Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elementary physical questions. This tag is intended for questions on modern mathematical methods used in quantum theory, general relativity, string theory, integrable system etc at an advanced undergraduate or graduate level.

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What is the energy required to add two systems together? [closed]

What is the energy required to add two systems together? Let's assume that each of those two systems have energy value which serves the purpose of the system just existing.
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1 answer
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I'm trying to represent $x^{2} = \dfrac{c^{2}}{2} + 4 \sum\limits_{j = 2}^{ \infty} \dfrac{J_{0}(\alpha_{j}x)}{\alpha_{j}^{2}J_{0}(\alpha_{j}c)}$

To expand, in the interval $(0 < x < c)$ $$f(x) = x^{2}$$ in a Fourier-Bessel function, I used: $$ f(x) = \sum\limits_{n = 1}^{\infty}a_{n}J_{0}(\dfrac{\alpha_{n}x}{c})$$ Where $$a_{n} = \dfrac{...
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Find the inverse Fourier transform of the following functions

Find the inverse Fourier transform of the following functions. enter image description here
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How to solve an integral equation????

How to solve this integral equation???? $$\large \int_{-\infty}^{+\infty}\frac{y(\tau)}{(t-\tau)^2+a^2}d\tau=\frac1{t^2+b^2}, \quad 0<a<b.$$
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Could someone help prove some properties of Legendre Polynomials?

I have already proved other properties of the Legendre polynomials, like: $$P_n(-x) = (-1)^n \, P_n(x)$$ $$P_{2n+1}(0) = 0$$ $$P_n(\pm1)= (\pm1)^n$$ But I can't get this one: $$P_{2n}(0) = \frac{(-1)^...
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The prerequisite of studying about differential forms and symplectic manifolds

I'm planning to read chapter 7-8 of Arnold's "Mathematical Methods of Classical Mechanics". I would like to know if there's anything I should know before reading it. I've studied ...
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1 vote
1 answer
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Explicit computation of Choi matrix for a qubit channel

I'm struggling with the explicit computation of the Choi matrix of a generic quantum channel ${\Phi}:\mathbb{C}^2\to\mathbb{C}^2$. I know that I can write the channel in the Bloch representation as ${\...
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What field of maths do I study if I am interested in Cosmic Topology? [closed]

I recently graduated with an undergrad mathematics degree, and I am really interested in pursuing mathematics research. My main interests lie in the geometry and topology of spacetime/ the universe in ...
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Application of Green formula

Please I need green formula to caculate this integral \begin{equation} \int_{\Omega} (-\Delta u)^{r} (x)v(x)dx \end{equation} where $r\geq 1$.
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How to get Lie algebra generators from infinitimal conformal transform?

In one of the lecture notes on conformal field theory, the author derives an infinitesimal version of conformal transformation for dimension $d \geq 3$ and its as follows, $x'^{\mu}= x^{\mu}+a^{\mu}+ ...
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1 answer
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The value of $\frac{1}{T}\int_{0}^{T} e^{ikt} dt$ when $T \rightarrow \infty$

I have this integral $\left(\dfrac{1}{T}\int_{0}^{T} e^{ikt} dt\right)$ when $T \rightarrow \infty$, in an equation that I am studying, and it seems that this integration gives a value of $\dfrac{\...
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How is $\nabla (u\cdot A) =u\cdot \nabla A+ u\times (\nabla \times A) $?

This was used in the answer here, in the derivation of the Lorentz force law from the Lagrangian. $u$ and $A$ are vectors, the velocity of the particle and the spacetime dependent Magnetic field As ...
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Do these conformal Killing vectors have a name? If not, what should we call them?

I have been investigating conformal Killing vectors on pseudo-Riemannian manifolds, that is, vectors which obey $$ \mathcal{L}_X g = \lambda g$$ where $g$ is the metric and $\lambda$ is some function. ...
1 vote
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Normalizing a function; Finding the solution to Burgers equation

I have been able to figure out the first half of the problem described below in the image. The part I am struggling on is finding the $C$ constant such that it will lead to a normalized solution of $\...
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Arnold Math Methods

I'm having trouble with this question on page 10 of V.I. Arnold's Math Methods book. A mechanical system consists of three points. At the initial moment their velocities (in some inertial coordinate ...
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Doubt in Linear vector space

Why would $<z_1|z_2>=\mathrm{Re}(z1^{*} z2)$ not be a legitimate definition of scalar product? It is equal to $\mathrm{Re}(z_2 ^* z_1)^* $. It satisfies the first property properties, question
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Natural $\sigma$-algebra and measure on a countable set?

Suppose $X$ is a countable set. I have seen many times in mathematical physics books the use of a Hilbert space $\mathcal{H} = L^{2}(X)$, the space of square integrable complex functions defined on $X$...
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1 answer
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Write $\int_0 ^\frac{\pi}{2}\tan(x)^\alpha\mathrm dx$ as Euler integral.

