# Tagged Questions

**1**

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**1**answer

229 views

### A question about complex polarization

Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar P ...

**3**

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**0**answers

48 views

### Closed form for 3j-symbol ratios

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...

**10**

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**3**answers

386 views

### Resource for learning quantum mechanics from the viewpoint of representation theory

Quantum mechanics is deeply connected with representation theory. Therefore, I'm looking for a textbook or article which presents quantum mechanics in a representation theoretic manner. Could anyone ...

**2**

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**1**answer

77 views

### An example to show that when $P$ is a complex polarization the subbundle $P+ \bar P$ is not necessarly involutive

I start my question with some motivation. A subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
$P$ is Lagrangian
P involutive
dim$P\cap\bar ...

**2**

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**0**answers

186 views

### a question about geometric quantization background

I don't understand why for geometric description of a regular system, we take always the classical phase space as a symplectic manifold?

**8**

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**4**answers

562 views

### Can the equation of motion with friction be written as Euler-Lagrange equation?

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 ...

**5**

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**2**answers

288 views

### C*-algebraic representation of observables vs self-adjoint operators one

I am trying to reconcile the "physicist" definition of an observable: self-adjoint operator on a Hilbert space, and the operational one as given by Strocchi in "An introduction to the mathematical ...

**5**

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**2**answers

208 views

### Certain partial integrations in quantum mechanics

In classical quantum mechanics (and specifically in the introductury texts on this topic) while calculating expectation values of certain operators in the Schrödinger approach we often have to do ...

**13**

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**1**answer

255 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 ...

**1**

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**0**answers

230 views

### How to estimate the quantum fidelity between two given states

There is a well-known theorem, firstly obtained by Denes Petz, in quantum information theory, which is described as follows:
$\mathbf{Theorem.}$ Let $\rho$ and $\sigma$ be two states on $\mathcal H$, ...

**3**

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**1**answer

125 views

### Three body problem with point interactions

I've studied the HVZ theorem for the three body problem interacting with regular potentials. I'd like to extend this result to the three body problem with point interactions (delta potentials).
Is ...

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**2**answers

302 views

### Translation of an article

I need to read this article
"On the spectrum of an energy operator for atoms with fixed nuclei in subspaces corresponding to irriducible representations of permutation groups"
authors:G.Zhislin, A. ...

**2**

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**0**answers

101 views

### Three body problem with two fermions and a different particle

I'm studying the three body problem with two fermions of unitary mass and another different particle. I need references of the HVZ theorem in this case. Is there someone who knows them?

**3**

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**1**answer

221 views

### Solvable models in quantum mechanics

Is there anyone who studied on the book "Solvable Models In Quantum Mechanics" by Albeverio? I don't succed in understanding the proof of page 116 about the eigenvalues of the Hamiltonian with point ...

**6**

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**7**answers

589 views

### Quantization of a classical system (e.g. the case of a billard)

There has been already several questions asking for an introduction to quantum mechanics
for a mathematician, but this ons is slightly different, and more restrictive. I know (some)
quantum mechanics, ...

**0**

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**2**answers

484 views

### Problem of quantization: state of the art

The "problem of quantization":
Find a vector space $Obs$ (as large as possible) of real-valued functions $f(p, q)$ on $R^{2n}$, containing the coordinate ...

**6**

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**2**answers

660 views

### Quantum mechanics basics [closed]

Hello. I'm thinking about where does the basic quantum mechanics things comes from. I mean the forms of operators and a Shroedinger equation. The more intuitive explanation is better.
To get forms of ...

**3**

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**1**answer

134 views

### Analytic continuation of instantaneous eigenstates of a time-dependent hamiltonian

We are considering the instantaneous eigenstates of an analytically time-dependent hamiltonian and I would like to know how legitimate it is to extend them to the complex plane.
Specifically, our ...

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**2**answers

1k views

### Classical Limit of Feynman Path Integral

I understand that in the limit that h_bar goes to zero, the Feynman path integral is dominated by the classical path, and then using the stationary phase approximation we can derive an approximation ...

**10**

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**2**answers

2k 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 ...

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**2**answers

594 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 ...

**5**

votes

**2**answers

932 views

### Space with 720° / not 2$\pi$ rotational symmetry?

Is there a space with a 720°, but no 360° rotational symmetry? Possibly one that can be mapped onto something more conventional like R(3) or R(3,1)?
The reason I am asking is because in quantum ...

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**2**answers

757 views

### Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?

I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.
Background
A simple consequence of the singular value decomposition is that any vector ...

**30**

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**3**answers

3k 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 ...

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**3**answers

1k views

### Justification for the matching condition for the wave function at potential jumps. Why is it both restrictive enough and sufficiently general?

Consider Schrödinger's time-independent equation
$$
-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi=E\psi.
$$
In typical examples, the potential $V(x)$ has discontinuities, called potential jumps.
Outside ...

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**1**answer

431 views

### What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors?

Construction
Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2:
$P^{(i)} = |a\rangle_i\langle a| + ...

**6**

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**3**answers

592 views

### Bounding a spectral gap: what proof techniques exist?

The following situation is ubiquitous in mathematical physics. Let
$\Lambda_N$
be a finite-size lattice with linear size
$N$. An typical example would be the subset of
...

**9**

votes

**1**answer

513 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 ...

**0**

votes

**1**answer

234 views

### The Quantum Operations On The Bipartite Systems

Given two distinct and noninteracting quantum mechanical
systems $\mathfrak{S}\_1$ and $\mathfrak{S}\_2$ with state spaces
$\mathcal H\_1$ and $\mathcal H\_2$, respectively, the state space
of the ...

**1**

vote

**0**answers

587 views

### Tensor products as isomorphic functors in category theory

An earlier question that I posed sought to define a category with a set of quantum channels as arrows and the C$^{*}$-algebra that these channels map from and to as the object. So, for example, my ...

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601 views

### Quantum channels as categories: question 1.

A quantum channel is a mapping between Hilbert spaces, $\Phi : L(\mathcal{H}_{A}) \to L(\mathcal{H}_{B})$, where $L(\mathcal{H}_{i})$ is the family of operators on $\mathcal{H}_{i}$. In general, we ...

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2k views

### Noether's Theorem in Quantum Mechanics

In classical mechanics:
If a Lagrangian L is preserved by an infinitesimal change in the state space variables qi -> qi + εKi(q) leads to only second order change in the Lagrangian:
$$ 0 = ...

**78**

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**41**answers

22k views

### 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 ...