Skip to main content

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.

Filter by
Sorted by
Tagged with
-3 votes
0 answers
86 views

Literature on representations of SU(2) on the direct product of irreps [closed]

I am in the situation where the action of SU(2) has lead my Hilbert space to be block diagonalised into a direct sum of irreps. For example I might have the tensor product of two qubits, i.e ...
East's user avatar
  • 143
6 votes
1 answer
366 views

On commutator of bounded operators

Let $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the bounded operators on $\mathbb H$. Let $J,K\in \mathcal B(\mathbb H)$ such that $ J=J^*, K=-K^*. $ Then the commutator $[J,K]$ ...
Bazin's user avatar
  • 15.3k
1 vote
1 answer
212 views

When is this operator positive semi-definite?

I have the following operator $$\Phi(\chi_A)=\int \text{d}\eta\, \text{d}\zeta\,\chi_A(\eta,\zeta)\,e^{i(\eta \hat{P}+\zeta\hat{Q})}.$$ With $\chi_A$ the indicator function associated to a set $A\...
Nicolas Medina Sanchez's user avatar
9 votes
1 answer
449 views

Quantum probabilistic method?

The probabilistic method uses arguments from probability to prove deterministic statements. This has been applied to diverse fields such as combinatorics, topology and number theory. In this method, ...
Riemann's user avatar
  • 645
2 votes
0 answers
56 views

Basis vectors using anti-commuting operators?

Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy: $$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
MathMath's user avatar
  • 1,275
4 votes
1 answer
160 views

Bound in terms of harmonic oscillator

I wonder if the following is true: Let $\alpha >0$ be a positive real number, do we have $$\Vert H^{\alpha} \psi''\Vert \le \Vert H^{\alpha+1} \psi\Vert,$$ where $H = -\frac{d^2}{dx^2} + x^2$ is ...
António Borges Santos's user avatar
0 votes
0 answers
29 views

Moment generating function for product states

In the sequel $B=M_\ell(\mathbb{C})$. For $M\in\mathbb{N}$ fixed and $N\geq M$ I consider the symmetrizer $\pi_{M,N}(x_M)\in B^{\otimes N}$, which is the symmetrized tensor product of $a_1$,...,$a_M$ ...
Kris's user avatar
  • 63
4 votes
0 answers
127 views

Is every pointwise-weakly continuous one-parameter group of automorphisms of B(H) given by a Hamiltonian?

Let $\mathcal H$ be a Hilbert space, $\mathscr B(\mathcal H)$ be the von Neumann algebra of all bounded operators on $\mathcal H$, and let $\sigma $ be a one-parameter group of automorphisms of $\...
Ruy's user avatar
  • 2,233
1 vote
0 answers
146 views

Recommendation to understand mean field theorem

I am studying Rodnianski and Schlein - Quantum Fluctuations and Rate of Convergence Towards Mean Field Dynamics. Everything was clear for me and I reproved everything before inequality (3.22) (except ...
Mr. Proof's user avatar
  • 159
2 votes
1 answer
170 views

Poisson quantization vs quantization in atomic physics

Is it possible to interpret quantization in atomic physics ( e.g. the quantization condition in hydrogen atom stated as exponential decay of wave functions at infinity and analogously for n-electron ...
0x11111's user avatar
  • 493
-2 votes
1 answer
125 views

Interpretation and validity of modified Heisenberg uncertainty principle in a metric context? [closed]

Considering the Heisenberg uncertainty principle, which states $\Delta x \cdot \Delta p \geq h$, I've explored a modified version by computing $(\Delta x + 1)(\Delta p + 1) \geq \Delta x \cdot \Delta ...
mathoverflowUser's user avatar
1 vote
0 answers
84 views

Definition of second quantization

The standard textbook for second quantization is Reed & Simon. However, I am a bit confused with their notation. They write: Let $\mathscr{H}$ be a Hilbert space, $\mathcal{F}(\mathscr{H})$ the ...
MathMath's user avatar
  • 1,275
4 votes
4 answers
432 views

Why computing $n$-point correlations?