Write given integrals in form of Euler integrals(Beta,Gamma) $\int_0 ^\frac{\pi}{2}\tan(x)^\alpha \mathrm dx$ after $\tan x=t$ subst. I get $\int_0^\infty t^\alpha(1+t^2)\mathrm dt$ which I don't know ...
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1 answer
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A limit for Dirac-delta function using real two sided exponential

If I have a real-value function (for $x_0 > 0$) $$ f(x; x_0) = \frac{1}{x_0}\exp\Bigg(-\frac{|x|}{x_0}\Bigg) $$ If I plot this for different values of $x_0$ it becomes more and more narrow with $...
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2 votes
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An unusual trigonometric series differential equation from quantum mechanics.

In the course of my work in experimental quantum physics (studying heteronuclear polarization transfer in coupled many-spin-1/2 systems) I derived a series of analytical expressions that describe the ...
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What does 'interior initial conditions' and 'interior boundary conditions' stand for?

My question concerns on the word 'interior' in the IVP and BVP problems. How do we describe the conditions with 'interior' mathematically? I know the difference between IVP and BVP, but it would be ...
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How to deal with the integral involving more than one Dirac delta function in high dimension?

My question is how to calculate $$ \int_{\mathbb{R}^n} f(x) \delta(h(\mathbf{x})) \delta(g(\mathbf{x})) \mathrm{d} \mathbf{x} $$ (In my case both $h(\mathbf{x})$ and $g(\mathbf{x})$ are linear ...
1 vote
1 answer
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Using the general formula from the Brachistochrone Problem on different functions.

When solving the Brachiostone Problem we reach the "general formula": $$T[y]=\frac{1}{\sqrt{2g}}\int^{\bar{x}}_{0}\sqrt{\frac{1+(\frac{dy}{dx})^2}{-y(x)}}dx$$ I am unsure about how to apply ...
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1 vote
1 answer
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Deriving a formula for Lie algebra of Conformal Field theory

I'm learning some conformal field theory. I'm trying to use the formula $$ \partial_{\mu} \epsilon_\nu+\partial_\nu \epsilon_\mu=\frac{2}{d}(\partial \epsilon) \eta_{\mu \nu} $$ to derive the equation ...
0 votes
0 answers
31 views

Find functional based on Euler-Lagrange Equation

I have the following equation for $u:D\subseteq \mathbb{R}^d\rightarrow \mathbb{R}$ $$-\Delta \Delta u=0$$ Based on this I have to find the functional where this u would be a extreme point. Since we ...
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2 votes
1 answer
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If $A \in O(3,1)$, why does $A^* gA = g$?

Let $O(3,1)$ denote the Lorentz group, i.e. the group of all linear transformations on $\mathbb{R}^4$ that preserve the following inner product: $$\Lambda(x,y) = x^0y^0 - x^1y^1 - x^2y^2 - x^3y^3.$$ ...
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1 vote
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Spin Representations

In the book Gauge Fields, Knots and Gravity, spin "j" representation is described as homomorphisms from SU(2) to the General linear group of space of polynomials of degree $2j$ in $\mathbb{C}...
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2 answers
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Is it true that two 3D bodies of different shapes cannot have the same volume to area ratio unless both have exactly the same volume and area?

I found this elegant physics question in the Q/A section of the research gate. My own judgment is that this is true except, and only except, for complete spherical shapes. . In other words, it applies ...
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0 votes
0 answers
22 views

Series representation of the exponential

I have a function such as $Z = \prod\limits_{r}\sum\limits_{N_{+}=0}^{\infty}\sum\limits_{N_{-}=0}^{\infty} \frac{\lambda_{+}^{N_{+}}}{N_{+}!}\frac{\lambda_{-}^{N_{-}}}{N_{-}!} \left(e^{-\beta\left(\...
-1 votes
1 answer
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How to view two small balls bumping into each other as passing through each other(2016 Pascal Contest Q24)?

Here is a question from 2016 Pascal contest(Waterloo, Canada). The question and its full solution are accessible via https://www.cemc.uwaterloo.ca/contests/past_contests.html Please find below ...
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0 answers
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Performing Maximum Likelihood Optimization in Source Range and Depth Estimation from Range Difference Measurements Paper

Background I have an array of $n \gg 4$ not necessarily uniformly-spaced collinear sensors with known locations, and a source with entirely unknown location which emits a signal at velocity $v$ at ...
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2 votes
1 answer
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What are the odds that De Vries' formula for the fine structure constant $\alpha$ is a numerical coincidence?

The dimensionless fine structure constant $\alpha \approx \frac1{137}$ has intrigued physicists for over a century. Whilst not currently a majority view, there is a school of thought that considers ...
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1 vote
1 answer
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Is the Curl $\nabla \times$ symmetric under inner product?

