Questions tagged [boundary-conditions]

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Proof that the momentum operator is hermitian without assuming the wavefunction approaches zero at infinity?

I am currently taking my second semester of quantum mechanics. For a number of proofs in the course, we have used the assumption that the wavefunction goes to zero at infinity. We have simply used the ...
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35 views

If the lagrangian density changes by a total derivative of the lagrangian density

When we derive energy momentum tensor current by actively transforming field. We see that lagrangian ( density) changes by a total derivative of the lagrangian. If a total derivative of the function ...
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28 views

Finding the potential off axis of a uniformly charged disk

This is problem 3.22 from Griffiths We know the potential at any point on the axis perpendicular to the center of the disk, I'm asked to find the potential at any point $(r,\theta)$ assuming $r<R$ ...
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34 views

Understanding Periodic and Anti-periodic boundary condition for Jordan-Wigner transformation

In the study of spin chains with periodic boundary condition ($S_{N+1}=S_{1}$) when one applies Jordan-Wigner transformation to map the spin chain to spinless fermion chain, one needs to make sure in ...
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2answers
94 views

Diffeomorphic but physically inequivalent spacetimes

In the last few years there has been a considerable endeavor in understanding the asymptotic symmetries of quantum gravity on Minkowski Spacetime. This has been tied to a study of the BMS group that ...
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8 views

Derivation of driven force on a string. How to prove maximum amplitud is achieved at resonant frequency?

I know if I have a driven oscillator of natural frequency $\omega$, applying a driven force $F_0 \cos (\Omega t)$ will result in a motion equation like this one (steady state/particular solution): \...
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25 views

Boundary conditions for an infinite line charge and grounded conducing plane

I'm not asking for a solution to the problem, I'm confused about what I should set the boundary conditions to, it's obvious that $V=0$ at $z=0$ because of the grounded $xy$ plane, but I don't know ...
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34 views

Neumann boundary condition in spherical coordinates

I'm trying to solve heat equation $$\nabla^2 u = \frac{1}{k}\frac{\partial u}{\partial t}$$ in the region $$ a \leq r \leq b, \ \ \ \ 0 \leq \varphi \leq 2\pi, \ \ \ \ 0 \leq \theta \leq \theta_0 $$ ...
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44 views

Dirac equation boundary conditions

In Schroedinger equation, which is second order differential equation, one normally, equates both $\psi(x)$ and $\psi'(x)$ across the boundary, as boundary conditions. However, the dirac equation ...
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2answers
48 views

Parallel plates capacitor, boundary conditions (paradox?)

Given a parallel plates capacitor with two dielectric as shown here: (dielectrics stacked in parallel). It's usually stated that the field is given by: $\vec{E}=\sigma/\varepsilon_i \hat{z}$ ...
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2answers
49 views

Closed end tube with anti-node

The experiment was about creating a sound wave inside a close ended resonance tube and finding the locations of maximum and minimum amplitude. (after adjusting the tube length that makes a standing ...
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1answer
46 views

Why does a wave reflect on the edge of an open tube? [duplicate]

Why does a wave reflect on the edge of an open tube? There is nothing solid to make the wave bounce. Then why is it reflected?
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53 views

Conflict of domain and endpoints in Noether's theorem and energy conservation

In the derivation of energy conservation, there is the transformation $q(t)\rightarrow q'(t)=q(t+\epsilon)$, whose end points are kind of fuzzy. The original path $q(t)$ is only defined from $t_1$ to $...
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1answer
46 views

Do we need to specify additional conditions in order to find the unique potential satisfying Laplace's equation?

Let us say we have a simple boundary value problem (BVP) in spherical coordinates : $$\Delta \phi = 0$$ along with $\phi=1$ at $\,r=1$, and $\,\phi =0\,$ at $\,\infty$. A surface sphere with radius ...
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13 views

What is the relationship between pressure and concentration in liquids in porous media?

