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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16 views

Is there any property for the eigenvalues of an Hermitian matrix on which a well-structured binary mask has been applied?

While working on a quantum-focused article, I came accross the following problem. Let $\rho$ be a positive, semi-definite, $2^{n+m}$-Hermitian matrix with unit trace ($\rho$ is a density matrix). Let $...
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77 views

Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...
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2answers
101 views

Massive dirac operator symmetric spectrum

Consider the Dirac operator $$ H = \begin{pmatrix} m & -i\partial_z \\ -i\partial_{\bar z} & -m \end{pmatrix},$$ where $\partial_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$ It is ...
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2answers
268 views

Energy levels of double well potential

Consider the (quantum) Hamiltonian on the real line $$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$ Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...
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63 views

Common core for unbounded operators

Suppose that $\mathcal H$ is a Hilbert space representing some physical system, $H$ is the Hamiltonian for the system, and $A$ is some observable for the system, that is, some unbounded self-adjoint ...
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2answers
261 views

Fourier transform of eigenvalue distribution of GUE matrices

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
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1answer
342 views

Basis of invariant tensors of rank n in three dimensions

[This is a question motivated by theoretical physics, so apologies if the language is rough...] In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...
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30 views

S-wave resonances in $V(r)=-c^2(r+1/r)^2, c \in \Re$

I need to know the s-wave resonances of the central potential $V(r)=-c^2(r+1/r)^2, c\in \Re$. Here we have a well attached to a barrier so we expect quantized quasibound/ metastable/ resonant states (...
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1answer
334 views

Is there a straightforward generalization of min(x,y) to positive-semidefinite Hermitian matrices?

This is an open-ended question I have. Is there a function of two positive-semidefinite hermitian operators $\min(A,B)$ returning another positive-semidefinite Hermitian operator such that: If A and ...
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4answers
1k views

Rigged Hilbert spaces and the spectral theory in quantum mechanics

I'm trying to learn some quantum mechanics by myself, and because of my mathematics background, I'm trying to understand it in a rigorous way. Since then, I've been intrigued by the use of rigged ...
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34 views

References for matrix variational problems on the unitary group

Let $U(d)$ denote the group of unitary $d\times d$ matrices. Let $\mathcal C_d$ denote the cone of Hermitian positive semidefinite $d\times d$ matrices. Fix an integer $r\geq 1$ and let $C(U(d),\...
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1answer
221 views

Is there a Hilbert space approach to commutative probability theory on locally compact spaces?

I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum ...
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Why does Riesz's Representation Theorem apply in quantum mechanics?

$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$. It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
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41 views

Domain issues regarding the Duhamel formula for the linear Schrödinger equation

I have some questions in succession regarding the rigorous domain issues about a Duhamel expansion formula (stated near the end of my post) for the linear Schrödinger equation. Consider a linear ...
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1answer
102 views

Hilbert space representation of a vector in terms of a continuous eigenbasis

Let $\mathscr{H}$ be a complex Hilbert space and $A$ be an Hermitian operator $A: \mathscr{H}\to \mathscr{H}$. Suppose, for a moment, that $A$ has a set of discrete eigenvalues $\{\lambda_{n}\}_{n\in \...
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1answer
298 views

Reference request for Deterministic $\subset$ Random $\subset$ Quantum

I hope this post is on topic as a reference request. I have seen somewhere the idea of (and saw it written just like this): $$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$ I am ...
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1answer
148 views

What is the precise relationship between real Poisson algebras and commutative $C^*$ algebras?

I've been teaching myself quantum mechanics, and I realized that I'm missing something fundamental. Namely, there are two pictures that I don't know how to reconcile: Quantum Mechanics generalizes ...
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1answer
86 views

Is there a Bell inequality for each of $2\times 2$, $3\times 1$, $2\times1\times1$ and $1\times1\times1\times1$ configurations?

There was no answer in https://physics.stackexchange.com/questions/600494/is-there-a-bell-inequality-for-2-times-2-and-1-times1-times1-times1-configur. Hence posting in mathoverflow on the possibility ...
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Convergence of steady states for Lindblad systems in infinite volume

In the physics of open quantum systems it is common to consider the Lindblad form. Which for a (super)-operator $\mathcal{L} \in B(B(\mathbb{C}^n ))$ means that \begin{align*} \mathcal{L}(\rho) = - i \...
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58 views

Generalized Ising Model

I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ...
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72 views

There is no dispersion free quantum state on $B(H)$

Let $H$ be a Hilbert space and $P(H)$ denotes the lattice of all orthogonal projections on $H$. The famous generalized Gleason theorem states that if $\mu:P(H) \to \mathbb{R}$ is a finitely additive ...
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107 views

How much Gleason type theorem do I need? Quasi states vs. states

Let $\varphi$ be a quasi state on $B(H)$. What does it mean? It means that $\varphi(cA)=c\varphi(A)$ for $c \in \mathbb{C}, A \in B(H)$, $\varphi(A) \geq 0$ for positive $A$ and $\varphi(A+B)=\varphi(...
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56 views

Gradient and Hessian

As we know gradient and Hessian of a map on Banach spaces are linear transforms (Frechet derivatives). In quantum control, control objective is a map which is defined on control fields as the ...
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2answers
196 views

Is the set of two-qubit absolutely separable states convex, and if so, what are its John ellipsoids?