I am trying to find a sufficiently convincing answer to this question, but it has been taking so much of my time and I can't get anywhere. It also follows my previous question on PSE. In axiomatic QFT,...
MathMath's user avatar
  • 1,275
4 votes
2 answers
210 views

Reference for rigorous interacting many-body quantum mechanics

Are there good references for (both zero and finite time) interacting systems of quantum many-body theory? More precisely, I would be interested in references discussing the following topics: Second ...
MathMath's user avatar
  • 1,275
1 vote
0 answers
143 views

Reconstructing the manifold from space of functions in quantum mechanics

Due to Banach–Mazur, every separable Banach space is isomorphic to a subspace of $C([0,1])$. But some spaces, like $C([0,1]^n)$ and generally $C(M)$ for $M$ a manifold, allow one to reason about the ...
0x11111's user avatar
  • 493
6 votes
0 answers
91 views

Conditions for completely positive maps to act homomorphically across multiple subalgebras

For a completely positive (CP) map $u: A \to A'$ of $C^*$-algebras $A, A'$, the concept of multiplicative domains characterizes the largest subalgebra of $A$ on which $u$ behaves as a $*$-homomorphism....
BiPolarBear's user avatar
1 vote
1 answer
128 views

Precise mathematical relation between chirality (or $\gamma_5$) and (spatial) orientation in $1+3$ Minkowski spacetime

This is a bit of a qualitative question, but I have great difficulty finding a reference that clarifies the point I have been confused about. So, I guess I need to ask here.. Let us restrict atttetion ...
Isaac's user avatar
  • 2,835
1 vote
1 answer
161 views

Inequalities involving entropy: quantum discord and mutual information

My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
truebaran's user avatar
  • 9,170
0 votes
0 answers
128 views

Generalized operator norm triangle inequality

Let $O_1, \cdots, O_n$ be Hermitian operators and $c_1, \cdots, c_n$ be complex numbers. If $\| \cdot \|$ denotes the operator norm, does the following inequality hold? $$\| \sum_{i=1}^N c_i O_i \| \...
curiousquantum's user avatar
12 votes
0 answers
750 views

Does this matrix norm inequality have interesting application in other areas of mathematics?

In my new paper, one of the main theorems gives an upper bound for the spectral distance of a general real symmetric matrix to diagonal matrices: Theorem 3. ‎Let $A=[a_{ij}]$ be a real symmetric ...
Mostafa's user avatar
  • 4,454
1 vote
0 answers
77 views

Definition of this formula for the $2p$ functions

I am reading this paper about constructive renormalization for fermions and I got a really basic question about it. There, the effective Lagrangian (with UV cutoff $\Lambda_{0}$ and IR cutoff $\Lambda$...
MathMath's user avatar
  • 1,275
0 votes
1 answer
123 views

From superoperator of unitary to the unitary itself

I have a question about super operators. Let's say I have the super operator of some unitary matrix $u$ called $SU$ where $SU = u^\ast\otimes u$ (here $u^\ast$ is the complex conjugate of $u$). If I ...
Mushahid Khan's user avatar
1 vote
0 answers
111 views

Sudden drop in complexity class due to the more general correlations

Recently I was asking about the impact of the groundbreaking result MIP*=RE on logic and proof theory (see this discussion). Surprising as it is I got confused with the following: MIP* is a ,,quantum''...
truebaran's user avatar
  • 9,170
5 votes
0 answers
244 views

Lie algebras, root systems and qubits

This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group ...
Malkoun's user avatar
  • 5,031
1 vote
0 answers
112 views

What is $H^*(\mathbb{CP}^{2^N-1}/\Sigma_n;\mathbb{Z})$ when $N=\binom{n}{2}$?

$H^*((S^3)^N/\Sigma_n;\mathbb{Q})$ is computed here. It makes a little more sense to compute $H^*((S^2)^N/\Sigma_n;\mathbb{Q})$ given that global phase is irrelevant. The proof is exactly the same. ...
Jackson Walters's user avatar
5 votes
1 answer
582 views

What is a particle in the context of QFT with interactions?

