"Mathematical physics consists of the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories." (from Journal of Mathematical Physics) This tag is intended for questions on methods used ...

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Proving commutation relation in Algebraic Bethe Ansatz

I have a problem with proving a certain commutation relation. For my Bachelor's thesis I give a more mathematically rigurous 'treatment' of a select set of chapters of a paper by L.D. Faddeev. Noting ...
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1answer
154 views

Applications of infinity in real life [duplicate]

I am writing a mathematical essay and would like to focus on the concept of infinity. I am not sure of any real life applications of infinity to write about or some way to narrow down the topics. Does ...
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80 views

A simple physics related algebra question.

This is making me feel like an idiot. I'm given this answer for a question but I don't understand it. $$y =\rm (-12.9\, m/s)(3.27\, s) + 1/2(9.81\, m/s^2)(3.27\, s)^2 = 105\, m = 0.11\, km$$ I ...
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41 views

pierre simon laplace and his knowledge of the (Laplacian) matrices

so as we all know, there is a graph matrix called the Laplacian that is used in some eigenvalue/eigenvector/graph theory/spectral theory problems. i'm wondering if the name of this matrix is ...
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1answer
107 views

Generators of Translation - Lie Algebra [duplicate]

I have just started learning Lie Groups and Algebra. Considering a flat 2-d plane if we want to translate a point from $(x,y)$ to $(x+a,y+b)$ then can we write it as : $$ \left( \begin{array}{ccc} ...
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15 views

Using Invariance of Lorentz interval and constant speed of light to prove the Lorentz transformations

By the invariance of the Lorentz interval and the fact that the speed of light is the same in both frames we have \begin{align*} -c^2 dt^2 + dx^2 = -c^2 dt'^2 + dx'^2 \end{align*} By considering the ...
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36 views

Help interpret Sommerfeld radiation condition.

I am studying the Sommerfeld radiation condition. In potential theory, a solution $u(r,\theta)$ to a partial differential (such as the Helmholtz equation $\Delta u(r,\theta)+\lambda^2 u(r,\theta)=0$) ...
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50 views

Why can't we say that all PDEs of a specified order require a fixed number of boundary conditions?

For an $n$th order ODE we always need $n$ boundary conditions (right?). But, as I've seen somewhere, for 2nd order PDEs there are many possible situations and a general answer to the question of ...
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38 views

Determine the motion for all time

In the frame $F=[0,\hat{k}]$, a particle of mass $m$, whose trajectory $[0,\infty)\xrightarrow{\rm r}\mathbb{R}$ is $r=z\hat{k}$ moves in response to a force ...
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28 views

An algebra associated with an important function

In the paper here the authors make a claim that the Natanzon potential (an implicit potential very important in mathematical physics) follows an $SO(2,2)$ algebra. This potential defined as : $$ ...
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48 views

Hermite Differential Equation - Non-integer values of $\lambda$

The Hermite differential equation, given by : $$ \frac{d^2y}{dx^2} - 2x \frac{dy}{dx} + \lambda y = 0 $$ has solutions of the $$ y(x) = \mathcal{H_n(x)} $$ when $ \lambda \: \epsilon \:\mathcal{Z_+} ...
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41 views

Angle that car is at after angular acceleration

A car starts from rest on a curve with a radius of $150m$ and tangential acceleration of $\displaystyle 1.5\frac{m}{s^2}$. Through what angle will the car have traveled when the magnitude of its ...
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64 views

Definition of logarithmic capacity

In the definition of logarithmic capacity of a compact set $E$ in the plane, the Robin constant is defined to be $V(E)=inf\int_E\int_E log\frac{1}{|z-w|} d\mu(z)d\mu(w)$ where $inf$ is taken over all ...
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95 views

Particle Motion

So this is a simple problem but I'm just getting stumped. The question is: A particle not connected to a spring, moving in a straight line, is subject to a retardation force of magnitude ...
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14 views

Fourier Operator and roots of Identity operator

I have seen that if Fourier operator is defined by $$ h(k) = \hat F(g(x)) = \frac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty} dx\:g(x)\:e^{ikx} $$ then $$ \hat F^2\{g(x)\}=g(-x) \implies \hat F^2 ...
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43 views