Let us order the four nonnegative eigenvalues, summing to 1, of a (by definition, $4 \times 4$, Hermitian, nonnegative definite, trace one) "two-qubit density matrix" ($\rho$) as \begin{...
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76 views

Creation and annihilation operators as operator-valued distributions

In QFT, one usually talks about operator-valued distributions. But let's take, for instance, $L^{2}(\mathbb{R}^{3})$ and its associated Fock space $\mathcal{F} = \bigoplus_{n=0}^{\infty}L^{2}(\mathbb{...
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57 views

Identify an ordered-eigenvalue simplex with tetrahedral dihedral angle $\cos ^{-1}\left(\frac{1}{3}\right)$ in volume, area formulas

Let us order the four eigenvalues ($\lambda_1 \geq \lambda_2 \geq \lambda_3 \geq \lambda_4$, $\Sigma_{i-1}^4 \lambda_i=1$) of a Hermitian, trace-one, positive-definite $4 \times 4$ matrix $\rho$ (a “...
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2answers
1k views

What determines the maximal dimension of the irreps of a (finite) group?

I am chemist and ask for apologies for all my mathematical inabilities when asking this question in advance, but after quite a bit of searching I found that this problem could be "open" or ...
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173 views

Derived geometry and theoretical physics

Is there any link between derived geometry and theoretical physics? for example with particle physics or quantum mechanics? Specifically something that included the obstruction bundle. If possible I ...
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1answer
202 views

Do pseudodifferential operators represent all physically meaningful quantities in quantum mechanics? [closed]

(Qualifier: I know virtually nothing about quantum mechanics) In classical physics, Newton's laws guarantee that any physically relevant quantity is a function of the position and momentum of the ...
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1answer
117 views

Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube

The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$ \begin{equation} \label{one} \int_0^1 \...
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107 views

Creation and Annihilation operators in QFT - Part II

Following some suggestions on my previous posts, I'm trying to reformulate my question in a more specific way. This is a continuation of my original post. Since the mentioned post, I think I've ...
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3answers
1k views

Is there a 'certainty' principle?

Heisenberg's uncertainty principle is a restriction on which probability distributions can describe the position and momentum of a quantum particle. In mathematical terms it says that if $\psi\in L^2$ ...
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1answer
153 views

What are the John ellipsoids for a pair of (9- and 15-dimensional) convex sets of $4 \times 4$ positive-definite matrices?

What are the John ellipsoids (JohnEllipsoid) for the 9- and 15-dimensional convex sets ($A,B$) of $4 \times 4$ positive-definite, trace-1 symmetric (Hermitian) matrices (in quantum-information ...
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81 views

Alfsen Shultz theorem-the space of states of $C^*$-algebra depends only on Jordan structure

According to the article on nLab the Alfsen Shultz theorem states that the space of states of a given $C^*$-algebra depends on somehow weaker structure namely on the so called Jordan algebra structure....
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2answers
480 views

Creation and annihilation operators in QFT

As I said before, I'm not a QFT expert but I'm trying to understand the basics of its rigorous formulation. Let's take Dimock's book, where the foundation of QM and QFT is discussed. If we consider, ...
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160 views

Canonical commutation relations-bounded vs. unbounded picture

Suppose that $Q,P$ are self-adjoint operators which satisfy the relation $$(1) \ \ \ \ \ [Q,P]=iI$$ One can easily show that in this case $P,Q$ cannot be bounded. However one can find unbounded ...
6
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1answer
270 views

Path integral as quantum mechanics on the tangent bundle

Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a ...
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62 views

Quantization of a non Hamiltonian flow

If one considers a classical Hamiltonian flow $\partial_t \boldsymbol{x} = \partial_p H(\boldsymbol{x},\boldsymbol{p})$, $\partial_t \boldsymbol{p} = -\partial_x H(\boldsymbol{x},\boldsymbol{p})$, the ...
6
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0answers
186 views

Eigenvectors of a symmetric sum of tensor products

Let $A$ and $B$ be two (finite-dimensional) Hermitian matrices and $n$ be a positive integer. We define the matrix $$ L_i = A\otimes \dots\otimes A\otimes B\otimes A\otimes \dots\otimes A~, $$ where ...
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321 views

Why in $S^2$ is there no spin structure? [closed]

For a Dirac fermion (spin half) on $S^2$, we have both the general covariant derivatives and the relativistic Hamitonian. What does the claim "in $S^2$ there is no spin structure" means? A reference ...
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1answer
593 views

Is there a physical reason that fields in QFT are globally defined?

I have been trying to read a physics textbook on Quantum Field theory. There seems to me to be a bit of a disconnect in most texts I have looked at between quantum mechanics and quantum field theory, ...
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1answer
229 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
78 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
88 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
495 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{...
2
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0answers
39 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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1answer
146 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
87 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 ...
4
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0answers
105 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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44 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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