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Questions tagged [boundary-conditions]

This tag is for questions regarding to the boundary conditions (b.c.) which expresses the behaviour of a function on the boundary (border) of its area of definition. The choice of the b.c. is fundamental for the resolution of the computational problem: a bad imposition of b.c. may lead to the divergence of the solution or to the convergence to a wrong solution.

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Minimum or Stationary Value of a Mixed Boundary Problem

Take the volume integral of the dissipated DC current in a finite volume $\mathcal V$ of conductivity $\sigma$ and stationary potential distribution $\phi$ while assuming charge conservation $\nabla \...
hyportnex's user avatar
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Robin conditions from action principle

Consider the Lagrangian density $$L(\tilde{\phi}, \nabla \tilde{\phi}, \tilde{g}) = \tilde{g}^{\mu \nu} \nabla_{\mu} \tilde{\phi} \nabla_{\nu} \tilde{\phi} + \xi \tilde{R} \tilde{\phi}^2$$ with $\...
Octavius's user avatar
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1 answer
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Possible boundary conditions in derivation of Euler-Lagrange equations

Given a Lagrange density $$\mathcal{L} = g^{ij} \phi_{,i} \phi_{,j} - V(\phi)\tag{1}$$ I have read (e.g. here) that the boundary term that occurs through variation of the action $$ \delta I = \int_V ...
Octavius's user avatar
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Bethe diffraction: surface charge density of an ellipsoid

I'm having a hard time following one part of Bethe "Theory of diffraction by small holes" paper, which can be found here: https://web.stanford.edu/class/ee349/Handouts/Bethe_PR1944.pdf At ...
Fernando's user avatar
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1 answer
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Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice

In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
Alex's user avatar
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0 answers
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Understanding certain boundary conditions of functionals of the form $\int_{p_0}^{p_1}f(x,y)\sqrt{1+y'^2}dx$

A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!) The General Question: Among all smooth curves whose end points $p_0$,$p_1$ lie between two ...
PhysicsIsHard's user avatar
2 votes
2 answers
100 views

Variation of the Lagrangian expressed as a time derivative of a function

In chapter 4.5 of Jakob Schwichtenberg's Physics from Symmetry, he expresses the variation of the Lagrangian $L = L\left ( q, \dot{q}, t \right )$ with respect to the generalized coordinate $q$ as $$\...
tugboat2's user avatar
2 votes
0 answers
68 views

Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?

Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. operators I'm interested in would replace boundary ...
0 votes
1 answer
32 views

Eigenstates of the Laplacian and boundary conditions

Consider the following setting. I have a box $\Omega = [0,L]^{d} \subset \mathbb{R}^{d}$, for some $L> 0$. In physics, this is usually the case in statistical mechanics or some problems in quantum ...
MathMath's user avatar
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Electromagnetic Field in a 3D Cavity with Lossy Boundary

I would like to find the electric and magnetic fields inside a cubic cavity with a lossy boundary (i.e. NOT a perfect conductor). I assume that the interior of the cavity is filled with a homogeneous ...
amrit 's user avatar
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Static pressure vs ambient pressure

If in a real scenario, a flat surface with a flush perpendicular closed duct of small diameter is exposed to a tangential fluid flow(laminar and naturally with the presence of boundary layer effect), ...
Sergio's user avatar
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2 answers
45 views

It's possible to have different potentials (boundary conditions) in the surface of a cylindrical conductor?

Edit I realized that my problem is not clearly stated. In general, I can solve the Laplace equation for boundary conditions $V(r,\phi, z=0) = f(r,\phi)$ (bottom of the cylinder), $V(r,\phi, z=L) = g(r,...
kurush's user avatar
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1 answer
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Boundary conditions in $\delta I=0$ to derive Einstein's equations -- why the derivatives of $g_{\mu\nu}$ are held constant?

