"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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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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68 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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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 ...
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72 views

Infinite dimensional reps of the rotation group

$\mathbf{Background:}$ The following is paraphrased from ``Representations of the rotation and Lorentz groups and their applications,'' by Gel'fand. Consider a finite-dimensional representation $T: ...
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65 views

How to convert FFT magnitude of square wave to dBm?

I wish to convert the FFT magnitude of square wave into dBm. I use FFT to covert voltage of square wave to a complex number, then i absolute the complex number into magnitude. Then i divide the ...
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64 views

A question about $1$-forms (on $S^3$)

What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form? [...Since $H^1$ of $S^3$ is trivial it follows that the ...
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83 views

How to perform this matrix integral?

Edit: some backgrouds added. In quiver matrix model which is reviewed DV or CKR, the path integral reduce to the matrix integral $$Z \sim \int \prod_{i=1}^r d\Phi_i \prod_{<a,b>} dQ_{ab} ...
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39 views

Proving invariance of $ds^2$ from the invariance of the speed of light

I've started today the book of Landau "Field theory". He starts from the invariance of the speed of light, expresses it as the fact that $c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$ is ...
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241 views

Lagrangian equations on Double Pendulum (Potential and kinetic Energy of double pendulum)

I have to prove the following equations using the Lagrangian equations, The figure shows the image, I know how to do lagrangian. I just don't know how to solve the kinetic and potential energies of ...
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55 views

Closure of numerical range contains spectrum

Let $A: D(A) \subset \mathcal{H} \to \mathcal{H}$ be a densely defined operator on a Hilbert space $\mathcal{H}$ with adjoint operator $A^{*}$. Given that $D(A) = D(A^{*})$ I'm trying to show that the ...
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88 views

A theorem about oscillation in Arnold's mathematical methods of classical mechanics

There is a theorem in page 100 of Arnold's Mathematical Methods of Classical Mechanics, which says that: If $\cfrac{dx}{dt} = f(x) = Ax + R_2(x)$, where $A = \cfrac{\partial f}{\partial x}|_{x = ...
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Proof of a theorem about oscillation [duplicate]

There is a theorem in page 100 of Arnold's Mathematical Methods of Classical Mechanics, which says that: If $\cfrac{dx}{dt} = f(x) = Ax + R_2(x)$, where $A = \cfrac{\partial f}{\partial x}|_{x = ...
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3answers
56 views

Positive unbounded operator with zero not as an eigenalue

I am currently doing Quantum Mechanics and I am supposed to show that zero is an eigenvalue of a positive operator. I have no knowledge of Functional Analysis at that kind of level, so I was wondering ...
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67 views

The Exponential decay.

I am studying semiconductor physics. there is a paragraph about Drude model in E.spenke's book "Electronic semiconductors" page 259 in art §9: "if on the average, a time $τ$ elapses between two ...
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64 views

When is it possible to construct ladder operators for a given Hamiltonian?

It is pretty cool (in my opinion) that one can solve Schrödinger's equation for the harmonic oscillator by using ladder operators, rather than just integrating it. In particular, it is possible to ...
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138 views

What is the simplest mathematical concept that does not map to a physical phenomenon?

One of my colleagues argues that everything in math proves something in the physical world. For instance, he claims that the existence of math to describe fractals proves the infinite divisibility of ...
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33 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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379 views

What is a particle mathematically?

In quantum field theory, what is a particle mathematically? How would you explain to someone who kows alot of math but no physics what a particle is? A simple example model would suffice.
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106 views

Proof that Legendre Polynomials are Complete

Can somebody either point me to, or show me a proof, that the Legendre polynomials, or any set of eigenfunctions, are complete?
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1answer
54 views

Relation between the eigenvalue density and the resolvent

Many texts (e.g. 1-2-3) on random matrices start with some variation of the identity: $$\rho_1(\lambda) = \frac{1}{\pi} \text{Im}\{\langle\text{Tr}(\mathbf{X}-\lambda\mathbf{I})^{-1}\rangle\}$$ ...
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2answers
92 views

closed form solution to the heat equation

Let smooth functions $f(x) , g(t)$ are given solve the heat equation on the semi infinite domain $(a,\infty) \times (0,T)$. for simplicity, we can let $a = 0$. \begin{eqnarray} &&u_t(x,t) = ...
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Status of a question from Freeman Dyson's 1972 article

In a famous article, Freeman Dyson mentions an interesting relationship between the $\tau$ functions of number theory and the dimensions of finite-dimensional simple Lie algebras (section 2). He ...
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123 views

Positivity of the Coulomb energy in 2d

Let $$D(f,g):=\int_{\mathbb{R}^3\times\mathbb{R}^3}\frac{1}{|x-y|}\overline{f(x)}g(y)~dxdy$$ with $f,g$ real valued and sufficiently integrable be the usual Coulomb energy. Under the assumption ...
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49 views

An $SU(3)$ isomorph in Clifford $G(5,0)$?

I am a computer scientist using the geometric (Clifford) algebras $G(n,0)$ over $\mathbb{Z}_3 = \{0,1,-1\}$ to describe distributed computations. My question concerns $G(5,0)$ with generators ...
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253 views

What is the exact and precise definition of an ANGLE?

