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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24 views

'Taylor Expansion' of Integral - Asymptotic expansion - Exponential function

I need to evaluate the following integral in the limit $\kappa \ll 1$ $$\int_0^\infty exp(-\kappa t) f(t)\, dt,$$ where $$f(x) = (1+x)(1-2x)\frac{u(x) \ln(u(x))}{u(x)^2 - 1},$$ $$u(x) = \frac{\sqrt{1+...
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30 views

A new type of manifold, is such a construction interesting? Is it relevant for the Euler-Lagrange equations

Recently, I've been wondering how to rewrite the standard Euler-Lagrange equations: \begin{align} \dfrac{\partial L}{\partial q^i} - \dfrac{d}{dt} \left(\dfrac{\partial L}{\partial \dot{q}^i}\right) &...
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0answers
54 views

Differentiating a “function-valued” function.

I'm a Physics undergraduate, deeply enamored of math too. I'm trying to make a proof of Noether's theorem more precise with minimal abuse of notation as well, since almost in every physics resource, ...
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1answer
56 views

Why does the trace show up in such expressions?

I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a purely-mathematical common step I do not understand: the showing-up of the trace. ...
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1answer
25 views

Computing a trace containing $\gamma$-matrices

I want to compute the following trace $$Tr \Big( Y(\not{\!p_1'}+m) \Big) \ \ (1)$$ Where $$\not{\!A} := \gamma^{\alpha} A_{\alpha} \ \ (2)$$ $$Y:= 4 \not{\!f_1} \not{\!p} \not{\!f_1} + m[-16(...
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1answer
22 views

If $V^{\mu}$ is a killing vector, then is $∇_μ V^{\mu} = 0$?

Working with the Levi-Civita Connection and a symmetric metric I want to show that if $V^{\mu}$ is a killing vector, then $∇_μ V^{\mu} = 0$. I am not sure how to show this fact, but I believe it to ...
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2answers
131 views

Angular integrals used in QED

I am reading a research paper and am stuck at a point where the author uses angular integrals. I don't have any idea about it and would like help. The angular integral is: $$I_k (y)=\int_0^\pi {\...
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25 views

Solutions for $\frac{\partial^2 u}{\partial t^2}=-\frac{1}{\hbar^2}\left(\frac{\hbar^2}{2m}\Delta(\cdot)-V(\vec{x})\cdot\right)^2u$ [closed]

If $A(\vec{x},t)+iB(\vec{x},t)$ is a solution for $$\frac{\partial^2 u}{\partial t^2}=-\frac{1}{\hbar^2}\left(\frac{\hbar^2}{2m}\Delta(\cdot)-V(\vec{x})\cdot\right)^2u$$ Is corret that $A(\vec{x},t)$...
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29 views

Problem with differentiating functions.. [closed]

How is $v\frac{dv}{dt}=\frac{d}{dt}\left(\frac{1}{2}v^2\right)$? Don't see it is obvious. Looked up several places but everywhere they show the derivation from the answer.
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33 views

Functional Derivative with Difficult Chain Rule

I am trying to evaluate the functional derivatives $\dfrac{\delta F[\phi(\textbf{r})]}{\delta n_{1}}$ and $\dfrac{\delta F[\phi(\textbf{r})]}{\delta n_{2}}$ where \begin{gather} F[\phi_1(\textbf{r}), ...
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1answer
21 views

Nonlinear vector calculus problem

Let $A$ be a vector field on $\mathbb{R}^3$. I am interested in finding solutions of $$ \nabla^2 A \times {\rm curl} A = 0,\\ \quad {\rm div} A = 0. $$ Are there any exact solutions with nonzero $\...
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30 views

Schrödinger Equation problem [closed]

Let $$\Psi(x,y,z,t)=A(x,y,z,t)+iB(x,y,z,t)$$ a solution for the Schrödinger equation $$i\hbar\frac{\partial\Psi}{\partial t}=-\frac{\hbar^2}{2m}\bigtriangleup\Psi+V(x,y,z)\Psi$$ Prove $$\frac{\partial}...
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0answers
15 views

Maximum/minimum principle for the laplace equation with robin boundary equation

Let $\Omega \subset \mathbb{R}^3$ and $u_0:\partial\Omega\to\mathbb{R}$. Then, $u(x)$ with $x\in\Omega$ is a solution to the Laplace equation with Robin boundary condition: $$ \Delta u =0 \ \ \text{...
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0answers
27 views

Counterexample for Korn inequality when $p=\infty$

I am currently studying the proof of Korn inequality for $1 < p < \infty$ for an exam, and a counterexample for $p=1$. I've read in the article that my professor gave me that this result doesn't ...
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1answer
27 views

Finding the trajectory of a charged particle in space in a magnetic field

I have been given that a charged particle of charge $q$, mass $m$ moving with a velocity $v_{0}(\vec{i}+\vec{j})$ enters a magnetic field $B_{0}\vec{i}$. At any time instant $t$, determine its ...
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34 views

A precise formula for the summation of an inner product

We have 2 strings $|v\rangle$ and $\langle u|$, $|v\rangle=|e_{1}\rangle^{np}|e_{2}\rangle^{n(1-p)}$ where $e_{1}$ occurs $np$ times and $e_{2}$ occurs $n(1-p)$ times and $\langle u|=\langle f_{1}|^{...
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0answers
46 views
+50

How to proceed in this Boundary value problem where Eigen values are calculated numerically?