I'm a bit of a novice, so bear with me. My understanding of the story is as follows. From Lagrangians to Irreducible Representations The story of the types of possible particles begins with the ...
Mehmet Coen's user avatar
1 vote
0 answers
239 views

Riemann hypothesis and zero of Mellin transform of eigenfunction of Schrödinger equation

The paper [1] shows that the eigenfunction of the Schrödinger equation $$ \left(x^2-\frac{1}{4\pi}\frac{d^2}{dx^2}\right)f_n=\frac{2n+1}{2\pi}f_n $$ satisfies the same functional equation as the ...
George's user avatar
  • 221
5 votes
1 answer
261 views

von Neumann algebra of canonical commutation relations

In quantum mechanics we have position and momentum operators $P$ and $Q$ acting on $L^2(\mathbb{R})$ in the usual way. I'm wondering what the von Neumann algebra generated by the bounded functions of $...
J_P's user avatar
  • 439
0 votes
0 answers
62 views

How to derive the formulas of the spin-weighted spheroidal eigenvalues (2.16a)-(2.16g) in arXiv:gr-qc/0511111?

I am reading the article "Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions", which is on https://arxiv.org/abs/gr-qc/0511111. I want to ...
amon xu's user avatar
4 votes
0 answers
105 views

Frobenius norm bounds on exponentials of anti-Hermitian matrices

Suppose $X$ and $Y$ are two anti-Hermitian matrices satisifying $\|X\|, \|Y\| \leq \pi$, where $\|\cdot\|$ is the spectral norm. I'm trying to prove the following bounds on the Frobenius norm of the ...
Haimeng Zhao's user avatar
1 vote
0 answers
151 views

A transformation game for natural numbers?

Consider the completely additive function $\eta(n) := \sum_{p\mid n} v_p(n)p$ defined on natural numbers, with values in natural numbers. For literature, on this function, see the corresponding OEIS ...
mathoverflowUser's user avatar
0 votes
0 answers
101 views

A variant of quantum harmonic oscillators

We have the following variant of harmonic oscillators. $$ \left\{ \begin{array}{**lr**} T = a + a^\dagger\\ a | n \rangle = \sqrt{[n]} |n-1 \rangle \\ a^\dagger |n\rangle = \sqrt{[n+1]} |n+1\...
Lili Si's user avatar
  • 105
10 votes
2 answers
462 views

Does approximate equality of quantum states imply operator inequality in a large subspace?

Let the trace norm of $X$ be $$\Vert X\Vert_1 := \operatorname{tr} \left(\,(X^\dagger X)^{1/2}\right)$$ and let the operator inequality $A \leq B$ denote that the operator $B-A$ is positive ...
Noel's user avatar
  • 165
4 votes
1 answer
1k views

What is Quantum Geometry supposed to be about?

There are meaningful questions we can ask about Euclidean geometry which could not have been posed in the time of Riemann or even of Hilbert, and which would have made no sense at all to Euclid. For ...
Babu's user avatar
  • 1,299
7 votes
2 answers
521 views

Weak convergence related to Hermite polynomial?

I am reading Griffiths's quantum mechanics book; in the section about harmonic oscillators, he wrote out the amplitude of wave function, and compared with the classical harmonic oscillators. He ...
Yuval's user avatar
  • 637
2 votes
1 answer
178 views

Manifold of entangled states

Given a specific density matrix $\rho$ that corresponds to an entangled quantum state, I would like to find a class of operators (that might be $\rho$ dependent) that tranform (with high probability) $...
Fabio's user avatar
  • 329
2 votes
0 answers
50 views

Coefficient growth upper bound of a recurrence relation

Consider the recurrence relation one can obtain from the radial Schrödinger equation for the hydrogen atom, where $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$: $$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$ ...
Godzilla's user avatar
  • 121
1 vote
0 answers
118 views

Probabilistic interpretation of von Neumann's approach to quantum mechanics

One of the basic postulates in the mathematical formalism of quantum mechanics is that the probability of a measurement of an observable $A$ in the state $\psi \in \mathscr{H}$ to return a value in a ...
MathMath's user avatar
  • 1,275
0 votes
0 answers
171 views

