# 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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### Physics applications of quantum logic

Are there any examples of quantum logic being applied to solve actual physical questions, in particular to predict the physical properties (spectrum etc.) of some quantum-mechanical system? (Note that ...
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### Observable nearly commuting with a “complete” set of commuting observables

Consider the Hilbert space $H = E^{\otimes n}$ where $E=\mathbb{C}^2$. On $E$ we have an observable $O$ (i.e. a Hermitian matrix) that is diagonalizable in the standard basis with eigenvalues $1$ and ...
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### How to calculate Fourier transformation of eigenstates in CV quantum information [closed]

The position $\hat{q}$ and momentum $\hat{p}$ has $[\hat{q},\hat{p}]=i$. And we set there eigenstates as $|s\rangle_q$ and $|s\rangle_p$ with eigenvalue s. In the paper [Phy Rev A. 79, 062318 (2009)],...
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### QFT and its notations

I know hardly anything about quantum field theory (QFT) but I'm giving a try to understand some ideas of it. As far as I understand, in QFT one is interested in studying measures such as: \begin{...
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### Textbook covering superoperators and tensor products

I am looking for a textbook to cover the following tensor product (and, of course, the theory around it): Let $\otimes_1$ denote the tensor product on Hilbert spaces. Let $\otimes_2$ denote the ...
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### Is the set of separable quantum states closed?

Let $\mathcal H,\mathcal H'$ be Hilbert spaces (not necessarily separable). A "separable state" is a trace-class operator of the form $\sum_i \rho_i\otimes\rho_i'$ where $\rho_i,\rho_i'$ are positive ...
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### q-difference equations and quantum mechanics

I have been trying to understand why the term quantum is so easily accepted for calculus based on q-numbers $[n]_q=\frac{q^n-1}{q-1}$ and q-analogs of classical operators (derivatives, integrals,...). ...
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### How can one integrate over the unit cube, subject to certain (quantum-information-theoretic) constraints?

To begin, we have two constraints \begin{equation} C1=x>0\land z>0\land y>0\land x+2 y+3 z<1 \end{equation} and \begin{equation} C2=x>0\land y>0\land x+2 y+3 z<1\land x^2+x (3 z-2 ...
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### Reference on iterated integrals against projection valued measures

I know (to some extent) how integration over $\mathbb{R}$ of a Borel-measurable function against a projection-valued measure works. Recently while reading a paper I came across calculations in which ...
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### Complete positivity with infinite dimensional auxillary spaces

The usual definition of complete positivity (e.g. Stinespring (1955), or Holevo's Statistical Structure of Quantum Theory) is that a linear map between (sub $C^*$ algebras of) the bounded operators on ...
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### Reference request: Energy variance of randomly drawn state

In quantum mechanics, given an $N$-qubit ($2^N$-dimensional) Hamiltonian $\hat{H}$, I'm fairly sure that the variance in energies of randomly drawn pure states (i.e. norm-$1$ vectors) may be ...
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### Localization of solutions for time-dependent Schroedinger equation

I've been playing around with numerical solutions to the Schroedinger equation and I came across something that feels very natural, but I was not able to prove it with the math I know. The ...
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### What role do Hecke operators and ideal classes perform in “Quantum Money from Modular Forms?”

Cross-posted on QCSE An interesting application of the no-cloning theorem of quantum mechanics/quantum computing is embodied in so-called quantum money - qubits in theoretically unforgeable states. ...
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### Decomposition of the group of Bogoliubov transformations

Consider the fermion Fock space $\mathcal{F}=\bigoplus_{k\ge 0}\bigwedge^k\mathfrak{h}$ of some finite-dimensional 1-particle Hilbert space $\mathfrak{h}$. The group $\mathrm{Bog}(\mathcal{F})$ of ...
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### Do any finite predictions of Quantum Mechanics depend on the set theoretic axioms used?

I was wondering if any of the finite predictions of Quantum Mechanics depend on what set theoretic axioms are used. We will say that Quantum Mechanics makes a finite prediction about an experiment if,...
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### Inductive limits of unitary groups and quantum mechanics

I'm curious if someone has seen concrete applications of $U(\infty)$ in quantum mechanics. Is it possible, for example, in some particular cases to write down the propagator as a limit of a sequence ...
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 ...