While solving a boundary value problem (background provided in the Context section) I reach the following variable separated two equations ($F(x)$ and $G(y)$) \begin{eqnarray} \lambda_h F''' - 2 \...
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26 views

Relationship between two series [closed]

I have these two series 1,7,13,19,25,31,37 1,2,3,4,5,6,7 I want to find the relationship between them, like a formula in which if i enter the number of the second series, I get the result of the ...
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39 views

Physics-related Differential Equation - Why do beams break?

I do have a physics-related differential equation question which I need help on. Given the beam deflection equation: $$F(x) = EI \frac{d^4u}{dx^4}$$ where $E$ is the Young's Modulus, $I$ is the ...
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0answers
24 views

Help with some gamma matrices trace identities

I need to understand the following derivation, but I can't understand which identities it used. Some help or more elaborate derivation will be great
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2answers
32 views

What does it mean to take the “ gradient with respect to the position $r_ i$”?

Let’s say we have a number of particles (charged, massive or anything that can create potential energy). The total potential energy of any particle can be given by $$U_i (\vec r_1, \vec r_2, ... \vec ...
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0answers
43 views

Disentangling and reordering operator exponentials from Lie groups

Consider a Lie algebra $\mathfrak{g}$ with elements $\{g_1, g_2,\ldots,g_N\}$, with a Lie group defined by the exponential map $\exp(g)$ for $g\in\mathfrak{g}$. Given an arbitrary general element $g=\...
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1answer
56 views

Formulating a differential equation for the radius of a falling raindrop as a function of time.

A spherical raindrop is falling with constant speed. As it falls it accumulates additional mass and its volume increases. The rate of change of the volume is proportional to its cross-sectional area (...
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1answer
61 views

Rigorous Mathematical Physics Books

There are some well-known books on mathematical physics that are commonly used in undergraduate courses around the world as an introduction to mathematical methods in physics and/or applied sciences. ...
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0answers
58 views

How can I find equationts for the EM field $E=(E_x,E_y,0)$ and $B=(0,0,B_z)$?

I am trying to find the motion's equation for a charged particle $q$ with mass $m$ in an EM field. First the 2 equations I have are: $$\frac{\mathrm dU_x}{\mathrm dt}= W_cU_y + \frac{W_cE_x}{B_z}\\ \...
0
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1answer
37 views

Double integral significance

If Integration $\int_{-1}^{1}\int_{-^{\sqrt{(1-x^{2})}}}^{^{\sqrt{(1+x^{2})}}}dy dx $ Gives area of a circle But what does this integral give? $$\int_{-1}^{1}\int_{-{\sqrt{(1-x^{2})}}}^{^{\sqrt{(1+...
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16 views

Understanding the formalism of Spin Sums

I am studying Spin Sums (Quantum Field Theory by Mandl & Shaw, section 8.2) and I have questions about the Mathematics. I'll give some context and then go for the specific questions. The Feynman ...
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25 views

Closed bounded orbits in central fields - Arnold

I am reading Arnold's book on classical mechanics and I didn't fully understand his proof of Bertrand's theorem on the central potentials for which bounded orbits of a point mass are also closed. His ...
2
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0answers
25 views

Well-definedness of the Einstein-Hilbert action

I am corrently working on mathematical general relativity and stumbled over the following question: The (vacuum) Einstein Hilbert action is defined to be $$\mathcal{S}_{\mathrm{EH}}(g):=\int_{\...
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1answer
31 views

Understanding how to get Maxwell's equations in standard form

I am studying Ex.1 of chapter $2$ in Quantum Field Theory's book by Peskin and Schroeder, whose solution is available. Note we're working with Maxwell's equations in vacuum. In section a) we are ...
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2answers
52 views

Book recommendations for fourier series.

I am a physics Student, i have finished both volumes of TOM M.APOSTOL calculus, but it doesn't have any chapter related to Fourier series, can you suggest me a book through which i can directly refer ...
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1answer
16 views

Minimum uncertainty wavefunction, quantum harmonic oscillator

Consider a particle of mass $m$ in a harmonic oscillator potential $V(x)=\frac{1}{2}m\omega^2x^2$. I am given that $\psi(x,t)$ is an eigenfunction of the ladder operator $a=\frac{1}{\sqrt{2m\hbar\...
2
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1answer
70 views

How to deal with this Dirac-Delta-function integral

Let me provide you all with some context first. I am studying how to get the differential cross-section formula (in the CoM frame) as explained in Quantum Field Theory's book by Mandl and Shaw (...
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1answer
34 views

