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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Where does a math person go to learn quantum mechanics?

My undergraduate advisor said something very interesting to me the other day; it was something like "not knowing quantum mechanics is like never having heard a symphony." I've been meaning to learn ...
70 votes
10 answers
11k views

The Planck constant for mathematicians

The questions Q1. What are simple ways to think mathematically about the physical meanings of the Planck constant? Q2. How does the Planck constant appear in mathematics of quantum mechanics? In ...
Gil Kalai's user avatar
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56 votes
6 answers
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Is the Mendeleev table explained in quantum mechanics?

Does anybody know if there exists a mathematical explanation of the Mendeleev table in quantum mechanics? In some textbooks (for example in "F.A.Berezin, M.A.Shubin. The Schrödinger Equation") the ...
Sergei Akbarov'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
50 votes
10 answers
10k views

How is the physical meaning of an irreducible representation justified?

This is maybe not an entirely mathematical question, but consider it a pedagogical question about representation theory if you want to avoid physics-y questions on MO. I've been reading Singer's ...
Qiaochu Yuan's user avatar
46 votes
3 answers
10k views

Quantum mechanics formalism and C*-algebras

Many authors (e.g Landsman, Gleason) have stated that in quantum mechanics, the observables of a system can be taken to be the self-adjoint elements of an appropriate C*-algebra. However, many ...
Naz Miheisi's user avatar
37 votes
4 answers
3k views

Representation theory and elementary particles

I have been looking for a clear expository mathematical text on the relation between the theory of elementary particles and the representation theory of $U(1), SU(2), SU(3)$, I was very disappointed ...
mathphys's user avatar
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37 votes
2 answers
2k views

Is Langlands reciprocity somehow analogous to the wave-particle duality of quantum mechanics?

Apologies for the vague question, and for the many inaccuracies (I am not a physicist and barely a number theorist). In physics, there is the notion of gauge group of a field theory. The gauge group ...
Bruno Joyal's user avatar
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36 votes
9 answers
5k views

Why is addition of observables in quantum mechanics commutative?

I am no expert in the field. I hope the question is suitable for MO. Background/Motivation I once followed a quantum mechanics course aimed at mathematicians. Instead of the usual motivations coming ...
Andrea Ferretti'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
33 votes
10 answers
5k views

Toy Models of Quantum Mechanics

Do toy models of quantum mechanics help us better understand "regular" quantum mechanics? For example, if we look at quantum mechanics over a finite field $F$ (e.g. $\mathbb{Z}_2$), can this lead to ...
NebulousReveal's user avatar
31 votes
1 answer
7k views

"psi-epistemic theories" in 3 or more dimensions

In their recent paper The Quantum State Can Be Interpreted Statistically, Lewis et al. end with a very nice mathematical question, one whose answer (either way) would have interesting implications for ...
Scott Aaronson's user avatar
30 votes
8 answers
3k views

On independence and large cardinal strength of physical statements

The present post is intended to tackle the possible interactions of two bizarre realms of extremely large and extremely small creatures, namely large cardinals and quantum physics. Maybe after all ...
Morteza Azad's user avatar
30 votes
3 answers
5k views

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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29 votes
2 answers
2k views

What determines the maximal dimension of the irreps of a (finite) group?

I am chemist and ask for apologies for all my mathematical inabilities when asking this question in advance, but after quite a bit of searching I found that this problem could be "open" or ...
Raphael J.F. Berger's user avatar
28 votes
6 answers
5k views

Any real contribution of functional analysis to quantum theory as a branch of physics?

In the last paragraph of this last paper of Klaas Landsman, you can read: Finally, let me note that this was a winner's (or "whig") history, full of hero-worship: following in the footsteps of ...
25 votes
3 answers
3k views

On the periods in the periodic table (or Why is a noble gas stable?)

Added 22, November: I've succeeded in making the question entirely unintelligible with all my additions. So I thought I would summarize it in the simplest form I could manage and add it to the title. ...
Minhyong Kim'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
24 votes
3 answers
1k views

Is there a 'certainty' principle?