Dirac derives Einstein's field equations from the action principle $\delta I=0$ where $$I=\int R\sqrt{-g} \, d^4x$$ ($R$ is the Ricci scalar). Using partial integration, he shows that $$I=\int L\sqrt{-...
Khun Chang's user avatar
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1 answer
58 views

Uniqueness Theorem and boundaries conditions

I was recently studying Jackson Electrodynamics and faced some issues directly. I have studied Griffiths Electrodynamics and I knew about the uniqueness Theorem from it. But in Jackson, they proved it ...
Charu _Bamble's user avatar
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1 answer
47 views

Why are Fresnel coefficients not symmetric?

When talking about reflection, we have the following coefficients for the electric field: $$r_{\perp}=\frac{n_1\cos(i)-n_2\cos(t)}{n_1\cos(i)+n_2\cos(t)} \\ r_{\parallel}=\frac{n_2\cos(i)-n_1\cos(t)}{...
Krum Kutsarov's user avatar
1 vote
1 answer
131 views

Question about boundary condition of electric field

Let us consider a positive point charge placed has electric field $E$, that decreases with distance $r$. When surface charge of $+q$ is placed in field, the electric field will get increase and ...
Rajesh R's user avatar
-1 votes
2 answers
89 views

Momentum Eigenstates for Particle in a Box [closed]

The following lines as attached as photos taken from Beiser Modern Physics (6th Edition): Now these equations and wavefunctions make no sense to me at all, first of all how are these wavefunctions ...
L lawliet's user avatar
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2 votes
1 answer
96 views

How does one define boundary conditions in electrostatics problems?

I am struggling to find a consistent method to approach electrostatics boundary condition problems. To motivate discussion, refer to Example 3.8, Problem 3.21, and Problem 4.24 from Griffith's ...
but_why's user avatar
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7 votes
3 answers
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Something fishy with canonical momentum fixed at boundary in classical action

There's something fishy that I don't get clearly with the action principle of classical mechanics, and the endpoints that need to be fixed (boundary conditions). Please, take note that I'm not ...
Cham's user avatar
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0 answers
25 views

Frame invariance of the conservation of linear momentum at a solid-fluid boundary

The Conservation of Linear Momentum (https://en.wikipedia.org/wiki/Cauchy_momentum_equation) states that $\rho \frac{D \mathbf{u}}{D t} = \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{f}$ (1) In the ...
Robert's user avatar
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2 votes
0 answers
53 views

Interpretations of wave numbers between open and periodic boundary conditions

I'm curious about the difference in physical interpretation between open and periodic boundary conditions (OBC and PBC) although they are identical in the thermodynamic limit. For simplicity, let's ...
Kitchen's user avatar
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3 votes
1 answer
147 views

Schrödinger equation, 2D delta function potential, and confusion

Apropos of nothing in particular, I thought I would play around with the Schrödinger equation in 2D with a delta function potential. To keep things simple I thought I would concentrate on the bound ...
bob.sacamento's user avatar
1 vote
3 answers
82 views

Heat equation in 1D with Robin conditions [closed]

I am tasked with analytically solving the boundary value problem as follows: the 1D heat equation for temperature $T \equiv T(x,t)$ in a solid extending from $x = 0$ to $x = L$ and located in air at ...
Dasty's user avatar
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0 votes
1 answer
30 views

Jump conditions and energy flux between moving block and floor

This question is asked from the viewpoint of continuum mechanics, its integral laws, and jump conditions. Consider an object with a flat bottom, say a cubic block of concrete, moving with friction on ...
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Zou He boundary condition for Lattice Boltzmann

I am utilizing this paper "https://arxiv.org/pdf/0811.4593.pdf" to implement the Zou-He boundary condition, aiming to enforce a velocity of 1 at the inlet of the complex geometry. The ...
Resa's user avatar
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0 votes
1 answer
48 views

