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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Matching conditions for perfect conductors in electro*dynamics*

I've been working through the problems on Wald's Advanced Classical Electromagnetism and recently I dealt with the scattering of a plane, circularly polarized wave off of a perfectly conducting ball (...
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How do the fall-off conditions follow from the finiteness of the charge flux?

I'm reading Stromingers "Lectures on the Infrared Structure of Gravity and Gauge Theory". There, they want to derive fall-off conditions for the electromagnetic field strength tensor from ...
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Stationary waves [closed]

Say we have a tube with one closed end and one open. Say that we have a hole at some point in the tube (not its half or a really specific height). The conditions which have to be imposed are that: we ...
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Converting a first order quantized boundary condition to a second quantized boundary condition?

Background So there is a recipe to convert a First Quantised $\leftrightarrow $ Second Quantised Theory (we are following these Ashok Sen's Quantum Field Theory I of HRI institute's notes). ...
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A contradiction in Nonrelativistic Quantum Field Theory

Reference : "Field Quantization" by W.Greiner & J.Reinhardt, Edition 1996. In the above reference as concerns the Hamilton density $\:\mathcal H\:$ and the Hamiltonian $\:H\:$ of the ...
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String vibration under periodic boundry conditions

I wish to known which are the vibration modes (in case that they exist) of an string vibrating under periodical boundary conditions (PBC). The question is related to molecular dynamics simulations of ...
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Coulomb gauge does not fix $\vec A$ uniquely. Is the solution to $\nabla^2{\vec A} = -\mu_0{\vec J}$ given in Griffiths unique?

With the Coulomb gauge $\nabla\cdot{\vec A}=0$, and $\nabla\times\vec A=\vec B$, the vector potential satisfies the Poisson's equation, $$\nabla^2{\vec A} = -\mu_0{\vec J},$$ which for $\vec J\to0$ ...
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Rationale for Continuity of Azimuthal Equation

To solve Laplace's equation $\nabla^2\psi =0$ or the Schrödinger equation: $$-\frac{\hbar^2}{2m}\nabla^2\psi+V\psi =E\psi$$ in spherical coordinates, we often separate variables as follows: $$\psi(r,\...
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Probability distribution of particle diffusion system with a source and absorbing boundaries

Consider a simple 1D particle diffusion process described by the SDE $dx=\sigma dW$, where $dW$ is a Wiener process. The forward Fokker-Planck equation can then be written as $$ \frac{\partial P(x,t)}{...
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No slip boundary condition in Hele-Shaw Cell

A Hele-Shaw cell can be used to visualise potential flow around a cylinder. See this image from Van Dyke Album of fluid motion with the z axis of the cylinder pointing out of the page: Potential ...
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What is the intuitive implication behind $L' = L + \frac{df}{dt}$ not affecting equations of motion?

I am referring to another post on the same question as this post: Lagrangian $L' = L + \frac{df}{dt}$ gives the same equations of motion In the first paragraph, the poster says: "It is well ...
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Is there more than one GR action (if we include boundary terms)?

The action of GR is proportional to the Einstein-Hilbert action $$S_1=\int \sqrt{-g}R dx^4.$$ Now, $R$, contains terms of the form $\partial^2 g$. Using integration by parts, one can write this ...
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Does this Fermion path integral have a solution?

This is a question on the mathematics of path integration. If we take the action density $S[\phi](t) = \frac{1}{2}\dot{\phi}\dot{\phi}$ and we take the path integral $$K_T(A,B) = \left. \int e^{-\...
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Path Integral Quantization in Peskin and Schroeder

I'm trying to learn the Feynman path integral quantization of scalar fields using Peskin and Schroeder, above eq. (9.14) on p. 282 in section 9.2 the book says: $$\langle \phi_b({\bf x})|e^{-iHT}|\...
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Open boundary condition in hubbard model

In case of open boundary condition, crystal momentum is not a good quantum number anymore. I mean we can construct an $N×N$ matrix in position basis and diagonalize it to obtain the eigenvalues, but ...
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Delta function: Intuitive way for boundary conditions

Giving the Schrödinger equation $$-\dfrac{\hbar^2}{2\,m}\,{\partial_x}^2\psi(x)+ V(x)\,\psi(x) = E\,\psi(x)$$ with potential $V(x) = V_0\,\delta(x)$. Solving this equation using an ordinary Ansatz ...
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How does one apply the phase change of $π$ on reflection at the rigid end of a string?

Consider a string, with a free end $P$ and another end $Q$ which is rigidly fixed. Now, we start oscillating the point $P$ (with $0$ initial phase difference) and a wave starts traveling(in the ...
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Boundary conditions for bulk partition function in AdS/CFT

In AdS/CFT, we are told that the bulk and boundary functions are equal: $$ \tag{1}Z_{bulk}[J]= Z_{CFT}[J], $$ where on the left hand side of the equality, $J$ is interpreted as a boundary condition at ...
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Constraining the metric on the boundary of space-time to get rid of unwanted equation of motion

I've posted this question on maths.stackexchange discussing the action I'm working with, which is (in Dirichlet boundary conditions) just the effective action of a minimally coupled free massive ...
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Boundary Condition for Free Wave Function

In Secion 3.5 of Quantum Field Theory An Integrated Approach, Fredkin, the author talks about Aharanov-Bohm Effect, where it says Define the wave function $$\Psi(\boldsymbol{r})=e^{i \theta(\...
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How does the boundary term matter in scalar field and in more general cases?

People always say that boundary terms don't change the equation of motion, and some people say that boundary terms do matter in some cases. I always get confused. Here I want to consider a specific ...
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Understanding boundary conditions for spherically symmetric dielectrics in Electrodynamics

I'm looking for some help understanding the reading in Classical Electrodynamics, JD Jackson, Chapter 4.4, Boundary-Value Problems. They state the potential as equations 4.48 and 4.49 as the separable ...
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Degeneracy of wavefunction in 1 dimension

Suppose we have a one-dimensional bound state, with the degenerate eigenstates given by $\psi(x)$ and $\phi(x)$. Using the Wronskian, we can show that there is no degeneracy, as the two functions are ...
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For the interface between two linear, isotropic, homogeneous dielectric materials with no free charge, does $E$ remain continuous?

The title basically says it all. Does the normal component of the electric field stay continuous across the interface of the two dielectrics? My intuition is that it would not stay the same as there ...
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What does "grounded" conducting sphere placed in an initially uniform electric field mean? Why the resultant field change?

Consider this example of an electrostatic boundary value problem. Let an uncharged hollow conducting sphere of radius $R$ be placed in an initially uniform electric field, $\vec E=E_0\hat{z}$, along ...
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3 votes
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Ising Model without periodic boundary conditions (PBC)

I try to calculate the correlation function $<\sigma_i \sigma_j>$ with the method of transfer matrices. I do understand how to use this method with PBC. But how can I do it without PBC? My ...
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Scalar theory with non-trivial boundary conditions. Green function

Let us consider the free scalar field theory $\varphi(x,t)$ in a space of dimension 1+1, with Minkowski metric $\eta_{\mu\nu}=diag(+,-)$. It is well known that it is described by the classical action ...
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First derivative boundary condition for scattering phase shift

My understanding is that in the partial-wave approximation we can solve for the phase shifts $\delta_{\ell}$ by (in my case, numerically) integrating the radial equation $$\frac{d^2 u_\ell}{dr^2} + \...
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Turbulent boundary layer: viscous sublayer definition

I have seen different definitions of the viscous sublayer within a turbulent boundary layer, through my searches. Ones say that the viscous sublayer area is for $y \leq 10 * \delta_v$ where $\delta_v$...
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Euler-Lagrange solution of $L=\ddot{q}^2$?

I'm new to Calculus of variations and have a very basic question. Suppose we want to solve the Lagrangian $L=\ddot{q}^2$ using the Euler-Lagrange equation. My intuition tells me that the solution ...
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Why does a ${\bf B}$-field follow a high $\mu$ core in an electromagnet?

I'm having a hard time trying to understand why ${\bf B}$-field lines tend to follow the path of a high $\mu$ material. Below is what actually happens when you apply a coil with some current around a ...
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What is boundary cosmological constant in boundary Liouville field theory?

In this paper, Liouville field theory with conformally invariant boundary is studied. The action is: $$ \int_{\Gamma}\left(\frac{1}{4 \pi}\left(\partial_{a} \phi\right)^{2}+\mu e^{2 b \phi}\right) d^{...
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Because Maxwell's equations are not all linearly independent, the six boundary conditions in the above equations are not all linearly independent

I am currently studying the textbook Microwave Engineering, fourth edition, by David Pozar. Section Fields at a Dielectric Interface of chapter 1.3 FIELDS IN MEDIA AND BOUNDARY CONDITIONS says the ...
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Why is the tangential component of electric field is equal on both sides of interface?