Heisenberg's uncertainty principle is a restriction on which probability distributions can describe the position and momentum of a quantum particle. In mathematical terms it says that if $\psi\in L^2$ ...
Oscar Cunningham's user avatar
24 votes
0 answers
1k views

conjectures regarding a new Renyi information quantity

In a recent paper http://arxiv.org/abs/1403.6102, we defined a quantity that we called the "Renyi conditional mutual information" and investigated several of its properties. We have some open ...
Mark M. Wilde's user avatar
23 votes
4 answers
3k 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 ...
MathMath's user avatar
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23 votes
5 answers
7k views

Can the equation of motion with friction be written as Euler-Lagrange equation, and does it have a quantum version?

My (non-expert) impression is that many physically important equations of motion can be obtained as Euler-Lagrange equations. For example in quantum fields theories and in quantum mechanics quantum ...
asv's user avatar
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21 votes
2 answers
3k views

States in C*-algebras and their origin in physics?

in $C^*-$algebras with unit element, there is the definition of a state, as a functional $\omega$ with $\omega(e)=||\omega||=1.$ Now, of course there is also in classical physics and quantum ...
Acuriousmind's user avatar
21 votes
4 answers
2k views

Why are quantum groups so called?

I've recently been to a seminar on quantum matrices. In particular the speaker introduced these objects as the coordinate ring of $2$ by $2$ matrices modulo some odd looking relations (see start of ...
Edward Hughes's user avatar
21 votes
2 answers
2k views

Infinite dimensional symplectic geometry

Could anyone comment on possible references concerning infinite dimensionsal symplectic manifolds?. I am mainly concerned with hilbert spaces, so i am not interested in the convenient analysis ...
Javier's user avatar
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19 votes
8 answers
2k views

A novice question on Quantum Mechanics

I'm currently working through Dirac's book The Principles of Quantum Mechanics. In it, he describes the nature of superpositions and at one point states: "... if the ket vector corresponding to a ...
Ryan's user avatar
  • 545
19 votes
3 answers
5k views

What is the relationship between algebraic geometry and quantum mechanics?

The basic relationship in algebraic geometry is between a variety and its ring of functions. Arguably a similarly basic relationship in quantum mechanics is between a state space and its algebra of ...
Qiaochu Yuan's user avatar
19 votes
2 answers
9k views

Mathematical "proof" of the stability of atoms?

I am trying to find proofs of the stability of an atom, says, for simplicity, the hydrogen atom. There are positive answers and negative answers in various atom models. The naive "solar system" model ...
user30830's user avatar
  • 213
18 votes
6 answers
3k 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 ...
MathMath's user avatar
  • 1,255
18 votes
3 answers
6k views

Classical limit of quantum mechanics

There is a well-known principle that one can recover classical mechanics from quantum mechanics in the limit as $\hbar$ goes to zero. I am looking for the strongest statement one can make concerning ...
dab's user avatar
  • 423
17 votes
4 answers
8k views

Noether's theorem in quantum mechanics

In classical mechanics: If a Lagrangian $\mathcal{L}$ is preserved by an infinitesimal change in the state space variables $q_i \to q_i + \varepsilon K_i(q)$, this leads to only second order change in ...
john mangual's user avatar
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15 votes
4 answers
2k views

Applications of Jordan algebras

Jordan algebras are non-associative algebras satisfying a somewhat strange (to me) list of axioms, see wikipedia. Basic examples are real symmetric and complex hermitian matrices with the product $A\...
asv's user avatar
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15 votes
2 answers
2k views

Is there an equivalent of Heisenberg's uncertainty principle in the decision sciences ?

From memories of a quantum mechanics class and Wikipedia: In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the ...
skillfeedback's user avatar
15 votes
1 answer
618 views

Approximate eigenvectors for a set of non-commuting self-adjoint operators

This problem is motivated by finding the right mathematical setting for expressing the compatibility of classical physics with quantum mechanics. Let $\mathcal H$ be a Hilbert space and $S$ a ...
David Mumford's user avatar
15 votes
0 answers
459 views

Squeezing physics out of formal deformation quantizations

I am reading various texts concerning the concept of "quantization". I am interested in quantization on Riemannian manifolds (as opposed to just on $\Bbb R ^n$); for absolute clarity, I am interested ...
Alex M.'s user avatar
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14 votes
4 answers
2k views

Meaning of a quantum field given by an operator-valued distribution

I am trying to grasp the basics of rigorous quantum field theory. Let me summise how the setup of non-interacting quantum field theories look like to me. Let $\mathcal{H}$ be a Hilbert space in which ...
Jannik Pitt's user avatar
  • 1,163
14 votes
2 answers
1k views

Is zero a hydrogen eigenvalue?