About the proof of Lebesgue decomposition theorem for Hilbert spaces

Let $\mu$ be a Borel measure on $\mathbb{R}$. By the Lebesgue decomposition theorem, there exists measures $\mu_\text{pp}$, $\mu_\text{ac}$ and $\mu_\text{sing}$ such that $\mu = \mu_\text{pp}+\mu_\...
MathMath's user avatar
  • 1,275
1 vote
0 answers
202 views

Characterization of the Hamiltonian's spectrum in quantum mechanics

This is basically the same question I made on physics stack exchange The spectrum of the Hamiltonian in quantum mechanics, but I got no answers so far and decided to move it to mathoverflow with some ...
MathMath's user avatar
  • 1,275
3 votes
1 answer
117 views

Time-dependent quantum dynamics as dynamical system and ergodic therorems

In full glory, a dynamical system is defined as a tuple $(T,M,\Phi)$ where $T$ is a monoid, written additively, $M$ is a set, and $\Phi$ is a function. $$ \Phi : U \subset T \times M \to M$$ with $ I(...
nervxxx's user avatar
  • 207
15 votes
4 answers
2k views

Meaning of a quantum field given by an operator-valued distribution

I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me. Let $\mathcal{H}$ be a Hilbert space in which ...
Jannik Pitt's user avatar
  • 1,191
2 votes
0 answers
45 views

Moduli spaces of 'generalized mutually unbiased bases'

Mutually unbiased bases in $\mathbb{C}^n$ with a chosen inner product are collections of orthonormal bases such that for each pair of orthonormal bases $e_i,f_i$, $i=1,\ldots,n$ we have $|\langle e_i, ...
Sergey Guminov's user avatar
8 votes
2 answers
544 views

Interpretation of spectral measures in quantum mechanics

Let us define a pure vector state of a quantum system as a vector $\psi$ in a Hilbert space $\mathscr{H}$ with norm $\|\psi\| = 1$. Let $\mathscr{B}(\mathscr{H})$ be the Banach space of bounded linear ...
MathMath's user avatar
  • 1,275
2 votes
0 answers
80 views

Evolution equation in renormalization group for infinitely-many variables

Let $\varepsilon > 0$, $L \gg 1$ and define the torus $\mathbb{T} = \varepsilon \mathbb{Z}^{d}/L\mathbb{Z}^{d}$. Let $K$ be a smooth, strictly decreasing function. To make things easier, consider ...
MathMath's user avatar
  • 1,275
6 votes
2 answers
429 views

Spectrum of operator involving ladder operators

The ladder operator in quantum mechanics are the operators $$a^\dagger \ = \ \frac{1}{\sqrt{2}} \left(-\frac{d}{dq} + q\right)$$ and $$a \ \ = \ \frac{1}{\sqrt{2}} \left(\ \ \ \!\frac{d}{dq} + q\...
António Borges Santos's user avatar
1 vote
0 answers
66 views

Moyal products of exponentials

I have trouble verifying a claim made by Sharan in his paper: https://doi.org/10.1103/PhysRevD.20.414. What he essentially claims is that the sequence of Moyal $*$-products $$\underbrace{\exp(-itH/n)*\...
Kostas's user avatar
  • 11
6 votes
0 answers
119 views

Can two eigenfunctions be almost linearly dependent in a region?

Consider the Schrödinger operator $H=-\Delta+|x|^a$ on $\mathbb{R}$, where $a>0$. Since the potential is growing at $\infty$, we have compact resolvent thus the eigenvalues are discrete and tend to ...
Tomas's user avatar
  • 869
4 votes
0 answers
117 views

Generalizing Kato-Seiler-Simon-type inequalities to diamagnetic operators

I recently learned about estimates one can perform with operators on $L^2(\mathbb{R}^n)$ given as $f(x)g(-i\nabla)$, see Chapter 4 in Trace Ideals and their Applications by Professor Barry Simon (the ...
garserdt216's user avatar
0 votes
1 answer
296 views

Dose density matrix with off-diagonal elements equal to zero has maximum von-Neumann entropy?

von-Neumann entropy I know von-Neumann entropy on density matrix $S=-{\rm Tr}(\rho \ln\rho)$ is similar to Shannon entropy $S=-\sum_i p_i\ln p_i$ in classical mechanics. And I want to get Bose-...
lbyshare's user avatar

1
2 3 4 5
8