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.
390
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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 (...
12
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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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4
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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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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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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 ...
4
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1
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359
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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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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 ...
3
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1
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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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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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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 ...
2
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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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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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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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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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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 ...
2
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1
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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 ...
0
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1
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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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0
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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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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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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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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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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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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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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 ...
6
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299
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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 ...
3
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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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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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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 ...
3
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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 ...
0
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1
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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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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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0
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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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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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1
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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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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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3
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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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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 ...
2
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0
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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 ...
6
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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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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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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,\...
5
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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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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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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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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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2
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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 $...
3
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2
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261
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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 ...
0
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0
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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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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 ...