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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If NP=P, then does quantum mechanics have a classical interpretation? [closed]

Is it true that $BQP\subset NP$, if $NP=P$?
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3 votes
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107 views

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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Does the von Neumann equation hold for continuous functions of the density matrix?

Let $\rho$ be a density operator obeying $$i\hbar \partial_t \rho = [H,\rho]$$ where $H$ is a time-dependent Hamiltonian and $\rho(x,0)=\rho_I(x)$. The evolution is given by $$\rho(t) = U(t) \rho_I U(...
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1 vote
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37 views

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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2 votes
0 answers
63 views

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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Wigner transform and logarithms

The Wigner transform $W$ of a density matrix $\rho(x,y) = \sum_{j=1}^{\infty} \lambda_j u_j(x) \overline{u_j(y)}$ is given by $$W[\rho] = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d} e^{-i\xi\cdot y}\...
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12 votes
1 answer
258 views

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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4 votes
1 answer
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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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48 votes
2 answers
4k views

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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3 votes
1 answer
177 views

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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1 vote
1 answer
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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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-3 votes
1 answer
109 views

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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Can we describe the transition affected by the measurement of a quantum mechanical observable in the language of probability theory?

Consider a quantum mechanical system $S$ with the state space being given by a $\mathbb C$-Hilbert space $H$. It is assumed each physically measurable quantity $\mathcal A$ is described by an ...
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2 votes
1 answer
110 views

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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292 views

“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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10 votes
1 answer
284 views

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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34 votes
5 answers
3k views

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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1 vote
1 answer
180 views

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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4 votes
1 answer
122 views

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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4 votes
1 answer
100 views

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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3 votes
1 answer
310 views

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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1 vote
1 answer
133 views

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 ...
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19 votes
2 answers
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John von Neumann's remark on entropy

According to Claude Shannon, von Neumann gave him very useful advice on what to call his measure of information content [1]: My greatest concern was what to call it. I thought of calling it '...
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3 votes
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203 views

What are quantum extremal surfaces from a mathematical viewpoint?

It is said that they are surfaces which locally maximize area and bulk entanglement entropy. It would be great if I could receive some introductory material on it and some prerequisites to understand ...
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14 votes
1 answer
742 views

Spectrum of matrix involving quantum harmonic oscillator

The quantum harmonic oscillator relies on two classical objects, the so-called creation and annihilation operator $$a ^* = x- \partial_x \text{ and }a = x+\partial_x.$$ Fix two numbers $\alpha,\beta \...
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6 votes
0 answers
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Two questions about Fock spaces

Let $\mathscr{H}$ be a complex Hilbert space and denote $\mathscr{H}_{n}$ the tensor product $\overbrace{\mathscr{H}\otimes\cdots\otimes\mathscr{H}}^{\text{n}}$. Denote by $\Pi_{\pm}$ the projection ...
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1 vote
1 answer
224 views

Spectral theorem and diagonal expansion for self adjoint operators

Asked by a physicist: In physics we use the fact that, given a self adjoint operator $A$, every element in Hilbert space can be written as "generalized linear combination" of the eigenstates ...
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4 votes
0 answers
265 views

Quantum mechanics outside $L^{2}$ spaces

To this day, it is known that a satisfying mathematical formulation of quantum field theory is far from sight, even though some noninteracting theories can be described in rigorous mathematical ...
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24 votes
1 answer
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Reading list recommendation for a hep-ph student to start studying QFT at a more mathematically rigorous level?

Edition On July 15 2021, the description of the question has been considerably modified to meet the requirement of making this question more OP-independent and thus more useful for general readers. ...
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9 votes
0 answers
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Is there any physics theory which is similar to these analogies?

Since I am doing this little "research" project on my spare time and in my physical neighborhood there are not many people to discuss these ideas, I wanted to share with you a small point of ...
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1 vote
0 answers
78 views

Are resurgent theory and the topological recursion truly different or are, in some sense, the same formalism?

In Large genus behavior of topological recursion by B. Eynard, it is described that the topological recursion invariants $F_{g}$ could be a resurgent series (or Borel resummable). Another connection ...
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2 votes
0 answers
181 views

Frontiers of QM and QFT

This is somehow a more mature version of an old question of mine. I'd like to have a more clear picture of the difference between QFT and QM from a mathematical point of view. Okay, so we begin with a ...
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18 votes
6 answers
2k views

What is the best place to learn about the mathematical foundations of quantum mechanics?

I'm looking for good references to learn about the mathematical foundations of quantum mechanics. By mathematical foundations, I do not mean rigorous quantum mechanics in general but the axioms behind ...
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2 votes
0 answers
43 views

Existence of quantum states given reduced states on subsystems

Suppose $$\mathcal{H}=\bigotimes_{i\in I} \mathcal{H}_i$$ is a tensor product of Hilbert spaces, where $I$ is some index set. Given a $J\subset I$, let $$\mathcal{H}_J=\bigotimes_{i\in J} \mathcal{H}...
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2 votes
1 answer
223 views

How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?

As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
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1 vote
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Is there any property for the eigenvalues of an Hermitian matrix on which a well-structured binary mask has been applied?

While working on a quantum-focused article, I came accross the following problem. Let $\rho$ be a positive, semi-definite, $2^{n+m}$-Hermitian matrix with unit trace ($\rho$ is a density matrix). Let $...
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4 votes
0 answers
100 views

Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...
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3 votes
2 answers
130 views

Massive dirac operator symmetric spectrum

Consider the Dirac operator $$ H = \begin{pmatrix} m & -i\partial_z \\ -i\partial_{\bar z} & -m \end{pmatrix},$$ where $\partial_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$ It is ...
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7 votes
2 answers
653 views

Energy levels of double well potential

Consider the (quantum) Hamiltonian on the real line $$H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x).$$ Let us assume that the potential $V$ is an even smooth functions with exactly two non-degenerate ...
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1 vote
0 answers
82 views

Common core for unbounded operators

Suppose that $\mathcal H$ is a Hilbert space representing some physical system, $H$ is the Hamiltonian for the system, and $A$ is some observable for the system, that is, some unbounded self-adjoint ...
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3 votes
2 answers
335 views

Fourier transform of eigenvalue distribution of GUE matrices

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
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10 votes
1 answer
403 views

Basis of invariant tensors of rank n in three dimensions

[This is a question motivated by theoretical physics, so apologies if the language is rough...] In three dimensions the spaces of invariant (or isotropic) tensors of rank $n$ have dimensions 1, 0, 1, ...
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1 vote
0 answers
33 views

S-wave resonances in $V(r)=-c^2(r+1/r)^2, c \in \Re$

I need to know the s-wave resonances of the central potential $V(r)=-c^2(r+1/r)^2, c\in \Re$. Here we have a well attached to a barrier so we expect quantized quasibound/ metastable/ resonant states (...
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12 votes
1 answer
402 views

Is there a straightforward generalization of min(x,y) to positive-semidefinite Hermitian matrices?

This is an open-ended question I have. Is there a function of two positive-semidefinite hermitian operators $\min(A,B)$ returning another positive-semidefinite Hermitian operator such that: If A and ...
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21 votes
4 answers
2k views

Rigged Hilbert spaces and the spectral theory in quantum mechanics

I'm trying to learn some quantum mechanics by myself, and because of my mathematics background, I'm trying to understand it in a rigorous way. Since then, I've been intrigued by the use of rigged ...
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1 vote
1 answer
288 views

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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8 votes
2 answers
2k views

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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1 vote
0 answers
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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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4 votes
1 answer
214 views

Hilbert space representation of a vector in terms of a continuous eigenbasis

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 \...
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