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0
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1answer
111 views

What are the boundary conditions for EM waves normally incident on the interface between two dielectric media?

An EM wave, amplitude $E_0$, frequency $\omega_0$, is incident upon a material with relative permittivity (dielectric function) $$\varepsilon \left( z \right) = \left\{ \begin{gathered}{\varepsilon ...
0
votes
2answers
50 views

When generalizing from discrete (but infinite) eigenstates to continuous eigenstates, Why do we change the definition?

The propagator function for discrete eigenstates is $$u(t)=\sum_{n=1}^{\infty}|E_n\rangle\langle E_n|e^{-iE_nt/ \hbar } \tag{1}\ .$$ But when we have continuous eigenstates, (like for the case of ...
0
votes
1answer
244 views

Quantum mechanics, operator commutes with Hamiltonian

My textbook said, if an operator $\hat{O}$ commutes with the Hamiltonian, then we can use the eigen vectors of the Hamiltonian as a basis of the Hilbert space, then express the operator $\hat{O}$ in ...
0
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0answers
23 views

Reference request for supersymmetric localization

I would like to ask for some readable introduction or maybe review of the technique of supersymmetric localization for $\mathcal{N}=1,2$ SUSY theories. I would like a different one than the one people ...
-2
votes
3answers
317 views

Given the Wikipedia notion of “arc length”, how is its manifestly real “signed variant” to be called and denoted?

I am dissatisfied with the presentation (not to say "definition") of "arc length", in its "Generalization to (pseudo-)Riemannian manifolds", as given in Wikipedia. (Who isn't?. But I'll sketch it here ...
0
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0answers
8 views

otto engine efficiency van der waals gas

I am trying to work out the efficiency of an Otto engine with the working substance being a Van der Waal's gas. I know that $ (P + Na^2 /V^2)(V-Nb)^α = const$ and that α= Cp/Cv. And then that $ ...
10
votes
5answers
2k views

Hubble's law and conservation of energy

If all distances are constantly increasing, as Hubble's law say, then lots of potential energies of form ~$\frac{1}{r}$ changes, so how is the total energy of the Universe conserved with Hubble's ...
0
votes
0answers
32 views

Completeness of solutions and the separation of variables method [migrated]

The method of separation of variables is introduced in every textbook on mathematical physics. A basic question is rarely addressed: does this method exhaust all the solutions? Is there any ...
22
votes
6answers
2k views

Is the converse of Noether's first theorem true: Every conservation law has a symmetry?

Noether's (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. Is the converse true: Any conservation law of a physical ...
2
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3answers
117 views

Expression of density operator

States in Quantum Mechanics can be thought of as density operators, i.e., positive semi-definite, normalized trace class operators on a Hilbert Space $\mathcal{H}$. In the case ...
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0answers
8 views

Is there any relation between single crystal elastic constants and plastic deformation?

The elastic constants are the tensor coefficient in Hook's Law. Which are valued only in elastic region.But, we are using elastic constants to find the hardness.The hardness is the measure of the ...
2
votes
0answers
31 views

Asymptotic Analysis of 1-D Schrödinger Equation [closed]

I'm looking to do a small personal project regarding the time independent Schrödinger equation in 1-D: $$y'' +V(x)y=Ey$$ $$y''=Q(x)y$$ where $ Q(x):=E-V(x) $. There is obviously nothing stopping ...
3
votes
2answers
57 views

Symplectic structure and isomorphisms

In his book Mathematical Methods of Classical Mechanics, V.I. Arnold writes To each vector $\xi$, tangent to a symplectic manifold $(M^{2n},\omega^2)$ at the point $\mathbf{x}$, we associate a ...
2
votes
1answer
77 views

Is the Hilbert space spanned by both bound and continuous hydrogen atom eigenfunctions?

As e.g. Griffiths says (p. 103, Introduction to Quantum Mechanics, 2nd ed.), if a spectrum of a linear operator is continuous, the eigenfunctions are not normalizable, therefore it has no ...
1
vote
0answers
27 views

Solving inhomogeneous Stokes equation

I want to solve the Stokes inhomogeneous equation, i.e. $$\nabla^2 \vec v -\nabla P = \vec f(r,\theta)$$ $$\nabla\cdot\vec v=0$$ where $\vec f$ is irrotational, i.e. $\partial_y f_x - \partial_x ...
0
votes
1answer
30 views

Fractional exponent in a scalar quantum field: Is energy and momentum conserved in this case?

