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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.

9
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2answers
405 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 ...
3
votes
0answers
234 views

Conserved Positive Charge for a PDE

Let $(x,t) \in \mathbb{R}^2$ and consider the following partial differential equation for the real-valued function $U(x,t)$: \begin{equation} \frac{\partial^2 U}{\partial t^2} = - \frac{\hbar^2}{4m^2} ...
2
votes
0answers
47 views

Quantum versus classical communication complexity

Problem. Is it true that any 2-party communication problem $f(x,y)$ of poly-logarithmic complexity in the quantum simultaneous message passing model ($Q''$) has complexity $n^{o(1)}$ (i.e., strongly ...
6
votes
1answer
84 views

Further Developments of Lieb-Schultz-Mattis theorem in Mathematics

The Lieb-Schultz-Mattis theorem [1] and its higher-dimensional generalizations [2] says that a translation-invariant lattice model of spin-1/2's cannot allow a non-degenerate ground state preserving ...
2
votes
1answer
157 views

PDE’s whose solutions can be presented using path integrals

It is well known that solutions of the Schroedinger equation and of the heat equation can be presented using path integrals: $$\psi(x,t)=\int K(x,t;y,0)\psi(y,0)dy,$$ where the kernel $K(x,t;y,0)$ is ...
3
votes
1answer
100 views

Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
3
votes
1answer
94 views

Reference on completely positive maps which are isometries

Let $\Phi:\mathcal{L}(H)\rightarrow \mathcal{L}(K)$ be a completely positive map sending positive self-adoint operators on a finite-dimensional Hilbert space $H$ to positive self-adoint operators on a ...
-2
votes
2answers
226 views

Adjunctions between Groupoids and Hilbert spaces

I am interested in any adjunctions between any of the familiar categories of Groupoids and the category of finite dimensional Hilbert spaces. Do any exist? Are there any well know monads on the ...
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0answers
38 views

Domain Monad on Density Operators Using Spectral Order

The spectral order for density operators is given in this paper Coecke Martin 2010. I won't give the full definition here. Essentially, it allows for a partial order of density matrices that forms a ...
1
vote
0answers
61 views

Integrability, quantum ergodicity, and observable algebra

Consider (for simplicity and definiteness) the Laplacian on a compact Riemannian manifold $M$. Let $\phi_k$, $E_k$ be its eigenfunctions and eigenvalues in increasing order. Quantum ergodicity is ...
4
votes
1answer
157 views

What is the “free symmetric monoidal category” 2-monad?

I have come across an n-category cafe post where someone describes a monad that generates symmetric monoidal categories. Can someone give details, like what is the base category, what exactly is the ...
2
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0answers
124 views

Multiset or Bag monad on Finite-Dimensional Hilbert Spaces

Edit: I will be happy if someone can get me the Bag monad on a 2-category of groupoids, regardless of any reference to Hilbert Spaces. (It's a fire sale!!) I am trying to create the quantum ...
6
votes
1answer
119 views

Can $S_n$ be partitioned into subsets containing an involution and satisfying $∀σ≠τ, ∃j$ s.t. $σ(j)≠τ(j),σ^{−1}(j)=τ^{−1}(j)$?

Background Let $\sigma, \tau \in S_n$. We will say that $\sigma$ and $\tau$ are locally orthogonal and write $\sigma \perp \tau$ if there exists $j \in \{1, 2, \ldots, n\}$ such that $\sigma(j) \neq \...
0
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0answers
70 views

The MultiSet (Bag) Monad on FinHilb

It was recently brought to my attention that the Bag monad, also known as the MultiSet monad, is not polynomial on Set, but is Polynomial on the category of Groupoids, 3.10 Examples. I then started ...
15
votes
1answer
274 views

Approximate eigenvectors for a set of non-commuting self-adjoint operators

This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics. Let $\mathcal H$ be a Hilbert space and $S$ a ...
1
vote
0answers
49 views

Proof — swapping sum with integral

Problem In Ceperley's 95 article on path integral Monte Carlo approach I have encountered $\hat{\rho}:L^{2}(R^{3N})\to L^{2}(R^{3N})$ $\hat{\rho} = e^{-\beta \hat{H}}$, where $\hat{H}$ is a ...
0
votes
3answers
131 views

Single quantum particle entropy

Consider a wave function of a single particle in free space, whose evolution is described by the (non-dimensional) linear Schrodinger equation $$i\psi _t (t,\underline{x}) + \Delta \psi=V(\underline{x}...
3
votes
2answers
317 views

Can one calculate the following operator? [closed]

Summary I recently defined some numbers which obey multiplication but not addition. To my surprise after some heuristic manipulations (ignoring convergence), it seems I can express the creation and ...
0
votes
0answers
67 views

Spectrum of a Hamiltonian on the real line

Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$ $$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$ Assume that $V$ is a smooth function and $V(x)\to +\...
3
votes
1answer
511 views

Is Quantum Mechanics (norm)-consistent?

