# 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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### 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) $...

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

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### 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 ...

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### 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_\...

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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 ...

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### 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(...

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### 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 ...

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### 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, ...

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### 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 ...

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### 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 ...

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### 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\...

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### 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)*\...

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### 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 ...

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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 ...

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### 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-...

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### References for applications of Young diagrams/tableaux to Quantum Mechanics

I am interested in knowing more about applications of Young diagrams and Young tableaux to Quantum Mechanics. A friend of mine suggested as a reference the following book:
Wybourne, B.G.; "...

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### Motivation for Heisenberg's modeling of observables

What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ...

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### Finding a unitary operator on L^{2}(\mathbb{R}^{2},dxdy)

I have a one parameter (r) family of self-adjoint representations of the universal enveloping algebra of some nilpotent Lie group on $L^{2}(\mathbb{R}^{2},dxdy)$ as follows:
$$\hat{X}^{r}=\hat{x}-i(r-...

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### Approximating e-bits by CHSH black boxes

In the CHSH game two parties Alice and Bob independently get a bit $x$ and $y$ which is $0$ or $1$ with probability $\frac{1}{2}$.
Without communicating each of them has to send a referee a bit $a$ ...

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### On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues

Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...

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### Generators of polynomial invariant ring of compact Lie groups

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$I'm a PhD student in physics working in the broad area of photonic quantum computing. My current project looks at the ...

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### Mathematical characterization of gravitational geons as reference request, and their properties as main question

I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...

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### Size of Hilbert space in geometric quantization from index theorem

In these notes on geometric quantization by Nair, on page 24, the Bohr-Sommerfeld rule in quantum mechanics is interpreted in terms of the Atiyah-Singer index theorem.
To be precise, the polarization ...

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### "Open systems" version of Stone's Theorem for one-parameter groups of quantum operations

Let $H$ be a Hilbert space, which we interpret as a space of quantum states.
If $U(t):H\to H$ is a unitary norm-continuous one-parameter group with $U(0)=I$, (essentially) Cauchy's functional ...

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### Fourier transform harmonic oscillator eigenstates

The normalized eigenfunctions of the quantum harmonic oscillator are
$$\psi_{n}(x)= \frac{1}{\sqrt{2^n n!}} e^{-x^2/2}H_n(x),$$
where $n \in \mathbb N_0$ and $H_n$ is the $n$-th Hermite polynomial, ...

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### What is the finite-temperature orthogonal/symplectic Tracy-Widom distribution?

The Tracy–Widom distributions admit many interpretations.
One of them is related to quantum mechanics: If we consider $N$ non-interacting fermions confined by the potential $V(x) = x^2$, then in the ...

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### Classifying endomorphisms of a direct sum Hilberts pace

Suppose I have a Hilbert space with a direct sum structure into "superselection sectors", i.e. $\mathcal{H} = \oplus_\alpha \mathcal{H}_\alpha$, where $\alpha$ labels irreps of some group $G$...

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### Analyticity of solutions to Schrödinger's equation

Take Schrödinger's equation on $\mathbb{R}$, $i\partial_t\psi(x,t)=H\psi(x,t)$. Assume that $\psi(x,0)$ has compact support. Using known integral formulas for the propagators, it is fairly ...

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### Mathematical investigation of preparation of states in QM

In his (excellent, imo) Lectures on the Mathematics of Quantum Mechanics (2015), G. Dell'Antonio writes:
"The preparation of states in Quantum Mechanics [...] is a foundational
problem [...]. ...

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### Ground state energy of anharmonic oscillator: algebraic or transcendental?

Consider the quantum anharmonic oscillator, with Hamiltonian $H=p^2/2+q^2/2+gq^4$ for some real $g\geq 0$, with $p$ and $q$ obeying the usual Heisenberg commutation relations. For $g=0$, the ground ...

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### Can Fock spaces be replaced by arbitrary Hilbert spaces under some hypothesis to justify path integrals?

I was reading this post from PSE and it reminded me an old question of mine, in which the use of creation and annihilation operators were discussed. Both questions got answers which agreed on the fact ...

