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Questions tagged [boundary-conditions]

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Is the principle of least action a boundary value or initial condition problem?

Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating: In analytic (Lagrangian) mechanics, the derivation of the Euler-...
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6answers
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Phase shift of 180 degrees of transversal wave on reflection from denser medium

Can anyone please provide an intuitive explanation of why phase shift of 180 degrees occurs in the Electric Field of a EM wave, when reflected from an optically denser medium? I tried searching for ...
32
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1answer
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Periodic vs Open boundary conditions

In condensed matter, people often use periodic boundary conditions to perform calculations about bulk properties of a material. It's generally argued that in the $N\rightarrow\infty$ limit the ...
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4answers
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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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5answers
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Normalizable wavefunction that does not vanish at infinity

I was recently reading Griffiths' Introduction to Quantum Mechanics, and I stuck upon a following sentence: but $\Psi$ must go to zero as $x$ goes to $\pm\infty$ - otherwise the wave function would ...
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4answers
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How to solve bound states of 2D finite rectangular square well?

I want to solve bound states (in fact only base state is needed) of time-independent Schrodinger equation with a 2D finite rectangular square well \begin{equation}V(x,y)=\cases{0,&$ |x|\le a \text{...
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2answers
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Stokes theorem in Lorentzian manifolds

I've fallen accross the following curious property (in p.10 of these lectures): in order to be able to apply Stokes theorem in Lorentzian manifolds, we must take normals to the boundary of the volume ...
14
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4answers
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Wave reflection and open end boundary condition intuition

I need to understand one seemingly simple thing in wave mechanics, so any help is much appreciated! When a pulse on a string travels to the right toward an open end(like a massless ring that is free ...
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6answers
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When/why does the principle of least action plus boundary conditions not uniquely specify a path?

A few months ago I was telling high school students about Fermat's principle. You can use it to show that light reflects off a surface at equal angles. To set it up, you put in boundary conditions, ...
14
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1answer
820 views

How fast does a wavefunction vanish at infinity?

When solving one-dimensional quantum mechanical systems, I find myself very confused about the behavior of wavefunctions at infinity. Let's first impose three reasonable constraints: The potential ...
14
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2answers
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Is there a physical interpretation of Neumann boundary conditions for the free Schrodinger equation on a domain?

Let $\Omega$ be a domain in $\mathbb{R}^n$. Consider the time-independent free Schrodinger equation $\Delta \psi = E\psi$.[*] Solutions subject to Dirichlet boundary conditions can be physically ...
14
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1answer
970 views

Asymptotic symmetry algebra

So after a lot of research, and tons and tons of papers that I've went through, I finally have some idea how to solve the equations that will give me candidates for the asymptotic symmetry group for ...
13
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1answer
952 views

How do I enforce the no-slip boundary condition in time dependent incompressible pipe flow?

This is a technical problem which must have been solved already. It won't be in beginners textbooks but there should be a solution somewhere. I welcome reading suggestions. Maybe someone with ...
12
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3answers
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Why is the pressure gradient zero at a wall?

It's accepted to impose a zero pressure gradient normal to a wall when solving the Navier-Stokes equation. Is there any mathematical reasoning for that? Which pressure (static pressure, total pressure....
12
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3answers
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Pressure standing wave nodes at the end of the open side of a tube

I do not understand why standing sound waves can be formed in a one-side or two-side open tube. Consider a one-side open tube. In particular how does the reflection of the wave at the open end occur? ...
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2answers
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How to deal with boundary conditions for path integrals?

For non-relativistic quantum mechanics, the boundary conditions are rather simple to deal with, they are just \begin{equation} \langle x_1, t_1 \vert x_2, t_2\rangle = \int_{x_1(t_1)}^{x_2(t_2)} \...
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3answers
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Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives

Suppose we have a Lagrangian that depends on second-order derivatives: $$L = L(q, \dot{q}, \ddot{q},t).\tag{1}$$ If we're working on the variational problem for this Lagrangian, then I know that we'...
11
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2answers
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What is the boundary condition for Ginzburg Landau equation?

I am trying to do some numerical calculation with Ginzburg-Landau (GL) equation for a superconductor. However, I am confused about the boundary condition of the GL equation. If we introduce the GL ...
10
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6answers
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Why is the electric field perpendicular to every point on the surface of a conductor?

