 views

### Can the von Neumann entropy of a positive, positive semi-definite, and unit-trace density matrix with equal on-diagonal terms be bounded by equalizing all off-diagonal elements to their highest/lowest value?

Statement of problem
Consider the density matrix $M = (m_{i,j})$ in $d$-dimensions with all positive elements: $m_{i,j} > 0$. From physics, a density matrix is Hermitian, positive semi-definite, ...

**5**

votes

**1**answer

700 views

### First Quantization is a mystery… but de-quantizing perhaps not

There is an well-known infamous DICTUM:
-Second Quantization is a functor, First Quantization is a mystery-.
Indeed, second quantization is the "Fock functor", which builds the Fock space in a ...

**5**

votes

**3**answers

932 views

### Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.
Background
A simple consequence of the singular value decomposition is that any vector ...

**3**

votes

**0**answers

190 views

### Bell polytopes with nontrivial symmetries

Take $N$ parties, each of which receives an input $s_i \in {1, \dots, m_i}$ and produces an output $r_i \in {1, \dots, r_i}$, possibly in a nondeterministic manner. We are interested in joint ...

**24**

votes

**10**answers

3k views

### Toy Models of Quantum Mechanics

Do toy models of quantum mechanics help us better understand "regular" quantum mechanics? For example, if we look at quantum mechanics over a finite field $F$ (e.g. $\mathbb{Z}_2$), can this lead to ...

**5**

votes

**1**answer

239 views

### Approximate analytic solution of Schroedinger equation with arbitrary power potential

I'm solving the following Schroedinger equation in the domain $r>0$
$\psi''(r) + \left(E-\frac{a}{r^b}\right)\psi(r)=0 $,
where $0 < b < 2$ and $a, E$ are positive constants. Primarily I'm ...

**32**

votes

**3**answers

3k views

### Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...

**7**

votes

**0**answers

308 views

### Quantizations in quantum mechanics

Perhaps this is not an appropriate question for MO, but having just discovered this site I wanted to ask it.
Is there a rigorous definition as to what "a quantization" of a Hamiltonian dynamical ...

**8**

votes

**1**answer

172 views

### Operator compression preserving lowest energy eigenspace.

I have a large ($10^6$ by $10^6$) sparse ($0.4$% nonzero) hermitian matrix $H$ arising from the discretization of an elliptic PDE. I would like to approximate $H$ with a smaller matrix $H'$ in such a ...

**9**

votes

**1**answer

1k views

### Wick rotation and the Riemann zeta function

The goal of this question is to conceptualize in some way the fact that the Riemann zeta function $\zeta(s)$, and other zeta functions like it, have analytic continuations.
Background
I have by now ...

**5**

votes

**1**answer

491 views

### Quantum probability experiment?

I am looking for an example (or definition) of a quantum probability experiment (if there is such a thing). Ideally it should have these properties:
Be purely mathematical; no mention of physics or ...

**1**

vote

**2**answers

589 views

### Quantum Error Correction

One can correct the errors in a quantum channel iff the coherent information of the input state is not reduced by the channel. This is analogous to sending quantum entanglement through a channel. If ...

**5**

votes

**1**answer

403 views

### What categorical mathematical structure(s) best describe the space of “localized events” in “relational quantum mechanics”?

In a recent (and to me, very beautiful) paper, entitled "Relational EPR",
Smerlak and Rovelli present a way of thinking about EPR which relies upon Rovelli's ...