# 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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### 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 ...

**9**

votes

**2**answers

440 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 ...

**4**

votes

**1**answer

378 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

**0**answers

52 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

**1**answer

86 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

**1**answer

162 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

**1**answer

112 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**

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**1**answer

107 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 ...

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votes

**2**answers

230 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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vote

**0**answers

39 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 ...

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vote

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63 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**

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**1**answer

160 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**

votes

**0**answers

127 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

**1**answer

123 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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**0**answers

72 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 ...

**16**

votes

**1**answer

284 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 ...

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vote

**0**answers

51 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

**3**answers

135 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

**2**answers

318 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

**0**answers

70 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

**1**answer

569 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

**1**answer

88 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

**2**answers

231 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

**1**answer

390 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

**1**answer

109 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

**3**answers

462 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

**2**answers

256 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

**1**answer

222 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 ...

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vote

**0**answers

130 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

**1**answer

396 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

**1**answer

203 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

**1**answer

128 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

**1**answer

104 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\...

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votes

**0**answers

102 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

**0**answers

63 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

**1**answer

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

**8**answers

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

**2**answers

963 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

**0**answers

212 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

**2**answers

859 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 ...

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**0**answers

67 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_{...

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votes

**0**answers

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 ...

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votes

**0**answers

74 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

**1**answer

331 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

**2**answers

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

**0**answers

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

**4**answers

299 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

**0**answers

54 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

**4**answers

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

**1**answer

261 views

### Invariance of sets under Schrödinger equations

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