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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11 views

Engineering question about magnitude in terms of frequency

How do I calculate magnitude of Vout/Vin as a function of w (frequency) when given the resistor and capacitive values?
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59 views

Is algebraic geometry actually used in string theory?

Many times I have heard string theorists say that string theory has a lot of algebraic geometry, but physicists seem to have identified complex differential geometry with algebraic geometry and ...
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29 views

How can I solve this PDE in cylindrical coordinates?

I got the following equation from physics. $$C \sin \theta \frac{\partial T}{\partial \theta} = \frac{1}{r} \frac{\partial }{\partial r} \left( r \frac{\partial T}{\partial r} \right)$$ where ...
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1answer
59 views

How to solve this integral of an ODE?

https://www.symbolab.com/solver/integral-calculator/%5Cint%5Cint%5Cleft(Ax%5E%7B2%7D%2B%5Cfrac%7BB%7D%7Bx%7D%2B%5Cfrac%7B2%7D%7Bx%5E%7B2%7D%7D%5Cright)dxdx I have used this symbolic solver to solve ...
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1answer
16 views

How to reduce differential operator to a 4th order ODE of the standard Euler type?

I read this article https://mae.ufl.edu/~uhk/STOKES-DRAG-FORMULA.pdf How do we actually arrive to a 4th order ODE below?
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Poincaré bundle, a way of understanding it.

So I will call the Poincaré bundle $(FM,\pi, M)$ the principal fiber bundle that has the Poincaré group as a structure group, the space of linear frames as total space and M as the Riemann-Cartan ...
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36 views

Road map Like github [closed]

Can someone create map like application for physics and mathematics like Github is for software. It is a mess at the moment.
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4answers
95 views

Is this workaround used by my book to find the value of $1/0$ legitimate?

I was doing a physics problem and found that I reached an interesting equation: Question To cross the river directly from A to B (making a right angle with the velocity of the stream), how must the ...
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90 views

Is this workaround used by my book to find the value of $1/0$ legitimate? [duplicate]

I was doing a physics problem and found that I reached an interesting equation:- To cross the river directly from A to B (making a right angle with the velocity of the stream), how must the fisherman ...
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0answers
154 views

Moments of inner products for Haar random matrices

Let $\mathbf{U}$ be a $n \times n$ Haar orthogonal matrix, $\mathbf{D}$ be a fixed diagonal matrix with half of its entries $+1$ and the remaining half $-1$ and $\left \langle\cdot, \cdot \right \...
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Difference between dual representation and the adjoint

The dual or contragradient representation from a vector space $V$ on $V^*$ (the dual vector space of $V$) is defined as the linear operator $$ (\pi^{-1})^T(g): V^*\rightarrow V^*, $$ where I ...
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1answer
196 views

What is the definition of moduli space, in math vs in physics?

It is easy to find that there are many questions regarding moduli space on MSE: https://math.stackexchange.com/search?q=what+is+moduli+space But it seems to me that this phrase, moduli space, may mean ...
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21 views

Principal axes of a 2D plane

Can some one please explain me whether Asymmetrical 2D planes contain principal axes? I'm aware that axes of symmetry can be taken as principal axes and if an object has more than two symmetrics it ...
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Simplifying integration over Gaussian [closed]

\begin{equation} \int d\underline{x} \exp\left(-\frac{1}{2\sigma^2}\underline{x}^T\underline{x}\right) \exp\left(-\frac{1}{2}(\underline{a}^T\underline{x}-b)^2\right)f(\underline{a}^T\underline{x}) \...
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How can we use the density Theorem to integrate functions with respect to this measure?

Let me denote by $x=(x_{0},x_{1},x_{2},x_{3})$ a generic point in $\mathbb{R}^{4}$ and by ${\bf{x}} = (x_{1},x_{2},x_{3})$ the "spatial coordinates" of $x$ in $\mathbb{R}^{3}$, so that $x \...
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2answers
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Any software recommendations to draw such graphs? [closed]

I am looking any alternatives software to draw similar graph images i posted here.
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1answer
56 views

Constructing a connection $1$-form from local forms.