E. Artin theorem? (Ergodic theory)

In the framework of mathematical cosmology, Bianchi IX model has great importance due to its stochastic properties. I'm reading a publication in which is claimed The use of the invariant measure ...
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28 views

Holonomy of Maurer-Cartan 1-form

I am studying the book Sternberg (2012): Curvature in Mathematics and Physics; I am also doing research on LQG. I was wondering: if on a 4-dimensional spacetime one defined the Maurer-Cartan 1-form, ...
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19 views

Differential Realizations of certain algebras

I'm a first year graduate student in Mathematical Physics, and I am trying to generalise a certain method involving the so-called "Differential realizations" of certain algebras. The problem I'm ...
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1answer
94 views

Will a knot tied in a hanging, frictionless rope slip out under the force of gravity?

I am overall just curious about what keeps knots where they are in a rope. Another related question you might be able to answer is: What happens if you tie a bowline on the bight in a frictionless ...
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2answers
75 views

Selfadjointness of Coulomb Hamiltonian in d>=3 dimensions

I know that the Coulomb-Hamiltonian $H=-\Delta - |\cdot|^{-1}$ is self-adjoint with $dom(H)=H^2(\mathbb R^3)$. This follows by the Kato-Rellich-Theorem. Has the corresponding quadratic form a form ...
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30 views

Algebraic formulation of quantum mechanics and unbounded operators

Posted in the physic site: In AQFT one specifies the structure of the observables as a $C^*$-algebra. This seems to excludes algebras that don't have a norm, such as the Heisenberg algebra. ...
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72 views

Geodesics Through a Singularity

A singularity on a manifold with metric is defined to be a point at which some geodesic cannot be continued through. For example in Schwarzchild spacetime, $r=0$ defines such a point. Is it the case ...
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1answer
598 views

Solving a distance/time/speed problem using the quadratic formula. [closed]

"The distance between Toronto and Ottawa is 352.72 km. The speed on a road trip from Ottawa to Toronto was double of the return, and therefore the drive took 2 hours less. What was the speed on the ...
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37 views

What does “The Hilbert space carries a representation of […] group” means?

Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta). Effectively, I know that this means ...
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46 views

Theorems on orthonormal bases and spectrum

Are there theorems similar to the following: If $T$ is symmetric and $D(T)$ contains an ONB of eigenvectors of $T$, then $T$ is essentially self adjoint and the spectrum of $\bar{T}$ is the closure of ...
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83 views

Contour Integration - Quantum field theory

I am a physics student. In calculating transition amplitude for Klein-Gordon real-scalar field, I encountered the integral, $$ \frac{-i}{2(2\pi)^2\Delta x} \int^{\infty}_{-\infty} \,dk ...
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1answer
60 views

Fourier transform of Legendre

I am trying to figure out the Fourier transform of Legendre polynomial $P_\ell [\cos(\theta-a t )]$: $Q(\omega)=\int_{-\infty}^\infty P_\ell [\sin\phi\cos(\theta-a t )] e^{i \omega t} dt,$ where ...
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2answers
71 views

Obtaining explicit solutions of the differential equation $\left(\frac{dy}{dx}\right)^{2}=\frac{1}{ay^2+by+c}$

I'm trying to see if it is possible to obtain an explicit form of the following differential equation $$\left(\frac{dy}{dx}\right)^{2}=\frac{1}{ay^2+by+c}$$ where $a,b$ and $c\in\mathbb{R}$\{$0$}
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1answer
88 views

Center of mass in a straight rod

I got an assignment to prove that in a straight homogeneous rod, you can always choose a coordinate system in such a way that $$\int_S x_1 \, dx_1 \, dx_2=0 $$ $$\int_S x_2 \, dx_1 \, dx_2=0 $$ ...
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539 views

Stacking Cylinders Mechanics Question (from brilliant.org)

Three cylinders, all of the same mass, are stacked on a table as shown in the figure. There is enough friction between the cylinders and the table such that the cylinders remain at rest. Let Fh be the ...
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1answer
71 views

Fourier transform involving a dirac delta function

I know that $\int \delta(x-a)f(x) dx =f(a) $ , the fundamental defining property of the delta function. How does this change if we no longer consider $x-a$ but $a^2 -x^2$, such that the integral is ...
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367 views

Mathematical and Theoretical Physics Books

Which are the good introductory books on modern mathematical physics? Which are the more advanced ones? I already read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods ...
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33 views

