Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [mathematical-physics]

DO NOT USE THIS TAG for elemetary 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.

6
votes
1answer
227 views
+50

Catenary Cable Problem: Timoshenko (2 solvers since last year only)

I was doing this amazing problem Chapter 4, Problem 10 from book Engg Mechanics Revised 4E by Timoshenko, and here is the link having the modified problem which resembles a lot from book. ...
20
votes
10answers
15k views

Do infinity and zero really exist? [closed]

From the first day that I entered college, I was wondering about the relationship between some basic mathematical abstract concepts and nature. I'm going to explain them here and you may find them a ...
0
votes
0answers
37 views

Integral of a multivalued function

I am trying to calculate an integral of a multivalued function, which has the form: $\begin{align} I=&\int_0^{\infty} d\tau \ \left( \frac{1}{v_c \tau+\alpha} \right)^{\frac{1}{4}(K_c+1/K_c)} \...
2
votes
0answers
6 views

Counting unitary transformations in $SU(N)$

I'm referring to the following article 1, in particular to section 6. The goal is to estimate the number of unitary transformations in $SU(N)$, identifying unitaries within balls of radius $\epsilon$....
0
votes
1answer
842 views

Moment of Inertia of Rectangular Prism about one of its edges

Question: What is the moment of inertia of a rectangular prism with dimenions $l\times w\times h$ represented by $a\times b\times c$ about one of its edges? Are my bounds correct, and what is $r$? ...
1
vote
0answers
31 views

Klein-Gordon PDE infinite line with periodic boundary conditions

In two of my mathematical physics textbooks on QFT, when solving the Klein-Gordon equation over the infinite line, the authors (Wald and Parker/Toms) take the Klein-Gordon equation and place the field ...
0
votes
1answer
56 views

Solving an equation for angle $\theta$

The shape of the ellipse can be represented parametrically in terms of the angle $\theta$ around the cycle, assuming $\theta$ is measured clockwise from the point $(V_0-(V_0-V_\mathrm{max}),P_0)$: $$...
2
votes
0answers
22 views

Ehrenfest Theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
1
vote
0answers
18 views

Determine minimizer of variational problem

Let $p>3/2 $ and define the energy functional $\mathcal E$ by $$\mathcal E(\psi) := \int_{\mathbb R^3} G \lvert \psi(x) \rvert^p - Z \frac{\lvert \psi(x) \rvert^2}{\lvert x \rvert} \, dx \quad \...
3
votes
1answer
103 views

Tautological one form assigns a numerical value to the momentum $p$ for each velocity?

In the wiki, it is written that : the tautological one-form assigns a numerical value to the momentum $p$ for each velocity $\dot {q}$, and more: it does so such that they point "in the same ...
3
votes
0answers
31 views

Hamiltonian Flows and Heisenberg picture of quantum mechanics

I am a math bachelor student studying Quantum Mechanics and I was very briefly introduced to the Heisenberg picture. (Hence many of the following may be trivial) In particular what I know is that: ...
2
votes
0answers
28 views

Discretization of a path integral and cylinder set measures

I'm studying Gaussian measures and in particular the Wiener measure, and in the process I've read about cylinder set measures. As in Wikipedia's page they are defined as: Let $E$ be a real ...
1
vote
0answers
40 views

equilibrium point in an inverted pendulum

The unstable upright position of an Inverted Pendulum on a cart does this corresponds to a Hyperbolic equilibrium point or a non- hyperbolic equilibrium point? what are some of the properties of a ...
3
votes
1answer
77 views

Cotangent lift of an action and its effect on the moment covector

If I have an action of a Lie group on a configuration space $G\to Diff(M)$ $g \mapsto \rho_g$ $\rho_g : q \mapsto \rho_g(q)$ (for example a rotation). Then when we consider the phase spaces $T^*M$, we ...
3
votes
1answer
76 views

Isomorphic Lie algebras and their Representations

What feature(s) of isomorphic Lie algebras distinguish between their respective representations? When will two isomorphic Lie algebras have the same or different representations? My particular case ...
1
vote
1answer
58 views

On the proof of BPS bound

In the wikipedia page on BPS bound (https://en.wikipedia.org/wiki/Bogomol%27nyi%E2%80%93Prasad%E2%80%93Sommerfield_bound) there is a proof, but I do not understand the final step: $$E\geq \pm \dfrac ...
1
vote
1answer
22 views

Square root of a probability

It's well known that if in quantum mechanics the quantity $$ J=\int_0^{\infty}dx|\Psi(x)|^2 $$ satisfies $$J=1$$ the $$|\Psi(x)|^2$$ represents the probability to find a particle in $x$. In ...
1
vote
0answers
35 views

Is there a spinor category?