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### Can any POVM be induced by a quantum instrument?

I suspect this is the obvious result of something in operator algebras, but that's far outside my field.
Recall that a projection-valued measure is a map $E$ from a sigma-algebra $\mathcal{F}$ on ...

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### Find $\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right)$

I am doing a quantum optimization where the final problem has the following form
$$\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right),$$
where $V \in \mathbb{C}^{d\times ...

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### Orthonormal basis of eigenvectors of Hamiltonian - Is there any theorem justifying the physicist approach?

In his book The Principles of Quantum Mechanics, Dirac states:
"We call a real dynamical variable whose eigenstates form a complete set an observable."
To Dirac, any observable has a ...

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### Mathematical explanation of orbital shell sizes: why is it sufficient to consider single-electron wave functions?

Motivation
The question "Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?" asks for an explanation of the sequence 2, 8, 8, ...

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### MIP^*=RE and quantum computation

I recently learned about the MIP^*=RE result. I have to admit that I don't understand big parts of this paper and I am barely familiar with quantum physics. I hope my questions below make sense.
I ...

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### Is there a good mathematical explanation for why orbital lengths in the periodic table are perfect squares doubled?

$\DeclareMathOperator\SO{SO}\newcommand{\R}{\mathbb{R}}\newcommand{\S}{\mathbb{S}}$The periodic table of elements has row lengths $2, 8, 8, 18, 18, 32, \ldots $, i.e., perfect squares doubled. The ...

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### Why does the CHSH game need complicated bases to show advantage?

The CHSH game is the standard example of a game where two cooperating players Alice and Bob who cannot communicate, but who nevertheless can get an advantage by measuring an entangled quantum state in ...

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### Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy

We consider the wave equation
$$\left\{
\begin{array}{ll}
u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\
u=0, \quad u \in \partial \Omega, t>0 \\
u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \...

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### Axiomatic string theory?

There have been many proposal of a mathematical definition of Quantum Field Theory, for instance through Wightman or Osterwalder-Schrader axioms. Were there any efforts toward doing the same for ...

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### SU(2) and entangled particles [closed]

We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...

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### Different quantum computation models equivalence

There are different models of quantum computing like quantum circuits, adiabatic or annealing. Another thing to mention is the complexity class BQP. It is pretty much a given that the different models ...

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### “Taylor series” is to “Volterra series” as “Laurent series” is to _________?

Preamble
My question is similar to an earlier MathOverflow question:
“Taylor series” is to “Volterra series” as “Padé approximant” is to _________? which I just answered (hopefully my first ever ...

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### What are the predictive implications of conditional non-commutative probability?

To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$.
In this context a state $S$ is a positive semi-definite ...

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### What are the strongest arguments for a genuine quantum computing advantage?

Despite having become a fairly mature field with enormous sums of money dumped into research and development, there does not as yet exist a formal proof that quantum computation actually provides an ...

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### Does $\mathcal{KL}(D)$ admit the "yanking" axiom

Bob Coecke made the "yanking" axiom famous as he applied it to teleportation in Quantum Computing:
This is normally presented on the category of Hilbert spaces, and so here is a derivation ...

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### Axiomatizing projective Hilbert spaces

This question arises in connection to trying to take a different (more intrinsic) perspective on the foundations of quantum mechanics, in which projective Hilbert spaces naturally arise, e.g. see ...

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### An introductory reference for tensor networks

I found a good reference on Tensor Networks: https://arxiv.org/abs/1912.10049. But I need an introductory reference with detailed proofs on Tensor Networks. Do you know another reference?

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### Has the von Neumann entropy ever been used in classical mechanics?

After going through an application of the von Neumann entropy(from quantum information theory) to certain problems in computational neuroscience [2], it occurred to me that this entropy might have ...

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### limit of Riemann-Stieltjes sums as an integral on $\mathscr{H}$

I was reading Leon Takhtajan's book on quantum mechanics and, at some point, he states the J. von Neumann Theorem. The first part of this theorem is as follows.
For every self-adjoint operator $A$ on ...