I am working on a model in porous media. In particular, it is about the formation damage due to the precipitation. I assigned Dirichlet boundary condition for the concentration near the well-bore, ...
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1answer
41 views

Current density boundary condition

Suppose that $$ {\nabla \bullet J= 0} $$ What I know about the boundary condition are for normal direction $$ J_{1n}=J_{2n} $$ for tangential direction $$ J_{1t}/\sigma_1=J_{2t}/\sigma_2 $$ ...
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87 views

Weak Solutions to the Einstein Equation across a Junction

Consider the principle part, i.e., the part which contains the highest derivatives of the metric (which is the $2^{nd}$ derivative) is $$\mathcal{P}\{R_{ab}\}=\frac{1}{2}g^{cd}\left(\partial_{a}\...
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12 views

Boundary condition for a bulk-surface and bulk-bulk diffusion reaction system

Consider this simple example below and the corresponding geometries. I simplified these equations from the real system. Geometry 1 The first geometry is a sphere. Inside this sphere a species $b(t,...
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50 views

Neumann Boundary Conditions for the Open string and the energy momentum tensor

I read in Polchinski's book, "String Theory", page 56, that for the open string the energy momentum tensor satisfies equation (2.6.26) at a boundary $$ T_{ab}n^a t^b=0 \,, $$ where $n^a$ and $t^a$ are ...
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2answers
41 views

How are standing wave patterns created in tubes?

I get how standing wave patterns are created in strings: At a certain resonance/natural frequency of the string, a standing wave is created, increasing the amplitude of the sound, and standing waves ...
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23 views

Why pressures on interface between two liquids are the same

Assume we have a cylindrical container that has a lower part filled with some liquid, say, water, and an upper part filled with another liquid, say, air. Assume that the interface between them is a ...
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38 views

Numerically solving unbounded Fokker-Planck equation

I am wanting to solve a 3D Fokker-Planck equation of the form: $$\partial_{t}p(\mathbf{x}, t) = -\nabla \cdot \mathbf{J}$$ where $\mathbf{J} = \mathbf{v}(\mathbf{x})p(\mathbf{x}, t) - D\nabla p(\...
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34 views

Clamped spherical pressure vessel

I am struggling to find the right terms to search my questions, so even just pointing me to reference material would be appreciated. 1) Suppose I have a spherical cap (thin), clamped at the edges, ...
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1answer
32 views

Boundary condition for partial reflection

I want to solve a wave equation for the wave $\psi(x,t)$. One boundary is moving, therefore I impose the velocity $$v(x=0)=v_a\cos(\omega t)$$ the other boundary is fixed, but reflecting. If the ...
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1answer
36 views

Diffusion equation with walls (if possible with gravity), analytical solution

The solution of diffusion equation $$ \partial_t\rho=D\nabla^2\rho$$ with a point source $$ \rho(0,z)=\delta(z)$$ is in 1 dimension $$ \rho(t,z)=\frac{1}{\sqrt{4\pi Dt}}e^{-\frac{z^2}{4Dt}}$$ My ...
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1answer
107 views

Why do bound charges not appear in boundary conditions on a surface between two dielectrics?

I'm studying electromagnetism, more specifically dielectrics. However, the concepts of bound charge and free charge have been somewhat confusing for me yet. The Griffiths' book deduces one of the ...
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33 views

Dirichlet boundary conditions Polyakov action

The most general solution for the equations of motion for Dirichlet is given by: $$ X^{\mu}=a^{\mu}+\frac{1}{\pi}\left(b^{\mu}-a^{\mu}\right) \sigma+\sqrt{2 \alpha^{\prime}} \sum_{n \neq 0} \frac{\...
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138 views

Importance of an extra total derivative term in Liouville theory

In this paper on boundary Liouville theory, the authors have introduced an extra term, $-\partial_{\sigma}^2\phi$, (the last term in the equation below) in defining the stress tensor of the Liouville ...
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2answers
174 views

Why is $x^n \psi^{(m)}(x)$ zero at infinity? [duplicate]

Prove: Product of any polynomial of $x$ with $\psi(x)$ or any of its derivatives goes to zero in the limit $x\to\pm\infty$. This comes from a footnote written by the professor in his quantum ...
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2answers
45 views

In band gap theory, why can we use periodic boundary conditions

In band gap theory, why can we use periodic boundary conditions when finding the wave function of free electrons in a conductor? Why do you think it is smoothly connected at both ends of the conductor?...
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29 views

Understanding boundary conditions and forced system - Wave Equation

I'm trying to solve the wave equation for a infinite string which is attached to a mechanism that moves as $y(t)=A\cos(\omega t)$ at $x=0$. The doubt I have is: can I see this system as an ...
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3answers
126 views

Initial conditions for wave equation

One of the common initial conditions given for the wave equation, $$\frac{\partial^2 p}{\partial t^2} - \nabla^2 p = 0,$$ is $p(\overline{x},t=0) =0$ and $p^\prime (\overline{x},t=0) =0$. What is ...
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1answer
21 views