Assuming that I would have the following term in the Lagrangian for a scalar boson field $$L=\int d^4x g (\phi^{2-p} \phi^{\dagger 2+p}+\phi^{\dagger 2-p} \phi^{2+p}))$$ with a fractional number $p$. ...
0
votes
1answer
36 views

A conjecture about the Møller operator

Consider the Møller operator $$ \Omega_+ = \lim_{t \rightarrow -\infty } e^{i H t } e^{- i H_0 t } , $$ Now, suppose a state $\psi $ is located far away from the potential $V = H- H_0$. I feel that ...
4
votes
2answers
148 views

Isomorphism of rigged Hilbert spaces

In connection with the statement that QM can be formulated in terms of separable complex (rigged) Hilbert spaces, the fact that all infinite dimensional separable complex Hilbert spaces are isomorphic ...
0
votes
0answers
48 views

Does this Hamiltonian have point spectrum?

Consider such a Hamiltonian $$ H = - \frac{1}{2} \frac{\partial^2}{\partial x^2} - F x + V(x) ,$$ with $F$ being some constant, and $V(x)= V(x+L)$ being some periodic potential. Does this ...
2
votes
0answers
33 views

What is the essential concept behind the difference in the fundamental solutions of the Stokes and Poisson equations?

The fundamental solutions, i.e., the solution with a point source, of the Poisson's equation and the Stokes equations in 3D are: $$\nabla^2 f=\delta(\boldsymbol x) \ \Longrightarrow\ G(\boldsymbol ...
0
votes
0answers
29 views

How is SO(2) compact according to this definiton? [migrated]

According to MathWorld, a compact Lie group is a group whose parameters vary over a closed interval. I'm not sure if this definition is rigorous enough. I've also seen a similar definition here: ...
5
votes
1answer
156 views

Noether's Theorem: Lie algebra, Lie groups

I've had a brief look through similar threads on this topic to see if my question has already been answered, but I didn't find quite what I was looking for, perhaps it is because I'm finding it hard ...
2
votes
2answers
126 views

Wave functions as $x$ goes to infinity

This problem emerged when I was going through some QM exercises: I've been asked to find the commutator $[A,B]$ where $A,B$ are defined as $$A\psi(x)=x\frac{\partial }{\partial x}\psi(x),$$ ...
1
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0answers
25 views

Necessity of infinitesimal steps for reversible processes

Is there a mathematical proof for why a reversible process is one that has many infinitesimal equilibrium steps, rather than fewer large steps? Maybe something along the lines of this?: ...
7
votes
1answer
316 views

Reference Request: Classical Mechanics with Symplectic Reduction

I am trying to find a supplement to appendix of Cushman & Bates' book on Global aspects of Classical Integrable Systems, that is less terse and explains mechanics with Lie groups (with dual of Lie ...
2
votes
1answer
123 views

Is there any physically relevant example of constructing series solution about infinity of an ordinary differential equation?

I was reading about how to test if a given second order ordinary differential equation has singularity at infinity from Arfken and Weber. I understood the steps mathematically but I could not find its ...
3
votes
0answers
86 views

Chocolate dynamics

Now I have found a possible model on how to describe chocolate when it is chewed. It has to do with geometrical transformations when a curve $\gamma$ intersects a manifold $M$. The chocolate is ...
0
votes
0answers
12 views

Are centrifugal fans subject to triboelectric effects, and if so can the increase in charge be predicted and measured?

A centrifugal blower is a device that moves gas, usually under power of an electric motor. The blower housing and impeller may be metallic, but often are made of plastic materials. My question is ...
1
vote
0answers
44 views

Is it possible to generalize quantum gauge theories? [closed]

I know that there are nonabelian gauge theories and their supersymmetric extensions. Mathematically, gauge theories basing on the fact that one can introduce a fiber bundle with a Connection. From ...
2
votes
1answer
195 views

Resources for theory of distributions (Generalized functions) for physicists

I am looking for tutorials, articles or books containing theory of distributions in context of mathematical physics. Please suggest.
0
votes
0answers
33 views

Algebraic number theory and physics [duplicate]

I would like to ask if there are any aspects of algebraic number theory related to physics (for example in string theory or Moonshine etc). I am thinking of attending a course on algebraic number ...
2
votes
0answers
40 views

Rheological behavior of chocolate

If someone eats chocolate, the chocolate goes through the following configurations: $\chi_0:$ chocolate is solid and has a smooth Surface everywhere; the Riemann Tensor vanishes on every Point of the ...
2
votes
2answers
141 views

Why do some bound states disappear in a discontinuous way?

