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.
390
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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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1
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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 ...
4
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1
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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 ...
4
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1
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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 ...
2
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0
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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, ...
3
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1
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139
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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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0
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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 ...
12
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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 [...]. ...
4
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1
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156
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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 ...
5
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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 ...
3
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0
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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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0
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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 ...
3
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0
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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 ...
13
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1
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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 ...
4
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1
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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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1
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Estimate $\Vert \Delta u(t)\Vert_{2}$ in term of energy
We consider the wave equation
$$\left\{
\begin{array}{ll}
u_{tt}(x,t)-\Delta u(x,t)=0, x \in \Omega, t>0\\
u=0, \quad u \in \partial \Omega, t>0 \\
u(0,x)=u_{0}(x), \quad u_{t}(0,x)=u_{1}(x), x \...
4
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0
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Axiomatic string theory?
There have been many proposal of a mathematical definition of Quantum Field Theory, for instance through Wightman or Osterwalder-Schrader axioms. Were there any efforts toward doing the same for ...
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1
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SU(2) and entangled particles [closed]
We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...
2
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1
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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 ...
10
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0
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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 ...
10
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1
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327
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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 ...
34
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5
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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 ...
1
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1
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210
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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 ...
4
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2
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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 ...
4
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1
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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?
3
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1
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477
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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 ...
1
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1
answer
173
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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 ...
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3
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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 '...
3
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0
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249
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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 ...
14
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1
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818
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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 \...
6
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0
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278
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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
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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 ...
6
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1
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688
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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 ...
25
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1
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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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0
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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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0
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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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0
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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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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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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 $$\mathcal{H}_J=\bigotimes_{i\in J} \mathcal{H}...
2
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How to trap a particle without using potential field which is infinity at some point? (quantum physics) If impossible, how to prove it?
As we all know, the wave function of the stationary state a quantum particle trapped in a rigid box (with infinite potential outside the box) cannot have a non-zero value outside the box. So can we ...
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0
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Is there any property for the eigenvalues of an Hermitian matrix on which a well-structured binary mask has been applied?
While working on a quantum-focused article, I came accross the following problem. Let $\rho$ be a positive, semi-definite, $2^{n+m}$-Hermitian matrix with unit trace ($\rho$ is a density matrix). Let $...
4
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0
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Fourier transform without characters (Eigenfunctions of an operator)
Let's consider a very simple problem in quantum mechanics:
We have, in $\mathbb R,$ a potential barrier of the form
$$
V(x) = V_0 \mathbf 1_{[-a,a]}(x),
$$
where $\mathbf 1_{[-a,a]}$ denotes the ...
3
votes
2
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163
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Massive dirac operator symmetric spectrum
Consider the Dirac operator
$$ H = \begin{pmatrix} m & -i\partial_z \\ -i\partial_{\bar z} & -m \end{pmatrix},$$
where $\partial_{\bar z}$ is the Cauchy-Riemann operator and $m \ge 0.$
It is ...
7
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2
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983
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Energy levels of double well potential
Consider the (quantum) Hamiltonian on the real line
$$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 ...
1
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0
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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 ...
3
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2
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437
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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 $X\sim \mathrm{GUE} (d)$, ...
10
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1
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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, ...