# 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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### Is there a Hilbert space approach to commutative probability theory on locally compact spaces?

I was recently made aware (thanks to the answers on Why does Riesz's Representation Theorem apply in quantum mechanics?) that the $C^*$ algebra approach and the Hilbert space approach to quantum ...
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### Why does Riesz's Representation Theorem apply in quantum mechanics?

$\DeclareMathOperator\tr{tr}$One begins with a quantum mechanical system, i.e. a unital $C^*$-algebra $A$. It is common to begin the discussion with embedding $A$ into the algebra of bounded operators ...
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### Domain issues regarding the Duhamel formula for the linear Schrödinger equation

I have some questions in succession regarding the rigorous domain issues about a Duhamel expansion formula (stated near the end of my post) for the linear Schrödinger equation. Consider a linear ...
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Let $\mathscr{H}$ be a complex Hilbert space and $A$ be an Hermitian operator $A: \mathscr{H}\to \mathscr{H}$. Suppose, for a moment, that $A$ has a set of discrete eigenvalues $\{\lambda_{n}\}_{n\in \... 1answer 298 views ### Reference request for Deterministic$\subset$Random$\subset$Quantum I hope this post is on topic as a reference request. I have seen somewhere the idea of (and saw it written just like this): $$\text{Deterministic }\subset\text{ Random }\subset\text{ Quantum }.$$ I am ... 1answer 148 views ### What is the precise relationship between real Poisson algebras and commutative$C^*$algebras? I've been teaching myself quantum mechanics, and I realized that I'm missing something fundamental. Namely, there are two pictures that I don't know how to reconcile: Quantum Mechanics generalizes ... 1answer 86 views ### Is there a Bell inequality for each of$2\times 2$,$3\times 1$,$2\times1\times1$and$1\times1\times1\times1$configurations? There was no answer in https://physics.stackexchange.com/questions/600494/is-there-a-bell-inequality-for-2-times-2-and-1-times1-times1-times1-configur. Hence posting in mathoverflow on the possibility ... 0answers 23 views ### Convergence of steady states for Lindblad systems in infinite volume In the physics of open quantum systems it is common to consider the Lindblad form. Which for a (super)-operator$\mathcal{L} \in B(B(\mathbb{C}^n ))means that \begin{align*} \mathcal{L}(\rho) = - i \... 0answers 58 views ### Generalized Ising Model I am in very trouble with a particular expression. I leave the original pages in order to have everything available and what I am goin to leave are the first pages of nine chapter of Non Perturbative ... 0answers 72 views ### There is no dispersion free quantum state onB(H)$Let$H$be a Hilbert space and$P(H)$denotes the lattice of all orthogonal projections on$H$. The famous generalized Gleason theorem states that if$\mu:P(H) \to \mathbb{R}$is a finitely additive ... 0answers 107 views ### How much Gleason type theorem do I need? Quasi states vs. states Let$\varphi$be a quasi state on$B(H)$. What does it mean? It means that$\varphi(cA)=c\varphi(A)$for$c \in \mathbb{C}, A \in B(H)$,$\varphi(A) \geq 0$for positive$A$and$\varphi(A+B)=\varphi(...
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As we know gradient and Hessian of a map on Banach spaces are linear transforms (Frechet derivatives). In quantum control, control objective is a map which is defined on control fields as the ...
Let us order the four nonnegative eigenvalues, summing to 1, of a (by definition, $4 \times 4$, Hermitian, nonnegative definite, trace one) "two-qubit density matrix" ($\rho$) as \begin{...