I am just wondering how do spinors relate to category theory? Can a category of spinors be defined? If so how? Can the spinor space be defined by universal property (w.r.t a vector space) in the same ...
1
vote
1answer
41 views

Integral of total derivative vanishes

I am reading Bertrand Eynard's book on Counting Surfaces. In this book he mentions that the integral of a total derivative vanishes. What does he mean by this? Basically, I am trying to understand ...
3
votes
1answer
35 views

Finding the inverse Laplace transform of a physics related problem

I'm trying to find the inverse Laplace transform of: $$\frac{as+b}{s(cs+d)+g}\tag1$$ First of all I can expand the fraction: $$\frac{as+b}{s(cs+d)+g}=a\cdot\frac{ s}{s(cs+d)+g}+b\cdot\frac{1}{s(cs+...
1
vote
1answer
72 views

Extension of a Sobolev Function

Let $\Omega _1$ and $\Omega_2$ are smooth open sets and $A= \partial \Omega_1 \cap \partial \Omega_2.$ Let $\Omega ^\prime = \Omega_1 \cup \Omega_2 \cup A$ be an open set. Let $f$ is defined on $\...
1
vote
0answers
28 views

Covariant derivative of a section of a trivial(?) associated bundle

First I'll briefly outline the physical context in which my question arose: If one tries to do 2D quantum mechanics in polar coordinates one encounters a problem with a property of the covariant ...
2
votes
0answers
22 views

Term for coordinate systems that are linear spaces with restricted set of symmetries

I'm trying to find a word or definition to capture something I've seen in theoretical physics. Physicists work with coordinate systems a lot, and they frequently represent physical facts as ...
3
votes
0answers
27 views

Maxwell Equation + initial data - dissipative or dispersive?

In Aspects of Symmetry, Coleman says (p. 185) ''Most of the simple field theories with which we are familiar have the property that all of their non-singular solutions of finite total energy are ...
2
votes
2answers
87 views

Find $ \int_{0}^{\infty} e^{ix} \sin(x) \frac{e^{-3x}}{x} dx$

The second contribution in the Born approximation for the Yukawa potential in scattering theory leads to the following integral (for some given ratio of parameters): \begin{align} \int_{0}^{\infty} e^{...
0
votes
1answer
20 views

Path for fastest end velocity while accounting for friction

How would one calculate the path for the fastest velocity for a rolling object while accounting for frictions? Because ideally in the theoretical world, the path would not matter as long as there was ...
1
vote
0answers
54 views

What is this process/action called in English?

it is a fairly generate question regarding a terminology. People without science or engineering discipline makes an unfounded claim X, but people with such discipline start with proven facts A, B, C, ...
0
votes
1answer
52 views

Position representation of an operator

$$\langle x| M|x'\rangle=M(x)\langle x|x'\rangle=M(x)\,\delta(x-x')$$ I know this is true for if $M$ is a momentum operator or position operator, is this is true for a general operator $M $? $\...
0
votes
2answers
36 views

Max Velocity on a curved ramp and ideal ramp for highest Velocity

Is it possible to find the max exiting velocity of an object roll down a curved ramp? Since the brachistochrone curve has the path of the fastest time, does this mean it also has the highest exiting ...
0
votes
1answer
18 views

How to normalize (dimensionless) the relativistic Binet's equation for mercury

$$\frac{\delta^2 u}{\delta\theta^2}+u=\frac{\mu}{h^2}+3\mu u^2. $$ This is Binet's relativistic equation. I was trying to make perihelion precession path of mercury through python for doing this I ...
2
votes
1answer
50 views

Least-square fitting to data (sine function) : what is the error of the derived fit parameters?

I have a set of data. I want to fit it to a sine function of the form : \begin{equation} f(x)=A sin(\omega x+B)+C \end{equation} I use the least-square method to find the appropriate fit-parameters ...
0
votes
0answers
15 views

Conditions to determine if a matrix is a Lorentz matrix and if it is a Galilean matrix

Consider a generic 4x4 matrix $\Lambda $ $${\begin{bmatrix}{\Lambda ^{0}}_{0}&{\Lambda ^{0}}_{1}&{\Lambda ^{0}}_{2}&{\Lambda ^{0}}_{3}\\{\Lambda ^{1}}_{0}&{\Lambda ^{1}}_{1}&{\...
3
votes
2answers
38 views

Does adding a constant to the potential of Schödinger's equation always make the solution pick up a factor $e^{-iV_0t/\hbar}$?