I am following Section $10.1.3$ of Geometry, Topology and Physics by Nakahara, and have ran in to an issue regarding local connection forms. Consider a principal $G$-bundle, $P(M,G)$, and an open ...
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32 views

Logarithm term in “light-cone” operator expansion

I'm cross-posting from Physics site here because I don't know where this question fits better. I'm trying to understand how the logarithm in eq. (3.25) from this paper appears from the equation above ...
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1answer
17 views

Calculate the angle between a vector and a gravity pendulum

There is a physical device (sensor) which I can rotate. The device sends its X Y Z coordinates and a calculated acceleration vector in mg (mill mg). For example: X 116.0 Y 98.0 Z 151.0 MG 1031.0 X 117....
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27 views

Integrating an exponential containing a cosine function $\int_0^{2\pi} e^{i \nu \theta \pm ix\cos(\theta-\theta_0)} d \theta$

I'm trying to integrate $\int_0^{2\pi} e^{i \nu \theta \pm ix\cos(\theta-\theta_0)} d \theta$. According to a reference from the Journal of Mathematical Physics (https://doi.org/10.1063/1.5108599), ...
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35 views

Density and dimensionality of zeros in inverse square force fields of randomly distributed + and - charges in (at least) 1, 2 and 3 dimensions?

@mlk's answer to Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions? is pleasing and straightforward, switching to ...
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1answer
645 views

Density and dimensionality of zeros in inverse square force fields of randomly distributed sources in (at least) 1, 2 and 3 dimensions?

Background: In this answer to Are there places in the Universe without gravity? in Astronomy SE I did a quick finite 2D calculation for 20 random sources to see if there was at least one zero, and ...
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1answer
53 views

Electric potential : numerical value for the triple Integral

The function $\phi:L\to\mathbb{R}$ where $L={\{(x,y)\in\mathbb{R}^2:x^2+y^2=4\}}$ is defined as, \begin{align*}&\phi(x,y)=\\ &\int_{0}^{\pi}\!\!\!\!\int_{0}^{2\pi}\!\!\!\!\int_{1}^{2}\!\!\frac{...
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1answer
50 views

Cauchy problem for an ordinary equation not in normal form

Let's consider a one-dimensional physical system ($x$ is the position and $t$ is time) described by a first order differential equation. I'm aware of the fact that if the equation can be put in normal ...
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1answer
49 views

RLC circuit differential equation

Question: Consider the RLC circuit shown in Figure, with $𝑅 = 110 \Omega, 𝐿 = 1 H, 𝐶 = 0.001 F$, and a battery supplying $𝐸_0 = 90 V$. Initially there is no current in the circuit and no charge on ...
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1answer
55 views

Understanding mathematical concept behind phase space and phase portait

I'd like to expose the problem through an example, which was what made me think about it. It's a rational mechanics problem. Consider the one dimensional Cauchy problem $\begin{cases}m\ddot{x} = F(x,\...
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0answers
36 views

Help decipher the notation said to denote a common pattern in various branches of science in Prelude to Mathematics by W. W. Sawyer [migrated]

In Section 1.2 - Nature's Favorite Pattern? (excerpted below) of Prelude to Mathematics by W. W. Sawyer (1982), he said mathematicians used the notation $\nabla^2 V$ to denote a pattern that occurs &...
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1answer
67 views

Scaled Dirac Delta function: $ \delta (xe^r - y) $

I was reading on squeezed Gaussian states and stumbled upon this paper: Equivalence Classes of Minimum-Uncertainty Packets. II. It is mentioned after Eq. $\left(2\right)$ that $$ \left\langle x\left\...
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1answer
60 views

Solving the Compton Scattering

This question refers to the Compton Scattering. We have an elastic impact between a photon and an electron, so conservation of $E$ and $\vec{p}$ in a 2D plane: $$\begin{cases}E^i_p+E^i_e=E^f_p+E^f_e \\...
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1answer
15 views

find period from simple harmonic motion position

If an object suspended from a spring is displaced vertically from its equilibrium position by a small amount and re- leased, and if the air resistance and the mass of the spring are ignored, then the ...
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0answers
26 views

Confusion with Levi-Civita Symbol and Proof regarding Tensor Identity

Although my question is about some specific stuff about gravity, one does not need to know anything about it, because the main point of my question is a purely calculational issue, regarding ...
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2answers
41 views

Deficiency indices theorem (Von Neumann theory)

Let $T: Dom(T) \rightarrow \scr H$ be a symmetric operator. Prove the following: $T$ admits self-adjoint extensions iff $d_+ = d_-$, where $d_\pm = dim \ Ker(T^\dagger \pm i \mathbb{I})$ The proof I'...
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1answer
42 views

Bra-Ket Notation problem

I am currently working to find the mean position for a harmonic oscillator, but I've stumbled upon a mathematical problem. I have: $$\langle n|x^{2}|n\rangle=\langle n|aa^{\dagger}+a^{\dagger}a|n\...
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1answer
52 views

How fast is the plane [closed]

A jet plane flying at a certain height in a horizontal line passes directly above you. When it is directly above you, the sound of the plane seems to come from a point behind the plane, in a direction ...
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Error Approximation for Initial/Boundary Value Problem

I need help with a Matlab question regrading PDE's I have produced a code but it doesn't work well for this question can anyone please help!! I am attaching the image of the question please click on ...
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1answer
35 views

Is there a unit of measure for computational complexity; through quantum computers?

