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1
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1answer
367 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 ...
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8answers
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
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1answer
127 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
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2answers
623 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
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0answers
204 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 ...
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2answers
502 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
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2answers
442 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
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1answer
205 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
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2answers
679 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
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1answer
140 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
2answers
1k 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 ...
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
206 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
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3answers
385 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
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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
617 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
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1answer
854 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
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0answers
313 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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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
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1answer
586 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 ...
6
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5answers
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
1answer
400 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
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2answers
533 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
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2answers
984 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
476 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
670 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
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3answers
856 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
0answers
186 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
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
1answer
236 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 ...
30
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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
294 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
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1answer
166 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
479 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
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2answers
579 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
395 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
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2answers
372 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 ...
1
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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
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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 ...
2
votes
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
685 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
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2answers
784 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 ...
2
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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
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1answer
618 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 ...
3
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1answer
456 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| + ...
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3answers
610 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
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2answers
441 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 ...