Damping procedure for Gross-Pitaevskii equation applied to an ODE

Premise: The time-dependent Gross–Pitaevskii equation (GPE) is (https://en.wikipedia.org/wiki/Gross%E2%80%93Pitaevskii_equation) $$ i \partial_t \psi = -\nabla^2 \psi + g |\psi|^2 \psi $$ plus other ...
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0answers
25 views

Solid angles comparison

Suppose I have two sets of unit norm vectors $(S_1, S_2)$ in $d$ dimension. Every set of vectors can define a solid angle. And I'm only interested in finding out which set has a bigger solid angle....
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1answer
19 views

Quantum harmonic oscillator expectation values satisfy Hamilton's equations

Consider a particle of mass $m$ in a harmonic oscillator potential $V(x)=\frac{1}{2}m\omega^2x^2$. I can show that the expectation values for the position and momentum operator are $<x>=\sqrt{\...
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0answers
52 views

Derivation of non-arbitrage-free Black and Scholes equation

I am well aware of how the ordinary Black and Scholes equation is derived, under the assumption of an arbitrage free portfolio, $V=G-hS$. Here $S$ is the price of the underlying and $G$ is the option ...
2
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0answers
28 views

Wave equation solution $u(x,t)$ for $u_0(x)=x$ and $v_0(x)=0$

Let a string with linear density $\rho$ and tension $k$ Its left and right hand ends $ [-\pi,\pi]$ are held fixed at height zero (Maybe not at $t=0$, but it are for $t>0$). Initial velocity $...
2
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1answer
32 views

Derivative of Christoffel symbol

I was answering a question which required calculating a double covariant derivative of a vector ($ D_\mu D_\nu V^\rho$) But then I got stuck trying to apply a normal partial derivative to a ...
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0answers
63 views

Energy conservation and solving for unknowns with a one dimensional string

I am trying to solve for unknowns $R,T$ when I have the following wave ansatz: $u(x,t)=\Re((e^{iwx}+Re^{-iwx})e^{-iwt}$, for $x<0$, and $u(x,t)=\Re(Te^{iwx}e^{-iwt})$, for $x>0$. If I impose ...
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1answer
12 views

A discrete torus with $L^d$ sides

I am taking a course in mathematical physics, and we've just begun a section in the lecture notes where we want to describe free electrons in the lattice $\mathbb{Z}^d$ for some $d$, which, to have ...
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0answers
33 views

Solution of the Wave Equation by Separation of Variables

We know for $\left\{\begin{array}{cccc}\dfrac{1}{c^2}u_{tt}&=&u_{xx}\hspace{0.25cm}x\in [0,L]&\\ u(0,t)&=&u(L,t)=0\\ u(x,0)&=&f(x)\\ u_t(x,0)&=&g(x) \end{array}\...
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0answers
25 views

Easy way to compute Fourier tranform of a box function

I have a big dilemma about Fourier transform... I know that: $\prod \frac{t}{T} \rightharpoonup T sinc (ft)$ but if I have something like $\prod (2t)$ or $\prod \frac{2t}{T}$ how can I compute the ...
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0answers
13 views

How to write a flow map for simple pendulum?

I am trying to show that the simple pendulum preserves phase-space volume. Although there seem to be many approaches on how to show that, I want to try and solve this problem by writing the solution ...
1
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1answer
18 views

Help finishing arguement - Symmetry property of Green's Functions

i could use just a bit of help with the justification, i believe i know why this is true but a small explination beyond my intuition would be great. Consider the Helmholtz equation in the form $$(...
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0answers
21 views

doubt with an integral - electrical fields[Solved]

I'm reading serway vol2. Can someone explain the steps, please?. I do understand how $$ λ=\dfrac{q}{l} $$ but not other steps. $$ E=kλ \int_{a}^{l+a} \dfrac{dx}{x^2} = kλ[\dfrac{-1}{x}]_{a}^{l+a} = k\...
0
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2answers
45 views

Let a string with linear density $\rho$ and tension $k$. Find $u(x,t)$

Let a string with linear density $\rho$ and tension $k$ Its left and right hand ends $ [-\pi,\pi]$ are held fixed at height zero (Maybe not at $t=0$, but it are for $t>0$). Initial velocity $...
0
votes
1answer
27 views

Solve a d-dimensional integral using coordinate transformation

Let $c_n > 0, n \in \mathbb{N}$, $g:\mathbb{R} \to \mathbb{R}$ and $\varphi_n:\mathbb{R}^d \to \mathbb{R}$ defined by $$ g(t) = e^{-1/t}, t>0, $$ otherwise for $t\le0 \ \ g$ is equal to $0$, and ...
0
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0answers
36 views

Show that a convolution of two functions solves an ODE

Given a function $f\in C_c(\mathbb{R})$, meaning $f$ is continuous with compact support in $\mathbb{R}$, and function $\Phi(x) = \frac12|x|$, show that the convolution $u=f \ast \Phi$ is well defined ...
0
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1answer
32 views

Why homogeneous? [closed]

Why in the conservation laws ( I mean energy, linear momentum and angular momentum conservations) we consider time or space to be homogeneous _as the characteristic of inertial frame? Can anyone ...

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