I am reading Berkeley Physics Course, Volume 2 (Electricity and Magnetism by Edward M. Purcell). I am in chapter $3$, page $92$, and the book discusses conductors. The following is from the book: ...
10
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1answer
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Must the derivative of the wave function at infinity be zero?

I came across a problem in Griffiths where the derivative of the wave function (with respect to position in one dimension) evaluated at $\pm\infty$ is zero. Why is this? Is it true for any function ...
10
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2answers
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When “unphysical” solutions are not actually unphysical

When solving problems in physics, one often finds, and ignores, "unphysical" solutions. For example, when solving for the velocity and time taken to fall a distance h (from rest) under earth gravity: ...
10
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2answers
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Necessity of the Gibbons-Hawking-York (GHY) boundary term

The fundamental point of my question is whether the GHY-boundary term in general relativity is even necessary at all, and if yes, then why is it so, and what is its physical significance. Several ...
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Effect of boundary conditions on partition functions

While computing partition functions in statistical mechanics models (say) on a 2d lattice one usually makes use of "circular boundary conditions" which thus gives the lattice topology of a torus. It ...
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3answers
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Why there is a $180^{\circ}$ phase shift for a transverse wave and no phase shift for a longitudinal waves upon reflection from a rigid wall?

Why is it that when a transverse wave is reflected from a 'rigid' surface, it undergoes a phase change of $\pi$ radians, whereas when a longitudinal wave is reflected from a rigid surface, it does not ...
9
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1answer
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Why do we require quantum fields to vanish at infinity?

Classical fields, like the electrical field must vanish at infinity, because otherwise their energy would be infinite. This can be used in computations to exclude certain solutions. In quantum ...
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1answer
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Newton's law of cooling for the heat equation boundary condition

Newton's law of cooling says the temperature of an object satisfies $$ \frac{dT}{dt} = -k(T(t) - T_0),\tag{1} $$ where $T_0$ is the surrounding temperature. See these HTML notes for example. Now if ...
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How can the solutions to equations of motion be unique if it seems the same state can be arrived at through different histories?

Let's assume we have a container, a jar, a can or whatever, which has a hole at its end. If there were water inside, via a differential equation we could calculate the time by which the container is ...
8
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5answers
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Does light change phase on refraction?

I have seen a lot about when light undergoes a phase change when it is reflected. But does it undergo a phase change when refracted and if so why and if not why not?
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2answers
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Question about the apparent loophole in principle of least action

In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
8
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3answers
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Interpretation of boundary conditions in time-independent Schrödinger equation

The time-independent Schrödinger equation: $$\ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$$ is second order, so we should expect the solution to have two "degrees of freedom" which can ...
8
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1answer
317 views

Einstein's equations as a Dirichlet boundary problem

Can Einstein's equations in vacuum $R_{ab} - \frac{1}{2}Rg_{ab} + \Lambda g_{ab}= 0$ be treated as a Dirichlet problem? I am thinking of something along those lines: Consider a compact manifold $M$ ...
8
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0answers
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Is it possible to do a path integral between two boundaries analytically on a quantum lattice?

I have been trying to perform some path integral between two boundaries for a massless scalar field $$\int_{\varphi(t_a, \vec{x})}^{\varphi(t_b, \vec{x})} \mathcal{D}\varphi(x)e^{iS[\varphi(x)]}$$ ...
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1answer
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Am I missing a trick to solving a 3D potential well problem?

I was playing around with a 3-D potential $V$ such that $V_{(r)} = 0$ for $r<a$, and $V_{(r)} = V_0>0$ otherwise. By using the Schrödinger Equation, I showed that: $$\frac{-\hbar}{2m}\frac{1}{r^...
7
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5answers
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Infinite Wells and Delta Functions

In considering a delta potential barrier in an infinite well, I can just enforce continuity at the potential barrier-it doesn't have to go to zero. Why then does it need to go to zero at the walls of ...
7
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2answers
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Physically, why are high sound frequencies more easily absorbed than low sound frequencies?

If someone is playing music in a room, outside the room you generally hear a more pronounced reduction in high frequencies, compared to low. I know that sound is most easily absorbed by a material ...
7
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4answers
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Why can we neglect surface terms in Field Theory?