This question has been bugging me for some time. Take the hamiltonian for the hydrogen atom: $$\hat{H}=-\frac{1}{2}\nabla^2-\frac{1}{r},$$ acting on (a domain contained in) $L^2(\mathbb{R}^3)$. It is ...
Emilio Pisanty'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
  • 192
14 votes
4 answers
4k views

Topology on the space of Schwartz Distributions

If we equip the Schwartz space $\mathcal{S}$ with its usual Fréchet space topology, then the space of continuous linear functionals $\mathcal{S}^\ast$ is known as the space of Schwartz distributions ...
Jonathan Gleason's user avatar
14 votes
2 answers
694 views

Is a unitary Hamiltonian TQFT the same as a unitary axiomatic TQFT?

Introduction Axiomatic TQFTs An axiomatic $n$-dimensional TQFT is a symmetric monoidal functor $\mathcal{Z}\colon \operatorname{Bord}_n \to \operatorname{Hilb}$ from $n$-dimensional oriented ...
Manuel Bärenz's user avatar
14 votes
1 answer
854 views

Set theoretical realizations of the hidden variables program in quantum mechanics

The hidden variables program in quantum mechanics has been largely discredited by two powerful theorems, namely those of Bell and Kochen/Specker. Nonetheless, this program retains a certain ...
Alex Lupsasca's user avatar
13 votes
2 answers
2k views

$\mathrm{Bessel}^3$ Integral

I'm trying to calculate the following integral: $\int_0^\infty \mathrm{BesselJ}[l_0,k_0r] \cdot \mathrm{BesselJ}[l_1,k_1r] \cdot \mathrm{BesselJ}[l_0-l_1,kr] \cdot r\,dr$ ($\mathrm{BesselJ}[n,x]$ is ...
Mikhael's user avatar
  • 133
13 votes
3 answers
2k views

Linear algebra underlying quantum entanglement?

Hope this question is appropriate. I think I saw certain claims that quantum entanglement is a certain phenomena that can be explained (or modelled) in terms of tensor products in linear algebra. I ...
aglearner's user avatar
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13 votes
2 answers
1k views

Is quantum game theory reducible to classical game theory?

Quantum game theory is an extension of classical game theory to the quantum domain. It differs from classical game theory in three primary ways: Superposed initial states, Quantum ...
Řídící's user avatar
13 votes
1 answer
374 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, ...
Dan Romik's user avatar
  • 2,480
13 votes
1 answer
1k views

Which functions are Wiener-integrable?

I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them. Background The Wiener integral is an analytic tool to define certain "...
Theo Johnson-Freyd's user avatar
13 votes
1 answer
647 views

Reconciling two notions of geometric quantization.

Let $(M,\omega)$ be a compact symplectic manifold and $(L,\nabla)$ a prequantum line bundle. There are two schemes to quantize this data: Choose a polarization $P$ of $M$ and define the quantum ...
Eric O. Korman's user avatar
12 votes
2 answers
2k views

Is there any published physics article where $q$-mathematics is applied?

Excuse me for the concern, but I want to ask you a question. In 2002 Professor John Baez had published a few articles on his page regarding the possibility of applying $q$-mathematics in the science ...
Martin Bokner's user avatar
12 votes
3 answers
1k views

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 ...
Dominique Unruh's user avatar
12 votes
2 answers
3k views

What impact would P!=NP have on the characterization of BQP?

Many complexity theorists assume that $P\ne NP.$ If this is proved, how would it impact quantum computing and quantum algorithms? Would the proof immediately disallow quantum algorithms from ever ...
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