I edited a few small comments to the question in order to make it perhaps more comprehensible. Today I came across the following question in quantum mechanics. In Quantum mechanics it is common to ...
3
votes
1answer
84 views

Quantum tunneling on the line with non-symmetric double well potential

Consider the Schroedinger equation on the line $$i\frac{\partial \Psi(x,t)}{\partial t}=[-\frac{d^2}{dx^2}+V(x)]\Psi(x,t),$$ where one assumes that $V(x)\to +\infty$ as $|x|\to +\infty$, and $V$ has ...
5
votes
2answers
222 views

An integral involving three Bessel functions

I am looking for a closed form for the following integral $$ I = \int_0^\infty \mathrm{d} x \ x \ J_0(ax) J_0(bx) J_1(cx) $$ which can be thought of as a particular case of the more general integral ...
4
votes
1answer
375 views

Question on Witten’s paper “Supersymmetry and Morse theory”

EDIT. I am trying to read the article “Supersymmetry and Morse theory” by E. Witten (JDG 17 (1982)). This well known article applies some tools developed by physicists (e.g. path integrals) to ...
4
votes
1answer
108 views

KMS-states of Bost-Connes type system

I have some struggles with understanding theorem 25 in the paper "Hecke algebras, type III factors and phase transitions with spontaneous symmetry breaking in number theory". More precisely, there is ...
5
votes
3answers
455 views

Closed, sum-free form for the $n$-th derivative of $\operatorname{arcsinh}(\frac1x)$ in $x=1$

During research involving the Born–Jordan quantization I came across the expression $$ \frac{d^k}{dx^k}\operatorname{arcsinh}\Big(\frac1x\Big)\Big|_{x=1}\tag1 $$ for $k\in\mathbb N_0$. It is not too ...
8
votes
2answers
249 views

Matrix exponential, containing a thermal state

This question was originally posted on MSE, and I'm cross posting it here. Define an infinite matrix $$ M = \begin{bmatrix} 0 & -1 & 0 & 0 & \cdots \\ 1 & 0 & -2 & 0 &...
1
vote
0answers
152 views

Are creation and annihilation operators closed?

I'm trying to understand some rigorous results in the basic formalism of second quantization and following some lectures I found on the internet I ran into a little trouble in defining creation and ...
1
vote
1answer
221 views

The Domain Monad

Many different kinds of data structures can be captured as Monads. Lists and trees are two good examples. A domain (dcpo) is like a tree, with extra axioms. Definition. A directed subset of a ...
1
vote
0answers
128 views

Why is $\textbf{J}$ called angular momentum?(Quantum) [closed]

Why is $\textbf{J}$ called angular momentum operator? Can anyone explain why the expectation value of J is angular momentum? Here is how $\textbf{J}$ is defined: The rotation operator $$ U(\alpha)=\...
4
votes
1answer
394 views

Does eigenvalue exist in a Hilbert space? [closed]

In a lecture on Quantum mechanics, the professor concluded that if $a$ is a linear operator with $[a, a^\dagger] = 1$, where $a^\dagger$ is the adjoint of $a$ and $[a, a^\dagger] = aa^\dagger - a^\...
0
votes
1answer
169 views

Does the uncertainty relation of Fourier transforms also extend to linear operators?

In Fourier theory, the pair composed of a variable and its Fourier transform is called conjugate variables, and one crucial property between the two is the uncertainty relation. This relation tells us ...
3
votes
1answer
127 views

What is the best numerical algorithm for integrating the 1D Schrödinger equation?

I'm interested in numerical algorithms for 1-dimensional Hamiltonians of the form $$ H = -\frac{d^2}{dx^2} + V(x) \quad \quad (1) $$ defined on the line ($x\in\mathbb{R}$) or on the circle. The ...
0
votes
1answer
89 views

QUBO formulation of a discrete-variable Genetic Algorithm optimization problem

I am facing a non-linear, discrete optimization problem, which I can formulate in this abstract manner: I have a certain non-analytic real-valued function $f$ depending on a set of parameters $ \theta\...
4
votes
0answers
97 views

Properties of solution to Schrödinger equation

Given a Schrödinger equation with, let's say continuous, periodic potential $$-y''(x)+V(x)y(x)=\lambda y(x)$$ where $V(x+1)=V(x)$ and $V$ is even, i.e. for $x \in (0,\frac{1}{2})$ we have $V(x+\frac{...
2
votes
0answers
62 views

Is this Frobenius Monad left exact? Does it preserve equalizers?