Fourier transform involving a delta function [duplicate]

For ease let us suppose that I am trying to transform the function $f(t)=\delta(a^2 -t^2)$. Therefore the Fourier transform is given by $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} \delta(a^2 ...
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61 views

Finding Percentage Contribution of a Variable in an Equation

I have an equation, for example: $$ y=a-b+c $$ I am actually confused how exactly to find the contribution of the variables individually to the entire equation. Due to the negative sign, following ...
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25 views

negative eigenvalues for small potential

I'm reading Lieb's book "Stability of matter". On page 66 he states that for any arbitrarily small negative $V$, for $V\in L^{1+\epsilon}(\mathbb{R}^2)+L^{\infty}(\mathbb{R}^2)$ (case $d=2$) OR ...
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2answers
71 views

what does Uncertainty principle means

i did not understanding idea behind Uncertainty principle,which says that For instance, the more precisely the position of some particle is determined, the less precisely its momentum can be known, ...
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45 views

The median of infinity [closed]

Would it be logical to assume that 1 is the median of countable infinity since all the whole numbers can be also used as denominators of 1?
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30 views

Physical interpretation of categorical structures related to Dirichlet Branes

In Dirichlet Branes and Mirror Symmetry by Aspinwall et al, section 5.9 discusses various questions that remain open. In particular they say: "There are many constructions from homological ...
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1answer
60 views

Scaling a cup to have a certain filling volume

I created a cup in Autodesk Inventor using lathe/rotation, ie I defined the profile and rotated it around an axis. I measured it's volume. By using Patch and Sculpt I filled the inner volume(which ...
2
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1answer
108 views

Does every major genre of mathematics have applications?

I know that it used to be said, in praise by some and as criticism by others, that Number Theory had no applications. Now it is used in cryptography and Quantum Theory. Since the mathematics that ...
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1answer
109 views

Lowercase delta in differential-like equation

Preface: The following question comes from an expression seen in a biophysics paper published in Nature protocols. I'm aware that in pure mathematical notation $\delta$ is never used in the context ...
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64 views

Two properties about Bessel function

Let $J_\nu(x)$ be the Bessel function of the first kind. $\int_0^\infty J_\nu(x)dx=1 , (Re(\nu)>-1)$. $\lim_{\nu\to+\infty}J_\nu(x)=0$ for any fixed $x$. I think the above two properties of ...
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84 views

Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension ...
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1answer
83 views

Particles in Free Fields

For the state $\left|\vec{p}\right> = a_{\vec{p}}^{\dagger}\left|0\right>$ we have the energy $H\left|\vec{p}\right>=E_{\vec{p}}\left|\vec{p}\right>$ ...
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2answers
71 views

Boundary conditions for second order PDE

For a second order PDE, for example heat conduction equation $\frac{\partial T}{\partial t} = \frac{\alpha}{C_p} \nabla^2 T$, is it possible to determine the steady-state (or even transient) solution ...
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53 views

Ratio of Hankel functions

I am trying to evaluate the ratio $\frac{H_m'^{(1)}(z)}{H_m^{(1)}(z)}, m\in N$. $'$ indicates derivative with respect to $z$. For large $m$, each term in the numerator or denominator can overflow ...
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1answer
63 views

Cauchy's problem. Equation of mathematical physics

$$U_{tt} = \Delta U + x^3 - 3xy^2$$ $$U|_{t=0} = e^x \cos y$$ $$U_t|_{t=0} = e^y \sin x$$ Help me, please, with solution of this equation. Can you prompt me algorithm to find the ...
2
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1answer
105 views

Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
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2answers
80 views

Solve separation of variables problem

Originally I had $\frac{d^2y}{dt^2}=-A e^{y/B} (\frac{dy}{dt})^2$. Using a given hint: $\frac{dx}{dy}=\frac{dx}{dt}\frac{dt}{dy}=\frac{d^2y}{dt^2}\frac{1}{x}$ and $x=\frac{dy}{dt}$ I got: ...
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1answer
123 views

Putting Maxwell's Equations in Tensor Form. (Carroll Chapter 1 Question 11)

Simply put, if you look at https://en.wikipedia.org/wiki/Electromagnetic_tensor#Significance it says you can go from the traditional four "vector calculus" maxwell equations to two tensor Maxwell ...