# 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.

395
questions

-3
votes

0
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### 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 ...

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 $[J,K]$ ...

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\...

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, ...

2
votes

0
answers

56
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### 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}...

4
votes

1
answer

160
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### 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 ...

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 $a_1$,...,$a_M$ ...

4
votes

0
answers

127
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### 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 $\...

1
vote

0
answers

146
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### 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 ...

2
votes

1
answer

170
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### 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 ...

-2
votes

1
answer

125
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### 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 ...

1
vote

0
answers

84
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### 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 ...

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,...

4
votes

2
answers

210
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### 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 ...

1
vote

0
answers

143
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### 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 ...

6
votes

0
answers

91
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### 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....

1
vote

1
answer

128
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### 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 ...

1
vote

1
answer

161
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### 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 ...

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 \| \...

12
votes

0
answers

750
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### 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 ...

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 $\Lambda$...

0
votes

1
answer

123
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### 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 ...

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''...

5
votes

0
answers

244
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### 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 ...

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.
...

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 ...

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 ...

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 $...

0
votes

0
answers

62
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### 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 ...

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 ...

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 ...

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\...

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 ...

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 ...

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 ...

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) $...

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 $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$:
$$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$
...

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 ...

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_\...

1
vote

0
answers

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### 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 ...

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(...

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 ...

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, ...

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 ...

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 ...

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\...

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)*\...

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 ...

4
votes

0
answers

117
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### 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 ...

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-...