In this paper we see a Frobenius Monad in example 5.2. Suppose we take Hilb as the underlying category. Is this functor left exact? Does it preserve equalizers?
1
vote
1answer
101 views

Is this Frobenius Monad an internal category in [Hilb, Hilb]?

In this paper we see a Frobenius Monad in example 5.2. Suppose we take Hilb as the underlying category. Is this Frobenius Monad an internal category in [Hilb, Hilb]? If you can show that it is an ...
25
votes
8answers
2k views

On independence and large cardinal strength of physical statements

The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics. Maybe after all ...
11
votes
2answers
952 views

Is there any published article where $q$-mathematics is applied?

Excuse me for the concern, but I want to ask you a question. In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science ...
6
votes
0answers
211 views

Quantum Optimization as approximating $\mathbb{CP}^{2^n -1}$ with the orbits of a subgroup of SU($2^n$)

For example given a great circle within the sphere, we can think about computing the average distance of a point on the sphere from the great circle. Slightly more generally, given a subgroup $H \...
18
votes
2answers
832 views

Infinite dimensional symplectic geometry

Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis ...
4
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0answers
65 views

Fierz like identity for $\epsilon_{abc}\sigma^a_{ij}\sigma^b_{kl}\sigma^c_{pq}$

It is known that contracting over the vector indices of two Pauli matrix can be simplified to a bunch of delta functions. This is done via Fierz formula $$\delta_{ab}\sigma^a_{ij}\sigma^b_{kl}=\delta_{...
0
votes
0answers
34 views

The role of $dim H_n$ in the definition of asymptotically continous functions on vectors

When considering the asymptotic continuity of quantum states, one works with asymptoticly continuous functions. In the definition one has the following, a funtion f is asymptotically cts if for a ...
2
votes
0answers
72 views

Maslov canonical operator

Suppose $\Lambda$ is a Lagrangian submainfold of $M=T^*\mathbb{R}^n$. Let $x_i$ be the standard coordinate on the base manifold $\mathbb{R}^n$ and $\eta_j$ the coordinate on the dual. According to a ...
4
votes
1answer
309 views

Some questions about correlation functions and amplitudes in quantum field theory

I have been trying to learn some quantum field theory recently and I have a few questions which should be easy to answer for experts. I understand the basics of quantum mechanics / statistical ...
5
votes
2answers
259 views

Bounding a graph invariant

We are given a graph $G=(V,E)$, which has clique number $k$. The graph invariant in question is given by $$q_{\mathrm{a}}(G)=\min_T \min_{A\subset T} |T|-|A|$$ where $T$ is a transversal of the ...
2
votes
0answers
43 views

When is a 2D homogenous potential essentially self-adjoint? What about the potential $V(x,y)=x^4+y^4-\lambda x^2y^2$?

Suppose I consider the operator $$ -\Delta+V$$ for some potential $V(x)$ for $(x,y)\in\mathbb{R}^2$, as the closure of the corresponding operator on smooth compactly supported functions. If I assume ...
1
vote
4answers
290 views

Independence of two noncommutative observables

If two observables are free, you can find the joint distribution of these two observables. But, by Heisenberg's Uncertainty Principle it is impossible unless $X$ and $Y$ are such that $XY=YX$. Is ...
3
votes
0answers
53 views

Estimate the composition of a bounded multiplier with a trace class operator

Let $T$ be a trace class operator on $\ell^2 (\mathbb{N})$. Let $A$ be a multiplier on $\ell^2 (\mathbb{N})$ defined by a sequence $a=(a_n)_{n\in\mathbb{N}}$ in $\ell^{\infty} (\mathbb{N})$. That is, ...
30
votes
4answers
2k views

Representation theory and elementary particles

I have been looking for a clear expository mathematical text on the relation between the theory of elementary particles and the representation theory of $U(1), SU(2), SU(3)$, I was very disappointed ...
0
votes
1answer
261 views

Invariance of sets under Schrödinger equations

We are considering the Schrödinger equation on $\mathbb{R}^d \times [0,T]$ $$i \partial_t \psi(x,t)=-\Delta \psi(x,t) + u(t)V(x) \psi(x,t), t>0$$ $$\psi(x,0):=\psi(x_0) \in L^2(\mathbb{R}^d)$$ ...