If $\nabla\cdot u=0,$ and $\nabla \times(\nabla \times u)=\nabla(\nabla \cdot u)-\nabla^2 u,$ we know that $-\Delta u=\nabla \times(\nabla \times u).$ But I have a question, why $$(-\Delta u,v)_{L^2}=(...
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1 vote
1 answer
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Scalar fields with definite weight under conformal maps as sections of some bundle

Let $(M,g)$ be a smooth Riemannian manifold. A scalar field in $(M,g)$ is merely a map $\phi:M\to \mathbb{R}$. Under a diffeomorphism $f:(M,g)\to (M,g)$ it transforms to $\phi' = \phi\circ f$. We then ...
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0 votes
1 answer
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Is the Poisson bracket uniquely determined by its properties?

Let $D\subset \mathbb{R}^2$ be open, as $C^{\omega}(D)$ I denote the space of analytic real-valued functions on $D$. The poisson bracket is an alternating bilinear map: $[ \cdot, \cdot ]: C^{\omega}(D)...
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Geometric optic, displacement formula f=(L^2-d^2)/4L

Stupid question I am stuck at the displacement formula in geometric optic I am following the steps at schoolphysics. The steps are the same more or less. But I cant for my mind see the steps wich ends ...
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0 answers
42 views

What is my result in numbers?

The total resistance of two resistors connected in parallel is given by the formula: $$\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}}$$ What is the maximum possible error (absolute) if we have $R_{1} ...
-1 votes
1 answer
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Hausdorff's formula proof [duplicate]

How do we prove Hausdorff's formula in the following form by pulling the operator B in $e^A B$ to the left side in the following form ? $e^ABe^{-A} = B + [A,B]+\frac{1}{2!}[A,[A,B]]+...$ I made some ...
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Higher Dijkgraaf Witten Theory

I am trying to understand higher form symmetries in TQFT. In particular the higher form version of Dijkgraaf Witten Theory. It is know that for a 0-form symmetry we can specify the principal G-bundle ...
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2 votes
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casting electromagnetism for exterior differential calculus

I am trying to understand curved spacetime Maxwell's equations in terms of exterior differential calculus. I am surfacing this topic due to working with a flat, but non-constant metric and I am ...
1 vote
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The point of BV formalism

I just started reading about BV formalism and I have a question regarding the slogan/philosophy of taking the derived critical locus of a smooth function. Specifically, as explained here, the BV ...
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2 votes
2 answers
65 views

What is the formula for winding and unwinding a spool with changing radius due to the strap thickness being wound.

I have a servo motor and I'm trying to come up with a formula that relates the angle of the motor to the length of a strap that is wound on it. Since the radius at <360 degrees is the radius of the ...
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0 votes
1 answer
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Why do I get the wrong $\Delta$ in spherical coordinates, if I calculate the square of the gradient in spherical coordinates $\Delta \neq \nabla^2$?

I wanted to calculate the Laplacian operator $\Delta$ based on $$\nabla=\left(\frac{\partial f}{\partial r},\frac{1}{r}\frac{\partial f}{\partial \theta},\frac{1}{r\sin(\theta)}\frac{\partial f}{\...
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2 answers
61 views

Legendre transform preserves strict convexity

Let $f:A\to \Bbb R$ be a strictly convex function of Class $C^2$ in a open and convex subset $A$ of $\Bbb R^n$. Say that $g$ is its Legendre Transform defined as: $g(p) = \max_{A} (xp - f(x))$ Say ...
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q-form symmetries

I have heard the words 0-form, 1-form, and also q-form symmetries for some positive integer q. Could someone explain what are these symmetries and how they are differ from group symmetries? Thanks
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Solutions to Hill equation with two coefficients

I'm searching for quasiperiodic solutions of the Hill equation $$ H \psi := \frac{d^2 \psi}{d z^2} + V(z) \psi (z) = 0 $$ where $$ V(z) := \theta_0 + 2\sum_{n=1}^{\infty} \theta_n \cos{(2 n z)} $$ in ...
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0 answers
20 views

Swapping the order of a limit and a non-linear functional acting on a sequence of compact operators of increasing rank.

Let $\rho_{n}$ be some compact operator for all $n$ ( I know finite-dimensional operators are compact, forgive the redundancy) and let the rank of $\rho_{n}$ equal to $n$. Now, let $F$ be some non-...
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The trace-norm of an infinite dimensional compact operator bounded by a limit of finite dimensional trace norms.

Let $\rho$ be a trace class operator with infinite rank. Furthermore, let $\rho_{N}$ be a finite rank approximation, of rank N, of $\rho$ in the following sense. I will use Dirac notation. Note that $\...
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1 vote
1 answer
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How do you invert a tensor in the EFE equation such as the stress energy tensor or the Newtonian tensor?

Motivation: Tensors are built from vector spaces because a tensor satisfies the axioms of a vector space under the proper equipment of addition and multiplication. How do we define rigorously the ...
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1 answer
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A third order ordinary differential equation to solve (how to recover this integral solution)

Problem statement: I am trying to derive the integral solution of this third-order ordinary differential equation (ODE) for the function $y(x)$ of the variable $x$ $$y'''(x)=4xy(x), \ \ \ \ \ \ [1] $$ ...

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