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5
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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 ...
10
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2answers
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
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0answers
202 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
3answers
379 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 ...
29
votes
1answer
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
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2answers
594 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
1answer
818 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
0answers
310 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
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3answers
1k 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
1answer
556 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 $\{\lambda_i^0\}$ ...
29
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6answers
3k 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 ...
5
votes
5answers
1k 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 ...
2
votes
1answer
394 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
2answers
522 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
2answers
932 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
1answer
451 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
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1answer
650 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 ...
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2answers
758 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 ...
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0answers
185 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 ...
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10answers
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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
1answer
232 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 ...
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votes
3answers
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 ...
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0answers
291 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
1answer
165 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
1answer
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
1answer
477 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 ...
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vote
2answers
574 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
1answer
390 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 ...
7
votes
2answers
368 views

Does there exist a potential which realizes this strange quantum mechanical system?

I have done some courses on quantum mechanics and statistical mechanics in the past. Since I also do math, I wonder about converge issues which are usually not such a problem in physics. One of those ...
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vote
0answers
1k views

Quantum computation implications of (P vs NP) [duplicate]

Possible Duplicate: What impact would P!=NP have on the characterization of BQP? Before I begin, I had a similar post closed for mentioning the recently released (to be verified) proof that ...
11
votes
2answers
2k views

What impact would P!=NP have on the characterization of BQP?

Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever ...
1
vote
3answers
1k views

Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?

Consider Schrödinger's time-independent equation $$ -\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi. $$ In typical examples, the potential $V(x)$ has discontinuities, called potential jumps. Outside ...
12
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1answer
672 views

Fixed point theorems and equiangular lines

I've been thinking about the equiangular lines (or SIC-POVM) conjecture, and my conclusion is that the best means of attack would be through some kind of fixed point theorem -- I'm thinking ...
3
votes
2answers
754 views

Spectral decomposition for an arbitrary linear combination of position and momentum operators

Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by: Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn) Pi ψ(q1,q2,...,qn) = -i ...
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votes
6answers
1k views

Functional Analysis and its relation to mechanics

Hi I'm currently learning Hamiltonian and Lagrangian Mechanics (which I think also encompasses the calculus of variations) and I've also grown interested in functional analysis. I'm wondering if there ...
11
votes
1answer
606 views

Set theoretical realizations of the hidden variables program in quantum mechanics

The hidden variables program in quantum mechanics has been largely discredited by two powerful theorems, namely those of Bell and Kochen/Specker. Nonetheless, this program retains a certain ...
2
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1answer
431 views

What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors?

Construction Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2: $P^{(i)} = |a\rangle_i\langle a| + ...
6
votes
3answers
592 views

Bounding a spectral gap: what proof techniques exist?

The following situation is ubiquitous in mathematical physics. Let $\Lambda_N$ be a finite-size lattice with linear size $N$. An typical example would be the subset of ...
3
votes
2answers
438 views

Simultaneous time-frequency concentration of orthonormal sequences?

Does there exist an orthonormal basis of square-integrable functions (either $L^2(\mathbb{R})$ or $L^2(\mathbb{C})$) such that the sequence of functions has bounded variance, and also the sequence ...
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9answers
3k views

How is the physical meaning of an irreducible representation justified?

This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO. I've been reading Singer's ...
9
votes
1answer
514 views

Which functions are Wiener-integrable?

I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them. Background The Wiener integral is an analytic tool to define certain ...
0
votes
1answer
234 views

The Quantum Operations On The Bipartite Systems

Given two distinct and noninteracting quantum mechanical systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces $\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space of the ...
24
votes
6answers
2k views

Why is addition of observables in quantum mechanics commutative?

I am no expert in the field. I hope the question is suitable for MO. Background/Motivation I once followed a quantum mechanics course aimed at mathematicians. Instead of the usual motivations coming ...
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vote
0answers
587 views

Tensor products as isomorphic functors in category theory

An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my ...
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4answers
603 views

Quantum channels as categories: question 1.

A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$, where $L(\mathcal{H}_{i})$ is the family of operators on $\mathcal{H}_{i}$. In general, we ...
2
votes
3answers
515 views

Are the Gell-Mann matrices extremal when used as Kraus operators for a quantum channel?

Landau and Streater proved that a set of Kraus operators, Ai, is extremal if and only if the set $\{A_{k}^{\dagger}A_{l}\}_{k,l \ldots N}$ are linearly independent. I have seen very convincing ...
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2answers
2k views

Noether's Theorem in Quantum Mechanics

In classical mechanics: If a Lagrangian L is preserved by an infinitesimal change in the state space variables qi -> qi + εKi(q) leads to only second order change in the Lagrangian: $$ 0 = ...
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3answers
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What is the relationship between algebraic geometry and quantum mechanics?

The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...
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41answers
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Where does a math person go to learn quantum mechanics?

My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn ...