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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Schwinger-Dyson Equations and boundary conditions
Schwinger-Dyson Equations (SDEs) can be understood in a myriad of different ways. In the Canonical formalism, they come from the non-commutation between time differentiation and the canonical time ...
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2
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Use of the eigenfunction expansion of Green's function in physics and physical significance
In physics, we find extensive use of Green's function in almost all branches of physics. To find the Green's function of an ODE, we usually solve an equation of the form
$$L_xG(x,y)=\delta(x-y)\tag{1}$...
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Invariance of the action under a symmetry of 2D isotropic harmonic oscillator
I have a question on the invariance of the action under symmetry transformation.
As the simplest example, here I consider two dimensional Harmonic oscillator. After some rescaling, the Hamiltonian can ...
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Doubt in free end reflection of string wave [duplicate]
When a pulse on string reflects from free end, the resultant pulse is formed in such a way that slope of string at free end is zero.
What is the reasoning behind this statement? How can I prove it?
(...
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3
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Understanding periodic boundary conditions [closed]
For a free particle, the solution of the (time independent) Schroedinger equation is
$$\psi = \frac{1}{\sqrt{V}} e^{ip\cdot r /\hbar}$$
Since over infinite space this is undefined (not normalizable) ...
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2
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Electrostatic boundary conditions in dielectric media
Studying electrostatics, using Griffths, I got the following issue whose explanation I couldn't find anywere. Consider the image below to follow the problem.
If a have a chunk of a linear dielectric ...
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Can taut membranes and strings that are clamped at both ends propagate non-standing waves?
I'm going through an introductory course in acoustics and I'm struggling with the intuition behind standing waves.
When deriving solutions to the various wave equations we usually impose a boundary ...
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Ambiguity for boundary conditions after conformal transformation
Abbreviations
EOM = Equations of motion
BCs = Boundary conditions
CT = conformal transformation
Intro
I was playing around a bit with EOMs, action principle, CTs and BCs. There, I met a problem. ...
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Can't make sense of this step while self-studying greens function [closed]
so I was self-studying green functions from Mary L Boas' MMP book and couldn't make sense of this step where she finds the difference in change of the associated greens function.
My main querie is ...
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Unitarity in black hole final-state projection
Horowitz-Maldacena model, which proposes a resolution to the black hole information paradox by introducing a final-state boundary condition at the black hole singularity. From what I understand, the ...
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Boundary conditions for massive spring hanging vertically [closed]
I was trying to to find the poistion of the center of mass of a massive spring hanging vertically under the effect of gravity and I tried to model it with a finite number of springs attached ...
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Why in QM the solution to Laguerre equations are ONLY Laguerre polynomials?
I am studying eigenfunction methods to solve Fokker-Planck equations and I got stuck with a calculation that is related to some typical problems in QM. In particular, the radial part of an hydrogen ...
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Boundary conditions on Schwarzschild event horizon
Consider the variational problem for a scalar field in Schwarzschild spacetime $M$ with respect to Eddington-Finkelstein coordinates $(v,r, \theta, \varphi)$, i.e.
$$\delta I(\phi) = \int_M dV \ \Big(...
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Functional derivative under a path integral sign
Is it possible to have the analogous version of the Leibniz integral rule for the path integral? Something like
$$\frac{\delta}{\delta\chi}\int_{\phi|_{x=x'}= \chi} [\mathcal{D}\phi]e^{-S[\phi]} \sim ...
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EM junction conditions from Faraday tensor
In classical electrodynamics, we have the Maxwell's junction conditions:
$$
\mathbf{n}\cdot[\mathbf{B}]=0
$$
$$
\mathbf{n}\cdot[\mathbf{D}]=\Sigma
$$
$$
\mathbf{n}\times[\mathbf{H}]=\mathbf{K}
$$
$$
\...
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Cells in phase space in Maxwell-Boltzmann statistical analysis of thermodynamics
Can't the states overlap? For example, for one particle the value of spread of its $p_x$ is 1 unit from $p_x=0.5$ to $p_x=1.5$ whereas, for the other particle, is 1 unit from $p_x=1$ to $p_x=2$. Isn't ...