I've been going over problem 1.8 of [Griffiths, Introduction to Quantum Mechanics, 2nd edition], where it asks to prove the following: Say $\Psi(x,t)$ satisfies the time-dependent Schrödinger equation ...
0
votes
1answer
33 views

Is this the correct definition of a discretization of a functional?

Physicists define path integrals by limits of discretizations, instead of using measure theory and Wiener measures. Now, the issue is that most texts do not give clear definitions, so it seems almost ...
0
votes
3answers
44 views

Find the differential equation given a fundamental set with no exponential name

My question is, given the general solution $$Y=C_1x+c_2\dfrac{1}{x}$$ find the differential equation My attempt: I have derivated the equation and then look for the constants: $$Y'=C_1+c_2\dfrac{-...
5
votes
1answer
89 views

In what $precise$ sense is Minkowski space asymptotically flat?

I've brought this question over from the physics stack exchange, where it didn't generate interest. We say a manifold $(M,g)$ is conformally compact if it is the interior of some $(\overline M, \...
2
votes
2answers
89 views

What is the definition of a bounded operator in an infinite dimensional Hilbert Space?

I am struggling to understand the meaning of a bounded operator in a Hilbert Space. Does a bounded operator simply means that if it acts on an element of the Hilbert Space, the "result" is bounded?
5
votes
3answers
323 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 this....
1
vote
6answers
169 views

Meaning of the notation $L^2(\mathbb{R}^3)$ and its generalization

I'm not sure whether this question really belongs to this website. In quantum physics texts, and physics stackechange website, I have often seen the notation $L^2(\mathbb{R}^3)$. My glossary of ...
0
votes
2answers
48 views

Force is one form, but what does it mean to evaluate a force into a velocity?

I was reading an example from Lee's book to try understanding the two body problem (see below). Since the forces are conservative, there exists a $U : Q \to \mathbb{R}$ such that $F=dU$, so I ...
1
vote
0answers
22 views

Undetermined coefficients in a perturbative expansion

In order to familiarize myself with perturbation methods, I've been trying to derive the Lorentz transformations, given by \begin{align*} x \rightarrow \frac{x + vt}{\sqrt{1 - v^2}} & = (x + vt)(...
0
votes
0answers
34 views

Fisher Information in Statistical Mechanics

I am studying the canonical ensemble and it seems to me there is an analogy between derivatives of the partition function, which can extract energy momenta for the system and Fisher score /information....
10
votes
3answers
3k views

What makes the Cauchy principal value the “correct” value for a integral?

I haven't been able to find a good answer to this searching around online. There is a related old question here, but it never received much attention. Suppose I have some physical property that I ...
0
votes
1answer
46 views

Is there a way to mathematically prove $\psi (\mathbf{r})$ varies continuously (using the intuitive arguments provided below)?

Electric potential at a point outside the charge distribution is: $\displaystyle \psi (\mathbf{r})= \int_{V'} \dfrac{\rho (\mathbf{r'})}{|\mathbf{r}-\mathbf{r'}|} dV'$ where: $\mathbf{r}=(x,y,z)$ ...
0
votes
0answers
14 views

Derivative of Hankel functions and Bessel functions

Dose anyone know about the formulations of derivative of Bessel and Hankel function as below, because when I just used the derivative of Bessel function and Hankel function as in the following ...
0
votes
1answer
59 views

Author's derivation of time-independent form of Maxwell's equations

Laser Electronics, 3rd edition, by Joseph T. Verdeyen, gives the following: To describe an electromagnetic wave, we need two field-intensity vectors, $\mathbf{e}$ and $\mathbf{h}$, which are ...
2
votes
1answer
37 views

Is there a construction of the Wiener measure by discretization and limits which parallels the Physics ideas?

In Physics one constructs the path integral by a limiting process together with a discretization procedure. Now, in order to better paralell with the Wiener measure, consider this in Euclidean ...
18
votes
4answers
7k views

What's the Clifford algebra?

I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really is?...
0
votes
0answers
20 views

Potential and field relation

This question is my attempt to find a solution to the question here Electric potential at a point inside the charge distribution is: $\displaystyle \psi (\mathbf{r})=\lim\limits_{\delta \to 0} \int_{...
0
votes
1answer
21 views

How to properly define the “discretization of a functional”?

In the derivation of the path integral formulation of quantum mechanics, most Physics books end up finding the following (or similar) expression: $$K(q',t';q,t)=\lim_{N\to \infty}\int\left[\prod_{k=1}...