From Maxwell's equations (gauss's law) : $ \hat{n} \cdot (\vec{D_2}-\vec{D_1})= \sigma \implies D_{2\perp}-D_{1\perp}=\sigma : $ surface charge density. How does one go on proving further that $E_{1\...
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Path integral in quantum mechanics with definite momentum states as the boundary states

$\newcommand{\bra}[1]{\left\langle#1\right|}$ $\newcommand{\ket}[1]{\left|#1\right\rangle}$ Consider a quantum mechanical non-relativistic particle with a Hamiltonian $$\hat{H} = \frac{\hat{p}^2}{2m}+\...
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3 answers
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Contradiction in the boundary conditions for normal $\textbf{H}$-field

Consider the boundary between two magnetic materials with different relative permeabilities $\mu$. Using a small Gaussian pillbox at the surface and $\nabla \cdot \textbf{B} = 0 $, we can show that ...
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2 votes
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Periodic boundary condition (Born-Von Karman): does the size of the cube matter?

Periodic boundary conditions are chosen to faithfully represent the dynamical situation of electrons flowing through the metal. It's a better picture than the stationary wave one. We imagine a cube of ...
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Neumann boundary condition on the radial Schrodinger equation

For central-potential problems in textbook quantum mechanics, the boundary condition is physically motivated. For finite-range potentials, we match to free-particle solutions (tunneling or scattering)....
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Strings: Neumann and Dirichlet boundary conditions

As I understand it there are two possible boundary conditions for open strings. The Neumann boundary condition implies that the ends of an open string move freely with the velocity of light. The ...
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How are the Boundary conditions for wave guides derived?

So I understand that for a linear material we get the equations where 1 is the material above a boundary and 2 is the new material below the boundary. $$E_1^{\parallel} - E_2^{\parallel} = 0 $$ $$\...
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9 votes
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Why do we need a boundary condition in quantum field theories?

When we discuss quantum field theory defined on manifolds with a boundary, we always choose a boundary condition for the fields. And the argument usually says that we need the boundary condition to ...
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1 vote
2 answers
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Electrostatics Boundary condition for 2D problems [closed]

The 4 conditions above are what I am being taught, and I am aware of the usual boundary conditions of electric field and potential, but that links to surface charge density, and I was wondering how do ...
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Why doesn't a standing wave cancel out?

When a wave gets reflected, doesn't its phase change by 180 degrees. So shouldn't a wave and its reflected counterpart get canceled out as their phase differs by 180 degrees?
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Path of least action subject to an initial and a final conditions

I would like to find the path of least action subject to a initial and final condition. I don't know whether this is possible and meaningful at all, but here goes: Let us say we have a particle moving ...
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How to find the surface charge induced on a perfect conductor when Lorentz transforming electromagnetic fields?

Suppose that I have a perfect electrical conductor ($B=0$ inside conductor) in free space with a known magnetic field $\mathbf{B_s}$ outside of it, and no electric field. If I transform to a frame of ...
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How to apply the boundary condition in the derivation of the 2-point function in MAGOO?

In the famous AdS/CFT review, in section 3.3.1 the authors give the two-point function of the operator $\mathcal{O}$ for which $\phi_0$ is a source, we write $$ \langle\mathcal{O}(p)\mathcal{O}(q)\...
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7 votes
1 answer
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Open strings and branes

In string theory, open strings are attached to branes. What does "attached" mean? Do they just interact with the branes, or are they related to them in some other way?
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4 votes
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Is the momentum in the infinite square well observable or not?

I've read in posts such as this and this that the momentum operator is not self-adjoint in the infinite square well because the geometric space is a bounded region of $\mathbb R$, for example $[0,a]$ ...
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Do inviscid fluids have thermal conductivity?

Inviscid fluids are an idealization and they don't have viscosity. The no-slip boundary condition DOESN'T apply to them and consequently, these fluids won't display a hydrodynamic boundary layer. I've ...
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Boundary condition of dyadic Green’s function for planarly layered media

The dyadic Green’s function of free space in terms of orthonormal system for TE/TM polarized waves is: where, now by applying boundary conditions to the first equation, the dyadic Green’s function ...
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