Fermi Level Shift in PN Junction

The image below shows the Fermi level shift of the PN junction when the potential of the N region is positive with respect to the P region. My textbook (Semiconductor physics and devices by Donald A. ...
papij's user avatar
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0 answers
44 views

Floating Elevator

Balloons can be balanced to have just enough string to be suspended off the floor with the end of the string touching the floor. Another balloon could then also be stacked and top of that balloon in ...
user avatar
3 votes
1 answer
175 views

Question about Griffiths' proof that $\Psi$ stays normalized

In "Introducion to Quantum Mechanixs", at p. 16, Griffiths writes what follows: Now, if $\Psi$ is just assumed to be in $L^2(\mathbb{R})$, this does not imply that $|\frac{\partial\Psi}{\...
Uagi's user avatar
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1 vote
0 answers
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Boundary conditions for the stagnation point axisymmetric boundary layer equations

Good morning, I am trying to solve the Boundary Layer equations in the stagnation point in order to compute the stagnation point heat flux. In particular, the fluid is: -A continuum -In thermal and ...
Domenico Lanza's user avatar
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0 answers
80 views

On the validity of energy eigenvalues obtained when solving the Schrödinger equation for a particle in a 1D box

I'm having trouble understanding the legitimacy of solving the Schrödinger equation for a particle confined in an infinite square well. Aren't we supposed to solve it for the whole space and not just ...
Arjun's user avatar
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0 answers
61 views

Wigner's formula for the kinetic energy density in QM

In the Schroedinger equation the kinetic energy is represented by the operator $T = -\frac {\hbar^2} {2m} \Delta$ which acts on a wavefunction $\Psi$. If we multiply this by the complex conjugate of ...
M. Wind's user avatar
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1 answer
56 views

Boundary Conditions for Spherical Harmonics Problem

I have encountered a problem with electrostatics potentials. The problem is given as follows: A sphere of radius $𝑎$ has the potential $\Phi(a,\theta, \phi)$ at the boundary. Obtain expressions of ...
SaaN's user avatar
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0 votes
1 answer
38 views

Identification of the variation on the boundary and why $\delta S_{\partial V}=0$

I recently asked this question about variational principles and how it all works. The essential answer I got was to go read a book on the calculus of variations, which I did, and this helped me make ...
Alex Byard's user avatar
1 vote
1 answer
85 views

Boundary for 2D Laplace equation in electrostatics

I'm currently delving into the fascinating topic of electrostatics, specifically the distribution of potential in configurations involving conducting plates and charged wires. My focus is on a setup ...
SaaN's user avatar
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0 votes
0 answers
55 views

How to understand variational principles and the math underlying them? [duplicate]

I work in finance, and studied math in college. I'm trying to use QFT statistics to model some aspects of the market. (I've already made some progress by deriving the Black-Karasinski Hamiltonian for ...
Alex Byard's user avatar
2 votes
0 answers
26 views

Frequency, if any, of a string with two different thicknesses [closed]

Very brief question. Assume a string that is made half out of a thin rope and half out of a thick rope (the thick rope is heavier of course). A transverse mechanical pulse is applied at one end (...
glib1's user avatar
  • 21
2 votes
1 answer
78 views

Terms in the Israel Junction Conditions

I'm confused about the Israel Junction Conditions. I've seen them written several different ways so far, but here I'll use: $$K^-_{ij}-K^+_{ij}=8\pi(S_{ij}-\frac{1}{2}g_{ij}S).$$ My understanding is ...
user345249's user avatar
1 vote
1 answer
382 views

Is this image on harmonics and overtones wrong?

I saw this image and believed this to be the definition of what the relationship between harmonics and overtones to be in strings, closed pipes and open pipes. That the $n^{th}$ harmonic = $n-1^{th}$ ...
John Hon's user avatar
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0 votes
0 answers
33 views

Fermion boundary conditions vs projective representations

I have a basic confusion about the two different boundary conditions for a fermion around a compact direction, periodic versus anti-periodic. We learn in a QM class that a fermion wavefunction picks ...
mkn's user avatar
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1 vote
1 answer
61 views

In Rayleigh-Jeans radiation law, why are the values of $n$ taken to be non-positive only?