Interface boundary conditions of superconductor

Are the usual interface conditions for electromagnetic fields, i.e. $$\mathbf{n}_{12}\times(\mathbf{E}_1-\mathbf{E}_2) = 0,$$ $$\mathbf{n}_{12}\bullet(\mathbf{D}_1-\mathbf{D}_2) = \sigma_s,$$ $$\...
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1answer
35 views

Laplace equation outside sphere

When solving the Laplace equation on sphere coordinates you get: $$ u(r,\theta) = \sum_{n=0}^{\infty}\left( A_n\,r^n + \frac{B_n}{r^{n+1}} \right) P_n(\cos\theta) $$ And it is clear that if you have ...
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2answers
72 views

Boundary conditions in QM and statistical physics

I don't understand something about boundary conditions in problem that I discuss it below. in QM we solve the particle in Potential well and we obtain that we should have $k=\frac{n*pi}L$ that $n\in{...
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39 views

A simple question about equation of motion in polchinski's String theory?

In page 14 to get the equation of motion, it takes the variation of the action $$ S_P[X,\gamma]=-\frac{1}{4\pi\alpha'}\int_Md\tau d\sigma(-\gamma)^{1/2}\gamma^{ab}\partial_a X^\mu\partial_b X_\mu $$ ...
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1answer
47 views

Doubts in an introduction to classical field theory

I started to study classical field theory using the book "Field Quantization" of Greiner and Reinhardt, and I have some doubts. First, the book write the Lagrangian $L(t)$ as a functional of a field $\...
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17 views

Idea behind boundary states in BCFT

In Blumenhagen's book on CFT, in the BCFT chapter he introduces the concept of a boundary state. TO do this, he first explains how there is a duality between the one-loop open string worldsheet and ...
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26 views

Fields and gauge transformations vanishing at infinity

I find that, in field theory, it is very often assumed that the fields (classical) vanish at infinity. The same assumption is also applied to gauge transformations, for example, when saying that the ...
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2answers
88 views

Is speed of light continuous when entering a medium?

I know that light has the speed $c$ in vacuum and speed $c/n$ in a medium with refractive index $n$. I wonder how this exactly happens - is there some kind of smooth transition? If so, on which scale?...
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4k views

Why does the Schrödinger equation work so well for the hydrogen atom despite the relativistic boundary at the nucleus?

I have been taught that the boundary conditions are just as important as the differential equation itself when solving real, physical problems. When the Schrödinger equation is applied to the ...
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58 views

Constraints vs Boundary Conditions

I have a very broad question about how the mathematical framework that classical theories of physics utilize to solve problems. The question is: What are the intrinsic differences between the ...
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1answer
42 views

Why shear stress is assumed constant in the inner layer

In the derivation of the log-law and the viscous sub-layer velocity profiles, it is customary to assume that the shear stress is constant and equal to the wall shear stress. Is there any physical or ...
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1answer
44 views

Localization Principle (SUSY)

Mirror Symmetry p.200/201 Last section p.200/first p.201 It says, that the localization principle would not work if one would not impose periodic boundary conditions for the fermion integration, ...
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31 views

Non-viscous incompressible Fluid between two coaxial cylinder

Consider a non-viscous incompressible fluid lies between two coaxial cylinders. The domain occupied by the fluid is defined as $0<z<\xi$, $A<r<B$. The coaxial cylinders slowly rotate ...
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140 views

What does a negative end correction mean?

I was asked this question in one of my tests: "In an experiment to measure speed of sound by a resonating column a tuning fork of frequency 500 Hz is used. The length of air in the column is varied ...
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22 views

Elasticity free boundary terms

Suppose I have a 2D elastic body, and $\mathbf{u}(x,y)$ is a displacement field of the body. I am trying to derive the equilibrium equations for linear elasticity; I define an elastic energy $$E[\...
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1answer
108 views

Symmetric potential well different solutions

I have solved $H|\psi\rangle=E_{n}|\psi\rangle$ with $V(x)=0$ from $-a<x<a$ and $\infty$ otherwise. If I propose a solution of the form $\psi(x)=A_{n}e^{ikx}+B_{n}e^{-ikx}$ I arrive to the ...
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2answers
51 views

Boundary conditions for $\mathbf D$ and $\mathbf H$

I understand the derivation for the boundary conditions for $\mathbf B$ and $\mathbf E$ as it was explained to me in Griffiths, but Griffiths states the following: $$H_{\text{above}}^{\bot} - H_{\...
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37 views

Robin Boundary Conditions in Electrostatics

Are robin boundary conditions ever used in electrostatics? I can find three references on the internet that say they are: https://en.wikipedia.org/wiki/Uniqueness_theorem_for_Poisson%27s_equation "...