The quantum-mechanics tag has no usage guidance.

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### Cardinality of the set of Boolean subalgbras of the lattice of projections on a Hilbert space.

A simple question I've managed to gey myself quite confused about.
Given a Hilbert space H, what do we know about the cardinality of
(a) the set P(H) of projection operators onto H (equivalently, ...

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**1**answer

408 views

### Tensor product of generator of SU(n)

I'm doing research in quantum mechanics and got some trouble. Any help would be very much appreciated.
Let $\{\lambda_j\}$ be the set of generator of $SU(n)$. Consider the operator:
$K=\sum_j ...

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**2**answers

468 views

### $\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral:
$\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$
($\mathrm{BesselJ}[n,x]$ is ...

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**1**answer

91 views

### An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar ...

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**1**answer

83 views

### Set of orthogonal simplexes or partial mutually unbiased bases

I am interested in the existence of a set of vectors $\{ v_{ij} \}_{ij} \subseteq \mathbb{C}^N$ for $i \in \{1,\dots,N\}$, $j \in \{1,\dots,N+1\}$ such that $\left\vert v^*_{ij} v_{ij'} \right\vert = ...

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**2**answers

994 views

### Is Langlands reciprocity somehow analogous to the wave-particle duality of quantum mechanics?

Apologies for the vague question, and for the many inaccuracies (I am not a physicist and barely a number theorist).
In physics, there is the notion of gauge group of a field theory. The gauge group ...

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**1**answer

170 views

### Eigenvalues of a matrix constructed with simple logic

If a matrix can be constructed with simple bit-logic operations, is it also possible to find Eigenvalues with logic?
First I'll just say that my knowledge of logic is pretty much limited to ...

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**1**answer

408 views

### Do quantum “Sure-Shor separators” have a natural Veronese/Segre classification? (question inspired by Gil Kalai and Aram Harrow)

Aram Harrow asked: "Is there any place this is written up?"
Update Partly in answer to Aram's question, the thermodynamical properties of varietal dynamical systems now are written-up in ...

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**1**answer

495 views

### a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?

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**5**answers

2k views

### Can the equation of motion with friction be written as Euler-Lagrange equation, and does it have a quantum version?

My (non-expert) impression is that many physically important equations of motion can be obtained as Euler-Lagrange equations. For example in quantum fields theories and in quantum mechanics quantum ...

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**0**answers

95 views

### Is there a known generalization of the Schmidt decomposition based on a maximal set of “locally orthogonal” vectors?

I came across the following unusual generalization of the Schmidt decomposition in my work, which I describe below. I would like to know if this structure has been studied before so I can read more ...

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**1**answer

1k views

### What is the “Tangle” at the Heart of Quantum Simulation?

The following questions generalize and naturalize the question that was originally asked. Provisional answers largely due to Will Sawin are now included.
As was discussed in the question originally ...

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**0**answers

76 views

### Discretization model for Dirac equation in higher dimensions

I'm wondering if there are cases of discretization models (in lattices or graphs) that converge toward the non-linear Dirac equations? While there is a paper on the 2-d and 3-d cases ...

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**2**answers

534 views

### C*-algebraic representation of observables vs self-adjoint operators one

I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...

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**4**answers

383 views

### Prequantization and Hilbert space

In the definition of pre-quantization on a symplectic manifold $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ ...

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716 views

### Is quantum game theory reducible to classical game theory?

Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways:
Superposed initial states,
Quantum ...

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255 views

### Certain partial integrations in quantum mechanics

In classical quantum mechanics (and specifically in the introductury texts on this topic) while calculating expectation values of certain operators in the Schrödinger approach we often have to do ...

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**1**answer

140 views

### Find a generalized hypergeometric-based function yielding certain ratios of fifth-degree polynomials

Find a (presumably, generalized hypergeometric-based function $f(n,a,k)$), yielding for $n=1, a=\frac{1}{2}$,the rational function (ratio of fifth-degree polynomials)
\begin{equation}
...

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1k views

### Is there an equivalent of Heisenberg's uncertainty principle in the decision sciences ?

From memories of a quantum mechanics class and Wikipedia:
In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the ...

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**2**answers

279 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 ...

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**1**answer

334 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 ...

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**0**answers

374 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 $\mathcal H$, ...

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**1**answer

136 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 ...

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**0**answers

312 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 ...

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**2**answers

312 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. ...

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**0**answers

110 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?

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**1**answer

274 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 ...

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330 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) =
...

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805 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, ...

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### 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 ...

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**1**answer

488 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 ...

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**1**answer

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### 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 ...

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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 ...

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**1**answer

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### 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 $u \in \mathscr{S}(\mathbb{R}^n)$,
$$ ...

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**2**answers

686 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 ...

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### 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 ...

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### 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 ...

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**2**answers

644 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 ...

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**1**answer

271 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 ...

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730 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 ...

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**1**answer

152 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 ...

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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 ...

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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 ...

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### 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 ...

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**3**answers

404 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 ...

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**1**answer

6k 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 ...

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670 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 ...

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**1**answer

981 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 ...

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**0**answers

324 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 ...

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### 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. ...