On wikipedea I found a definition of an Angle as such: "In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g. with a pair of compasses. The length of ...
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27 views

Expectation value in a Quantum derivation

I'm reading a physics paper (John Bell's 1964 paper on the EPR paradox if anyone is physics-curious) and I'm having an issue following his derivation. It's the probability distribution stuff -not ...
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1answer
30 views

How has this equation been represented in the frequency domain

So, in my previous question Where does this formula for prediction of a multiple wave come from?, I get that using this picture: we have so far written the time it takes for a multiple to travel ...
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1answer
71 views

Bounded Self-adjoint Operator on Hilbert Space

I am trying to show that if $A$ is a bounded, self-adjoint and positive operator on a Hilbert space $H$, $0 \in \rho(A)$, the following inequality holds for all $x \in H$ with $\|x\| = 1$: ...
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Metastable solution for system of nonlinear equations

System of nonlinear equations: $$E_i=\epsilon_i+\sum_{j\neq i}^N \left(\frac{1}{1+\exp(E_j/T)}-\frac{1}{2}\right)\frac{e^2}{r_ij} \tag 1$$ where $T=0.05$, $r_{i,j}$ is given symmetric matrix with ...
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89 views

Maths-Physics question, can I solve this situation for $x$?

So Let's say I have an object going at velocity $V$, initially. Each second, the current velocity $v$ is reduced by $v/x$ . After $250$ (arbitrary) seconds the velocity has been reduced to below/equal ...
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38 views

Choice of the First Term in Legendre Polynomials

The two solutions of the Legendre's Differential Equation obtained by series solution method are : and Now according to my textbook, for the useful polynomial for n equal to a positive integer, ...
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What is Newton's theorem?

I'm reading a paper about mathematical physics at the moment and am wondering about the following: Let $w\colon\mathbb{R}^2\to\mathbb{R}$ be defined by $w(x)=-\log|x|$ and ...
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where $\nabla^2V = 0$ , evaluate $\int_S V d\Omega /4\pi$

Where $\nabla^2 V = 0$ in 3 dimensional Euclidean space, it is a well-known fact that $${\int_S V(\vec{r'}) d\Omega'\over 4\pi}=V(\vec{a})$$ where $\vec{a}$ is the center of a sphere $S$ of radius ...
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Why the method of separation of variables works? [duplicate]

The method of separation of variables is used in many occasions in the upper level physics courses such as QM and EM. But when it is used there is no clear reason why using it is permitted it except ...
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1answer
96 views

Understanding the intermediate field method for the $\phi^4$ interaction

In Rivasseau's and Wang's How to Resum Feynman Graphs, on page 11 they illustrate the intermediate field method for the $\phi^4$ interaction and represent Feynman graphs as ribbon graphs. I had to ...
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39 views

Avegaring a harmonic function over solid angle

It is well-known fact that if you average a harmonic function over the area of a sphere, you get the value of the harmonic function evaluated at the sphere's center. (Let's restrict the dimension to ...
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1answer
43 views

Hankel trasformation of acoustic wave equation

We consider a simplified version of acoustic wave equation \begin{equation} \frac{\partial^2 p}{\partial r^2}+\frac{1}{r}\frac{\partial p}{\partial r}+\frac{\partial^2 p}{\partial z^2}+k^2 ...
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Constructive weights to resum Feynman graphs

In Rivasseau's and Wang's "How to Resum Feynman Graphs", the weights of a spanning tree corresponding to a connected graph are defined as $$w(G,T) = \frac{N(G,T)}{|E|!},$$ where $N(G,T)$ is the ...
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1answer
85 views

Expectation value quantum mechanics momentum operator

What is the random variable that belongs to the expectation value of momentum in quantum mechanics. Or in general: Is there any way we can define the expectation values that occur in quantum mechanics ...
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164 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
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Mathematically rigorous text on classical electrodynamics.

Is there any textbook (preferably not written by a physicist) on classical electrodynamics which gives a rigorous (by the standards of pure mathematics) treatment of (a part of) the topics found in ...
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1answer
136 views

Navier-Stokes equations in tensorial form on a general coordinate system

How to write the classical Navier-Stokes equations in tensorial form on a general coordinate system? Any references?
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1answer
119 views

Linear Algebra in curved space

We know that Euclidean geometry and Newtonian Physics are special cases that only work in a flat space-time. Got to thinking about linear algebra and matrices. Is linear-algebra a special subset of ...
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1answer
106 views

Boundary Value Problem with Robin condition

How to solve the problem: $\left(3\right)$ \begin{cases} u_{tt}-a^{2}u_{xx}=f\left(x,t\right)\\ u_{x}\left(0,t\right)-h_{0}u\left(0,t\right)=g_{0}\left(t\right)\\ ...
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Why does something constant have a parabolic shape?

Consider an object dropped from a certain position, and the only force is acceleration due to gravity. The object accelerates the same throughout the free fall; not speeding up or slowing down. So ...
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Why is acceleration $\frac{1}{2}at^2$ halved when finding final height (distance)?

The final distance of an object dropped from a certain height is: $$S_f=S_0-\frac{1}{2}at^2,$$ $S_f=$ Final distance $S_0=$ Initial height from which the object was dropped $a=$ acceleration due ...
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Transpose of an operator $T$

How can I prove that a transpose operator is a basis-dependent? Is it true that I define transpose operator in this way: $ A^T= \Sigma_{i,j} \langle e_j|A|e_i\rangle$?
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57 views

Application of representation theory

I often read that one can use representation theory in the field of quantum physics or for the analysis of symmetries in physics or chemistry. Unfortunately I coundn't find a concrete example for ...
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116 views

Physical Meaning of Symplectic Vector Fields

The mathematics of symplectic (as well as Hamiltonian) vector fields is something that has been quite clear to me for some time, but recently I have been thinking much more about what certain ...
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1answer
285 views

Showing Jacobi identity for Poisson Bracket

We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where ...