Generally, we have the picture that as the parameter (say, the depth of a trap) of a system varies, the bound state gets more and more extended and disappears eventually at some critical parameter ...
2
votes
0answers
48 views

What is a modular tensor category / functor?

I have reads several answers here about this notion, especially regarding topological order, see e.g. this answer, but this notion sounds completely new for me. Also, I found nothing really helpful on ...
1
vote
1answer
66 views

How is this a gauge choice mathematically?

I've been reading an article about the "square cat", which is described as the system bellow Such system is a deformable body that can change $a$ and $\theta$ but has $b$ fixed. The article uses ...
1
vote
1answer
99 views

About the orthogonality of the Hamiltonian eigenstates for the the continuous energy spectrum

I would like first to describe a strange case that I encountered. $ \ \ - $ I solved the Schrodinger equation with a potential barrier (a potential well limited by a finite height wall which decrease ...
0
votes
1answer
32 views

A series of bound states covering an interval

Generally, the bound states (normalizable eigenvectors) of a Hamiltonian have discrete eigenvalues. Is it possible for the eigenvalues to cover an interval? Say, $(a,b)$? That is, for each $E \in ...
2
votes
2answers
117 views

If path integrals aren't well-defined, how can they have any physical meaning?

I am confused about a particular point about the nature of path integration. According to what I've read, what we really mean when we say functional integration is \begin{equation} ...
32
votes
9answers
4k views

Rigor in quantum field theory

Quantum field theory is a broad subject and has the reputation of using methods which are mathematically desiring. For example working with and subtracting infinities or the use of path integrals, ...
1
vote
0answers
34 views

Deriving general boundary conditions from first principles for elastodynamic scattering

It seems that most of the relevant books only give the linear case and the rest say something along the lines of "here are common examples of boundary conditions." What are the most general boundary ...
1
vote
1answer
579 views

Proof of equality of the integral and differential form of Maxwell's equation

Just curious, can anyone show how the integral and differential form of Maxwell's equation is equivalent? (While it is conceptually obvious, I am thinking rigorous mathematical proof may be useful in ...
6
votes
3answers
272 views

Coadjoint orbits in physics

I am looking for some application of coadjoint orbits in physics. If you know some of them please let me know.
6
votes
1answer
202 views

How to construct an isomorphism between the Complexified Special Linear Lie Group and the Special Unitary Group?

This may be an unenlightening question, but I'm just not sure about the result and hoping someone can help me varify it. $\\$ This question is related to these three questions. $\\$ I want to ...
1
vote
0answers
63 views

Non Hermitian Quantum Mechanics

I was just reading about Non-Hermitian Quantum Mechanics dealing with Hamiltonians $H$ that are not Hermitian operators. Then it is unclear that we get orthonormal eigenstates. Now, I was reading a ...
2
votes
1answer
113 views

Floquet quasienergy spectrum, continuous or discrete?

I haven't got a feeling about Floquet quasienergy, although it is talked by many people these days. Floquet theorem: Consider a Hamiltonian which is time periodic $H(t)=H(t+\tau)$. The Floquet ...
4
votes
1answer
91 views

Functional derivatives as distributions

I have asked this on math stack exchange, due to its mostly mathemtical content, but aside from one upvote and minimal views it has not garnered any attention, so I am trying here as well. This isn't ...
5
votes
3answers
563 views

Axiomatic statistical mechanics

Ive read a few courses on statistical mechanics, and while their textual explanations and example choices differ, the flow of information from microscopy to macroscopy seems the same, and reading ...
7
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2answers
93 views

Is there a reason why the subset of our Hilbert space that corresponds to a particle is a vector subspace?

I'm trying to gain some intuition behind the definition that states a particle is an irreducible unitary representation of the restricted Poincare group (or more specifically, its double cover). ...
1
vote
0answers
43 views

Can we avoid singularities by embedding the manifold on a bigger space, maybe Euclidean or Riemannian?

Can singularities be avoided by embedding Riemannian manifold on bigger space, or more specifically black hole singularities can be avoided or not by embedding in any other manifold. We don't have ...
3
votes
1answer
59 views

Is there a general theorem stating why the restricted Lorentz group's exponential map is surjective?

The exponential map for the restricted Lorentz group is surjective. An outline of why is shown on the wiki page Representation Theory of the Lorentz Group. Is there a more general theorem that states ...