My teacher says that the integral $$\int _{-\infty }^{\infty }\frac{\partial J^{\mu }}{\partial x^{\mu}}d^4x$$ that we met in QFT can always be neglected since $$\int _{-\infty }^{\infty }\frac{\...
7
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2answers
364 views

If $A^\mu$ is not determined uniquely by Maxwell's equations, what happens if we solve for it numerically?

Given a solution $A^{\mu}(x)$ to Maxwell's equations \begin{equation} \Box A^{\mu}(x)-\partial^{\mu}\partial_{\nu}A^{\nu}=0\tag{1} \end{equation} which also satisfies some specified initial conditions ...
7
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1answer
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Interpretation Born-Von Karman boundary conditions

The cyclic Born-Von Karman boundary condition says that if we consider a one dimensional lattice with length $L$, and if $\psi(x,t)$ is the wavefunction of an electron in this lattice, then we can say ...
7
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1answer
861 views

Why are periodic boundary conditions used for the derivation of phonons? [duplicate]

I am currently reading "Quantum Field Theory for the Gifted Amateur". In chapter 2 Phonons are introduced as solutions (in k-space) of a coupled harmonic oscillator. In real space the oscillator is ...
7
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2answers
236 views

Poincare invariance of Dirichlet and Neumann boundary conditions

The action which describes a string propagating in a $D$ dimensional spacetime, with given metric $g_{\mu\nu}$, is given by the Polyakov action $$S_{\text{p}}=-\frac{T}{2}\int \mathrm{d}\sigma\mathrm{...
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1answer
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Green function two solutions questions

I am having some trouble with Green functions in electrostatics What is the meaning of this trick: Given $$\vec{\nabla}^2 V(\vec{r}) = \frac{-1}{\varepsilon_0}\rho(\vec{r}) = \frac{-1}{\varepsilon_0}...
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2answers
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Intuitive Cause for End Corrections

I have looked for an intuitive description for the reasons for end corrections. I find most of them with mathematics far beyond my level (high school). I found two sites that attempted to explain it, ...
7
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2answers
183 views

Why are holomorphic boundary CFT2 primary operators massless in the AdS3 bulk?

I saw a claim in this paper that holomorphic boundary CFT$_2$ primary operators correspond to massless states in the AdS$_3$ bulk. Specifically, As always, we simplify the situation by assuming the ...
7
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1answer
608 views

Do spacelike junctions in the Thin-Shell Formalism imply energy nonconservation and counterintuitive wormholes?

The Thin Shell Formalism (MTW 1973 p.551ff) is used to properly paste together different vacuum solutions to the Einstein equations. At the junction of the two solutions is a hypersurface of matter – ...
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Using a time-like boundary as a computer?

Question and Summary Using classical calculations and the Robin boundary condition I show that one calculate the anti-derivative of a function within time $2X$ (I can compute an integral below) $$\...
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1answer
604 views

Boundary conditions on current carrying wire

I'm trying to simulate by finite elements method Maxwell equations for a current carrying wire. My 3d geometry consists of a cylinder and a box containing it. I will use a mixed formulation and ...
6
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1answer
415 views

How do you account for the fact that the field points in a particular direction when the charge density is uniform? [duplicate]

Suppose an electric field ${\mathbf E}(x,y,z)$, $$E(x)=ax, \quad E(y)=E(z)=0,$$ where $a$ is a constant. So we can calculate its charge density is $$\rho = \varepsilon \nabla\cdot{\mathbf E} = \...
6
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1answer
594 views

Quantum Field Theory: why fields are equal to zero on the boundary?

One of the first assumptions, when introducing the Lagrangian and Hamiltonian in an undergraduate course on QFT is $$ \phi(x)=0\,\text{on the boundary} $$ and this is widely used in many situations (e....
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2answers
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Conductors and Uniqueness Theorem

I'm working with Griffiths Electrodynamics, and he introduces a uniqueness theorem: First Uniqueness Theorem: The potential $V$ in a volume $\Omega$ is uniquely determined if (a) the charge density ...
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3answers
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What's the difference between “boundary value problems” and “initial value problems”?

Mathematically speaking, is there any essential difference between initial value problems and boundary value problems? The specification of the values of a function $f$ and the "velocities" $\frac{\...