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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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 ...
Deepesh Singh's user avatar
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1 answer
266 views

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 ...
user142929's user avatar
4 votes
1 answer
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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 ...
Mtheorist's user avatar
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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 ...
Yonah Borns-Weil's user avatar
2 votes
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147 views

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, ...
Pritam Bemis's user avatar
3 votes
1 answer
139 views

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 ...
LeechLattice's user avatar
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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 $G$...
Sam Makhoul's user avatar
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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 ...
J_P's user avatar
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1 answer
341 views

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 [...]. ...
Alessandro Della Corte's user avatar
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1 answer
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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 ...
Matt Hastings's user avatar
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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 ...
IamWill's user avatar
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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 ...
Yonah Borns-Weil's user avatar
1 vote
0 answers
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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 ...
Trong Duong's user avatar
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356 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 ...
MathMath's user avatar
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13 votes
1 answer
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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, ...
Dan Romik's user avatar
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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 ...
Ioannis Souldatos's user avatar
52 votes
2 answers
5k 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 ...
Eugene Stern's user avatar
4 votes
1 answer
625 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 ...
Frederik Ravn Klausen's user avatar
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1 answer
71 views

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 \...
Yuji's user avatar
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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 ...
Giafazio's user avatar
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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| ...
aldous99's user avatar
2 votes
1 answer
251 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 ...
H.C Manu's user avatar
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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 ...
Nike Dattani's user avatar
10 votes
1 answer
327 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 ...
Mehmet Coen's user avatar
34 votes
5 answers
4k 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 ...
user6873235's user avatar
1 vote
1 answer
210 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 ...
mathlete42's user avatar
4 votes
2 answers
232 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 ...
Keefer Rowan's user avatar
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1 answer
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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 ...
Aidan Rocke's user avatar
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1 vote
1 answer
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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 ...
MathMath's user avatar
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30 votes
3 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 '...
Aidan Rocke's user avatar
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3 votes
0 answers
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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 ...
Siddharth Panigrahi's user avatar
14 votes
1 answer
818 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 \...
Kung Yao's user avatar
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6 votes
0 answers
278 views

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 ...
IamWill's user avatar
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1 vote
1 answer
323 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 ...
Rosario's user avatar
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6 votes
1 answer
688 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 ...
MathMath's user avatar
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25 votes
1 answer
2k views

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. ...
Qi Tianluo's user avatar
8 votes
0 answers
1k views

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 ...
mathoverflowUser's user avatar
1 vote
0 answers
92 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 ...
rgvalenciaalbornoz's user avatar
2 votes
0 answers
229 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 ...
IamWill's user avatar
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18 votes
6 answers
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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 ...
MathMath's user avatar
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2 votes
0 answers
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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}...
Josh Kirklin's user avatar
2 votes
1 answer
252 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 ...
owenBBT's user avatar
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1 vote
0 answers
82 views

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 $...
Tristan Nemoz's user avatar
4 votes
0 answers
133 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 ...
Ma Joad's user avatar
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3 votes
2 answers
163 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 ...
Landauer's user avatar
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7 votes
2 answers
983 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 ...
asv's user avatar
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1 vote
0 answers
123 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 ...
Isaac's user avatar
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3 votes
2 answers
437 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 $X\sim \mathrm{GUE} (d)$, ...
Michał Oszmaniec's user avatar
10 votes
1 answer
655 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, ...
Austen's user avatar
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