I'm concerned with trying to determine whether the same computational processes on a Turing computable algorithm can be ascertained for a quantum computer in some form of actual 'metric' for how many ...
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Is the covariant derivative of the inverse metric zero?

I know that covariant derivative of metric is zero [source, and my lecture notes confirm this]: $$\nabla _{\mu}g_{\alpha \beta} = 0,$$ How can I know if $$\nabla _{\mu}g^{\alpha \beta} = 0?$$
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1answer
31 views

Redundant term in Galilean transformation of time-dependent Schrödinger Equation

In my quantum physics book (Quantum Mechanincs Second edition B.H. Bransden & C.J. Joachain) in a chapter of Galilean transformation and Schrödinger equation there is a couple of weird equations. ...
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2answers
35 views

Describe a system in orbit [closed]

Suppose I have a function $f(x)$ (for drawing purposes, I will image it to be $2D$). Is there some sort of dynamics/set of differential equations etc that allows me to move around the orbit defined by ...
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18 views

Locally measured angle between two 4-vectors

(I am aware of thet Angle between two 4D vectors quesion, this is different.) I have an observer at some fixed radius away from a Schwarzschild black hole. It emits a photon at some angle $\alpha$ ...
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1answer
49 views

Decomposition of representation that has multiple copies of isomorphic irreducible representations, of a finite group.

To better show where the weightlifting point is, I'll use $D_{3d}$ group as an example. The problem is that a 24 dimensional reducible representation of $D_{3d}$ is given, which is a matrix form $r_{\...
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25 views

Understanding representation of Cauchy stress tensor for the simplest plane steady flow

Consider the simple problem of a flow between two plates, one at $x_2=0$ and one at $x_2=h$ with the bottom one held stationary and the top plate moving in the $x_1$ direction with velocity $V$. Also, ...
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25 views

Density of a graded algebra

I'm trying to prove the following proposition: If $ v \in V $ and $ Y \in \mathfrak{so} (V) $ then $[\dot\mu(Y), B(v)] = B(Yv)$. By definition $[\dot\mu(Y), B(v)] = \dot\mu(Y)B(v) - B(v)\dot\mu(Y).$ I ...
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Divergence of first Piola-Kirchoff stress tensor: how is it computed?

studying the bending of an incompressible elastic block of Neo-Hookean material, one finds out the first Piola-Kirchoff stress tensor $$S=\left(\frac{\pi + \mu_0}{\lambda_1} + \mu_0 \lambda_1\!\right)...
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1answer
61 views

Weight force or gravitational force?

Taking the case of a hydrogen atom with the relative masses of proton and electron the electrical interaction (module) $F^C$ is much more intense than the gravitational interaction (module) $F^G$ (on ...
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0answers
11 views

Mean energy in the isothermal-isobaric ensemble.

I'm dealing with the isothermal-isobaric ensemble, where the fixed parameters are temperature, pression and particle number: $T,P,N$. I konw that the expression for the mean value of the volume is ...
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0answers
11 views

Problems met in mastsubara frequency sum

I would like to calculate $\sum\limits_{\omega_{n},\vec{k}}(\ln(-i\omega_{n}+\xi_{\vec{k}})+\ln(-i\omega_{n}-\xi_{\vec{k}}))$, where $\omega_{n}=\frac{2n+1}{\beta}$ and $n=0,\pm1,\pm2,\dots$ Using the ...
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0answers
15 views

how to design a function to capture the boundary condition for $u(x=-1,t)=u(x=1,t)$ (for a PDE)?

I need to come up with a boundary condition function for a PDE solving method desribed here: https://github.com/sciann/sciann-applications/blob/master/SciANN-BurgersEquation/SciANN-BurgersEquation....
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1answer
170 views

A detail in the proof of Schur's lemma: the closures of the $\ker$ and $\operatorname{img}$ of the intertwiner.

Consider two irreducibles of a topological group $G$, acting in respective Hilbert spaces $\,\mathbb V\,$ and $\,{\mathbb{V}}^{\,\prime}\,$. Schur's lemma says: An intertwiner $\,M\,:\; {\mathbb{V}}\...
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2answers
44 views

Is there a proof that no rational number splits the octave equally?

In music circles, when the topic of tuning comes up, it is said that there is no rational number that splits the octave (the interval between a musical pitch and another with double its frequency, for ...

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