Questions tagged [quantum-mechanics]

For questions about mathematical problems arising from quantum mechanics, a branch of physics describing the behaviour of nature at very small scales, at the level of atoms and subatomic particles.

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1answer
183 views

Physics applications of quantum logic

Are there any examples of quantum logic being applied to solve actual physical questions, in particular to predict the physical properties (spectrum etc.) of some quantum-mechanical system? (Note that ...
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1answer
66 views

Observable nearly commuting with a “complete” set of commuting observables

Consider the Hilbert space $H = E^{\otimes n}$ where $E=\mathbb{C}^2$. On $E$ we have an observable $O$ (i.e. a Hermitian matrix) that is diagonalizable in the standard basis with eigenvalues $1$ and ...
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1answer
67 views

How to calculate Fourier transformation of eigenstates in CV quantum information [closed]

The position $\hat{q}$ and momentum $\hat{p}$ has $[\hat{q},\hat{p}]=i$. And we set there eigenstates as $|s\rangle_q$ and $|s\rangle_p$ with eigenvalue s. In the paper [Phy Rev A. 79, 062318 (2009)],...
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2answers
414 views

QFT and its notations

I know hardly anything about quantum field theory (QFT) but I'm giving a try to understand some ideas of it. As far as I understand, in QFT one is interested in studying measures such as: \begin{...
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0answers
34 views

Cwikel–Lieb–Rosenbljum inequality including zero resonances

The Cwikel–Lieb–Rosenbljum inequality asserts that, for any potential $V:\mathbb{R}^n\to\mathbb{R}$, we have $$(\mbox{number of eigenvalue} \leq 0\mbox{ , counted with multiplicity, of }-\Delta+V\,)\...
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39 views

Long-time evolution under Schroedinger equation followed by short-time free evolution

Suppose $\psi_t(x)$ solves the Schroedinger equation $$ i \partial_t\psi_t=H\psi_t$$ where $$H= -\Delta +V.$$ Fix $s>0$. Suppose $t$ is large. Is it true that under certain conditions, the vectors ...
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1answer
94 views

When is rank-1 perturbation to a positive operator still positive?

Let $A : \mathcal{H} \to \mathcal{H}$ and $B : \mathcal{H} \to \mathcal{H}$ be trace-class (hence compact) Hermitian operators on a separable Hilbert space. Assume that $A$ is strictly positive and ...
2
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1answer
79 views

Resonances for Schrodinger operators with radial potentials

Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...
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0answers
85 views

Symplectic geometry connects random density matrices?

This question arises from studying the following papers: Christandl et al. '14 and Mejia et al. '16. These two papers use a connection between symplectic geometry and reduced density matrix. In ...
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0answers
40 views

Trace Differentiation with Pauli operators, finding $\frac{d x}{d t}$ and $\frac{d z}{d t}$ from the master equation [closed]

I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $\theta$. For context this is the setup and derivation I have for continuous measurement along ...
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3answers
1k views

Linear algebra underlying quantum entanglement?

Hope this question is appropriate. I think I saw certain claims that quantum entanglement is a certain phenomena that can be explained (or modelled) in terms of tensor products in linear algebra. I ...
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1answer
119 views

Quantum entropy Venn diagrams

We know that in classical information theory the relation between different entropies can be depicted by Venn Diagram as given below. Can we create such Venn-diagrams for the quantum information ...
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0answers
78 views

Lippmann-Schwinger equation for the Coulomb potential

Let $H=H_0+V$ be a Hamiltonian on $\mathbb{R}^3$ where $H_0=-\frac{\Delta}{2m}$ is the free Hamiltonian and $V$ is a potential. Let us assume first that $V$ decays sufficiently fast at infinity and ...
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2answers
162 views

Relativistic scattering theory vs non-relativistic one

In relativistic scattering theory (e.g. in quantum electrodynamics) the existence of the $S$-matrix as well as of Moller operators is postulated as far as I understand (although at some stage it has ...
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1answer
159 views