In $k$-space the allowed values for standing waves in a cube of side length $L$ are given by $$\vec{k} = \left(\frac{\pi}{L}\right) (n_1, n_2, n_3)$$ where the $n_i$ are nonnegative integers. Why are ...
iman Bilal's user avatar
0 votes
0 answers
23 views

Fourier transformation of Hamiltonian with Alternate Hopping amplitude

How to perform the Fourier transformation of the series: $$ \sum_{<l,m>}(-1)^{l+1}(a_{l}^{+}a_{m}+a_{l}a_{m}^{+})$$ where the sum is over the nearest neighbours (l,m), l=1 to N,and there is ...
Ani Dutta's user avatar
1 vote
0 answers
29 views

Where did the idea for the approach to solving this boundary layer problem come from?

I found the derivation of Blasius-Equations here: A free stream velocity hits a flat plate and the goal is to derive boundary layer behavior. Everything in this tutorial is clear - but the most ...
MichaelW's user avatar
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1 vote
0 answers
67 views

Canonical commutation relation on the spatial boundary of the hypersurface

Consider the equal time commutation relation of a field given on a $d$ dimensional spacelike hypersurface $\Sigma$ of a $d+1$ dimensional manifold given by $$[\Pi(t, x), \Phi(t, x')] = i\hbar\delta^{(...
Dr. user44690's user avatar
0 votes
1 answer
74 views

On the boundary conditions of the Casimir effect and quantization of the wave vector

I'm reviewing the famous Casimir effect. I'm uploading an image with the starting setup and frame of reference. The electric field field operator is: where $\textbf{e}$ is the polarization vector, $\...
Giuliano Artale's user avatar
0 votes
1 answer
61 views

Problems about "boundary conditions and topology"

In the book Field Theories of Condensed Matter Physics by Fradkin In Page 311, when discussing the effects of boundary conditions on $Z_2$ lattice gauge theory, in the weak coupling phase, Fradkin ...
xiang sun's user avatar
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1 vote
2 answers
175 views

Hermiticity of a radial momentum operator $\hat{p}_r$ and the spectral theorem

In Nolting's QM book (Theoretical Physics 7), in the chapter on central potentials, a radial momentum operator $\hat{p}_r$ is defined as \begin{equation} \hat{p_r} = -i \hbar \Big( \frac{\partial}{\...
EM_1's user avatar
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2 votes
0 answers
106 views

Gauge symmetries, isometries of spacetime and asymptotic symmetries

I am having a hard time understanding the physical meaning of asymptotic symmetries in General relativity. I think I understand the mathematics, but the meaning eludes me. I'll try to write the things ...
P. C. Spaniel's user avatar
6 votes
1 answer
491 views

Periodic boundary conditions: torus or infinite images?

I have a "philosophical" question regarding the use of periodic boundary conditions (PBD) in modeling and simulating systems of particles. Let us consider a system of $N$ classical particles ...
Michele Pellegrino's user avatar
0 votes
1 answer
131 views

Boundary Layer Momentum Integral Equation for velocity profile $U/U_e = 2 y/\delta - (y/\delta)^2$

Assuming a 2D boundary layer on a flat surface with no pressure gradient (i.e. $dP/dx=0$), suppose the $x$-component of velocity, $U$, has a profile $$\dfrac U{U_e}=\dfrac {2y}\delta -\dfrac{y^2}{\...
user256872's user avatar
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3 votes
5 answers
315 views

How does light know the destination?

According to Fermat's principle, light travels the fastest path from dot A to dot B. I wondered how light knows which path is the fastest, and found out that light actually goes all path, but non-...
tneserp's user avatar
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