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When Does a Singularity Become Naked in Space-Time?
A singularity within a space-time 𝑀 is typically hidden behind an event horizon, preventing any information from the singularity from reaching distant observers. However, a singularity may be "...
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Non-linear Eigenvalue problem with an ODE to solve for the Linear Stability Theory of a Boundary Layer
I am working on the Linear Stability Theory (LST) for analysing the stability of a boundary layer in fluid mechanics. I have conducted CFD simulations and extracted the base flow data. To form the LST ...
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Problem with asymptotic behavior of metric
In the Hamiltonian description of asymptotically flat spacetimes, the metric should deviate at infinity from the flat metric by terms of order 1/r
$$g_{ij} = \delta_{ij} + \frac{\overline{h}_{ij}}{r} +...
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Boundary conditions in quantum mechanics
I'm having some trouble understanding the role of boundary conditions in (non-relativistic) quantum mechanics.
EDIT:
The following text talks a bit about Bloch's theorem, but this was just supposed to ...
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How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ will give the same EL and EoM for corresponding coords? [duplicate]
How does $\dot{q}_i p_i - H = \dot{Q}_i P_i - K + \frac{d}{dt}F$ give the same Euler-Lagrange equations and Equations of motion (EoM) for corresponding coordinates and allow us to determine a ...
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Physical meaning of Cahn-Hilliard boundary conditions
Consider the 1D Cahn-Hilliard equation for a two-component mixture, on an interval $x\in[a,b]$:
$\frac{dc}{dt} = -\frac{d}{dx}j(x)$ where the flux $j(x) = -D\frac{d}{dx}\left(c^3 - c - \gamma\frac{d^...
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Difference between boundary conditions in thermodynamic limit
Consider a model for a spin chain. I somehow am able to find a general formula for the expectation value of some observable in both periodic and open boundary conditions. ie.,
under PBC, I have
$\...
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From what equations is magnetic field uniquely determined for a given current distribution?
The Maxwell equations for magnetostatics in the absence of time varying electric field state that -
$$
\mathbf{\overrightarrow{\nabla}} \cdot \mathbf{\overrightarrow{B}} = 0
$$
$$
\mathbf{\...
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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 \...
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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 $\...
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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
$$ \delta I = \int_V ...
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Bethe diffraction: surface charge density of an ellipsoid
I'm having a hard time following one part of Bethe "Theory of diffraction by small holes" paper, which can be found here: https://web.stanford.edu/class/ee349/Handouts/Bethe_PR1944.pdf
At ...
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Classical open string in Polchinski -- consistency of Neumann boundary conditions with gauge choice
In Section 1.3 of String Theory, Volume 1, Polchinski derives the open string spectrum from the Polyakov action with Neumann boundary conditions, by first considering the classical open string in ...
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Understanding certain boundary conditions of functionals of the form $\int_{p_0}^{p_1}f(x,y)\sqrt{1+y'^2}dx$
A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!)
The General Question:
Among all smooth curves whose end points $p_0$,$p_1$ lie between two ...
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2
answers
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Variation of the Lagrangian expressed as a time derivative of a function
In chapter 4.5 of Jakob Schwichtenberg's Physics from Symmetry, he expresses the variation of the Lagrangian $L = L\left ( q, \dot{q}, t \right )$ with respect to the generalized coordinate $q$ as
$$\...
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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 ...
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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 ...
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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 ...
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0
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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), ...
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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,...
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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 $$I=\int L\sqrt{-...
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Uniqueness Theorem and boundaries conditions
I was recently studying Jackson Electrodynamics and faced some issues directly. I have studied Griffiths Electrodynamics and I knew about the uniqueness Theorem from it. But in Jackson, they proved it ...
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Why are Fresnel coefficients not symmetric?
When talking about reflection, we have the following coefficients for the electric field:
$$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)}{...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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0
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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 ...
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1
answer
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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 ...
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3
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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 ...
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1
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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 ...
- 3,753
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0
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45
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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 ...
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1
answer
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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. ...
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