Scattering theory for Coulomb potential

Both physical and mathematical theories of quantum scattering seem to be well developed in the case when the potential (or a more general perturbation of the Laplacian) decays fast enough at infinity ...
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0answers
62 views

Partitioning the set of Pauli words into abelian pieces

Let $\sigma_x,\sigma_y,\sigma_z$ be the Pauli matrices. A Pauli word of length $n$ is defined as the tensor product $\otimes_{i=1}^n\sigma_i$ of operators $\sigma_1,\dots,\sigma_n\in\{\mathbf 1,\...
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2answers
266 views

Is there a reasonable notion of spectral theorem on a pre-Hilbert space?

I'm trying to understand how bad things could possibly get without Cauchy completeness as a criterion for Hilbert spaces in quantum mechanics. Obviously, doing calculus on a pre-Hilbert space would be ...
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6answers
3k views

Any real contribution of functional analysis to quantum theory as a branch of physics?

In the last paragraph of this last paper of Klaas Landsman, you can read: Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
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0answers
130 views

Internal logic in topos theory, monoidal categories, and quantum mechanics

To obtain the internal logic of a topos (roughly speaking), we associate a type of free variable with an object, and a statement about such a variable with a subobject of that object. Intuitively, the ...
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1answer
88 views

Can quantum codes have more than $c \cdot \sqrt{N}$ correction distance for N encoding qbits?

I'm not an expert in quantum computing at all, but recently I've started to learn it (read Shen-Vyalyi-Kitaev's book and looked up some other literature here and there). There are few remarkable ...
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2answers
167 views

Estimate of a solution of Schroedinger equation for a free particle

Let $\psi(x,t)$ be a solution of the Schroedinger on the line $$i\frac{\partial \psi}{\partial t}=-\frac{1}{2m}\frac{\partial^2 \psi}{\partial x^2}.$$ One assumes that $\psi(x,0)$ "behaves well" as $...
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2answers
214 views

Explicit form of S-matrix on the line

Consider the Hamiltonian $H$ on functions on the line with \begin{eqnarray} H=H_0+V,\\ H_0=-\frac{1}{2m}\frac{d^2}{dx^2} \end{eqnarray} where $V$ is a potential vanishing outside of a bounded interval ...
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0answers
47 views

References for a proof or interpretation of deficiency indices theorem (von Neumann)

I am looking for a proof or some interpretation around why the domain of the new extension $D(A_U)$ in the Theorem below is given by its specific formula. I have already searched in papers and here ...
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0answers
96 views

Pants algebra $M_n$ as a dagger-special symmetric Frobenius algebra and $CP^*$

I'm looking at the paper Categorical Quantum Mechanics II: Classical-Quantum interaction by Coecke and Kissinger (arxiv link), and I'm having difficulty with one particular aspect. Throughout the ...
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0answers
107 views

Product of sines to sum

I hope this is a research level question; it is to me at least, I'm a beginning researcher in the field of the Bethe Ansatz. In the expressions I'm considering, I stumbled across the following ...
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1answer
53 views

Perform certain constrained integrations over an ordered subsection of a 3-simplex, yielding “absolute separability” probabilities

Let us order the four points $\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4 \geq 0$ of a 3-simplex, $\lambda_1+\lambda_2+\lambda_3+\lambda_4=1$, giving us a subsection $L$. Integration over $...
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0answers
85 views

About geometric quantisation and application to real system

Quantisation is a important step to properly define a quantum system from a classical one. In a nutshell : On a symplectic manifold $(M,\omega)$ and an algebra of function $f$ on $M$, one defines an ...
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237 views

What is the status of the Born Rule in axiomatic QM?

While physicists have tried multiple times and failed to derive the Born Rule (for example: https://arxiv.org/pdf/quant-ph/0409144.pdf). I was wondering what axiomatic Quantum Mechanics had to say ...
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1answer
122 views

What is the measures monad for FDHilb?

I am labouring under a particular assumption that, perhaps, needs to be corrected. I believe that FDHilb, the category of Finite Dimensional Hilbert spaces and general linear maps is a category of ...
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10answers
9k views

The Planck constant for mathematicians

The questions Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant? Q2. How does the Planck constant appear in mathematics of quantum mechanics? In ...
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72 views

Schrödinger operators : current status between ballistic motion and absolutely continuous spectrum?

Regarding Schrödinger operators $H= \Delta + V$ on $\mathbb{R}^d$ or $\mathbb{Z}^d$, it is said that $H$ manifests ballistic motion if the square root of the second moment of the position operator ...
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0answers
43 views

Domain of definition of a hamiltonian with delta(contact) potential

I am having a hard time making sense of the so-called "delta function potential well" in quantum theory. The Hamiltonian operator is defined as (with $\mathscr D_H\subset \mathscr H=L^2(\mathbb R)$) $$...
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0answers
80 views

Conditions on the Hamiltonian of a classical system that yeild essentially self-adjoint quantum Hamiltonian

What are the conditions on the Hamiltonian of a classical system that under these conditions the quantum Hamiltonian obtained via Weyl quantization will be essentially self-adjoint in $L_2(\mathbb{R}^...
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1answer
102 views

Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values

To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
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0answers
79 views

On relationship between cryptography and operator algebras [closed]

Does quantum cryptography connect two different areas of math operator algebras and Cryptography?
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0answers
122 views

Quantum Scattering Experiments: C-Modules, N-Modules and Their Monads

I am working on a theory of particle physics where we use monads. I have a few conjectures that I need to check. The cateogory of $\mathbb{C}$-Modules is monadic over set The category of $\mathbb{N}...
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0answers
114 views

Textbook covering superoperators and tensor products

I am looking for a textbook to cover the following tensor product (and, of course, the theory around it): Let $\otimes_1$ denote the tensor product on Hilbert spaces. Let $\otimes_2$ denote the ...
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3answers
481 views

Is the set of separable quantum states closed?

Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable). A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
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2answers
308 views

q-difference equations and quantum mechanics

I have been trying to understand why the term quantum is so easily accepted for calculus based on q-numbers $[n]_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...). ...
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1answer
133 views

How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?

To begin, we have two constraints \begin{equation} C1=x>0\land z>0\land y>0\land x+2 y+3 z<1 \end{equation} and \begin{equation} C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...
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0answers
44 views

Reference on iterated integrals against projection valued measures

I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which ...
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1answer
89 views

Complete positivity with infinite dimensional auxillary spaces

The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on ...
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1answer
49 views

Reference request: Energy variance of randomly drawn state

In quantum mechanics, given an $N$-qubit ($2^N$-dimensional) Hamiltonian $\hat{H}$, I'm fairly sure that the variance in energies of randomly drawn pure states (i.e. norm-$1$ vectors) may be ...
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1answer
91 views

Localization of solutions for time-dependent Schroedinger equation

I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know. The ...
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0answers
256 views

What role do Hecke operators and ideal classes perform in “Quantum Money from Modular Forms?”

Cross-posted on QCSE An interesting application of the no-cloning theorem of quantum mechanics/quantum computing is embodied in so-called quantum money - qubits in theoretically unforgeable states. ...
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98 views

Decomposition of the group of Bogoliubov transformations

Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of ...
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1answer
385 views

Do any finite predictions of Quantum Mechanics depend on the set theoretic axioms used?

I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used. We will say that Quantum Mechanics makes a finite prediction about an experiment if,...
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0answers
44 views

Inductive limits of unitary groups and quantum mechanics

I'm curious if someone has seen concrete applications of $U(\infty)$ in quantum mechanics. Is it possible, for example, in some particular cases to write down the propagator as a limit of a sequence ...
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0answers
103 views

Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”

I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
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2answers
668 views

Harmonic oscillator in spherical coordinates

It is probably the most well-known result in quantum mechanics that the harmonic oscillator can be solved by supersymmetry. More precisely, the operator $$-\frac{d^2}{dx^2}+x^2$$ can be ...

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