Questions tagged [boundary-conditions]

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Conditions to determine the Green's function for scattering phenomena

Consider the elastic scattering of particles by a potential $V$ in Quantum Mechanics. In the zone of influence of the potential the Hamiltonian may be written as $$H = H_0 + V,$$ being $H_0$ the ...
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2answers
301 views

Missing Hypothesis in Electromagnetism Texts

In the Feynman Lectures, Chapter 21, I find the statement We have solved Maxwell's equations. Given the currents and charges in any circumstance, we can find the potentials directly from these ...
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1answer
499 views

Is every solution of Einstein field equations unique?

Einstein's equation is $$8 \pi T_{ab} = G_{ab},$$ where the left side contains the stress-energy tensor and the right side contains the Einstein tensor. Is there exactly one unique stress-energy ...
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0answers
134 views

What is the exact value of the constant in the similarity solution for blast waves?

Recently I used the Rankine-Hugoniot equation to reason that the limit of the speed of shock waves for extremely strong shocks is $\bigl(\frac{6P}{5r_0}\bigr)^{1/2}$ where $P$ is the pressure of the ...
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436 views

Standing Waves in a String of two Linear Densities

Given a string with sections of two linear densities like this: Does the point where the two linear densities meet have to be a node if a standing wave is produced? Are there alternatives? Could I ...
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1answer
45 views

Normalization of states of continuos spectra with complicated boundary conditions

Let's consider the following Schrödinger equation: $$\psi''(x)+k^2\psi(x)=0$$ with the following boundary condition $$\psi(0)+a\psi'(0)=0$$ $k$ is supposed to be larger that $0$. This equation is ...
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82 views

Perodic boundary conditions vs Dirichlet?

I have been working through several examples recently involving particles in boxes (when finding the partition function of an ideal gas for example or looking at photon gases). I have seen two ...
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1answer
60 views

Size of box vs. discrete-ness of state of the system

From Statistical Physics, 2nd Edition by F. Mandl, pg. 36: A sufficiently large box (say 10 light-years across) will clearly not affect the properties of our system, ion plus electron sitting ...
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1answer
130 views

Why doesn't $σ_xσ_p$ change with the width of the well in the infinite square well problem (intuition)?

I calculated that the product of the uncertainty in position $\sigma_x$ for the ground state of an infinite square well of width $L$ with the uncertainty in the momentum $\sigma_p$ for the same state, ...
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1answer
139 views

Problem with boundary condition for the differential equation of a cooling Cube and Sphere

Lets considere a sphere of radius $R$ in a temperature $u_0$ which is cooling in an environment of temperature $u_\infty$ (Note: I already solved it). I have to solve a diferential equation and one of ...
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2answers
3k views

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 ...
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0answers
224 views

Eigenvalue problem $−\psi''(x) − (ix)^ N \psi(x) = E\psi(x)$ in complex plane

To find the eigenvalue in the complex plane of $x$ for one dimensional Schrodinger equation $$ −ψ''(x) − (ix)^ N ψ(x) = Eψ(x). $$ where $N$ can be any real number, the boundary condition $ψ(x) → 0$ ...
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2answers
83 views

Could someone please explain the physics as to why two trash-bins could be more stable if placed together instead of apart during windy conditions?

I recently asked the following question on another StackExchange community(Home Improvement: Preventing trash bins from falling by placing apart of side-by-side, on a windy day?), but I would like ...
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3answers
554 views

What is the electric field exactly on the surface of a conducting sphere?

Within a conducting sphere, the electric field is 0, but is the electric field still 0 exactly on the surface?
3
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1answer
551 views

Why must this boundary condition be met? (Electromagnetic wave at interface between two mediums)

My textbook says that The laws of Electromagnetic Theory (Section 3.1) lead to certain requirements that must be met by the fields, and they are referred to as the boundary conditions. ...
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0answers
234 views

(boundary conditions) Interface between two lossless media

I'm wondering why there's usually no free charges nor free currents in the interface between two lossless media? no free current "I guess" is due to the insulating nature of a lossless media but why ...
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78 views

“Simple” Variation of the gravity action with boundary

I'm concerned with the derivation of the quasi-local stress tensor (getting from eqn 2.4 to eqn 2.6 in this paper: http://arxiv.org/abs/hep-th/0508218). As is the case with all the references I have ...
4
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1answer
205 views

Solved Gauss' Law for $\vec{E}$ without boundary conditions?

Why can I solve for the electric field of a point charge Q at the origin without boundary conditions? $\nabla\cdot\vec{E}=\rho/\varepsilon_0 = \delta(\vec{r})/\varepsilon_0$ is a 1st order ...
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162 views

on fundamental 2D conductivity equation boundary value problem

Consider the following homogeneous boundary value problem for a function/potential $u(x,y)$ on the infinite strip $[-\infty,\infty]\times[0,\pi/4]$ w/positive periodic coefficient/nductivity $\gamma(x+...
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2answers
664 views

Why is the no-slip condition valid for fluids but not for solids?

You can obviously move a solid at a different speed along the surface of another solid, so how come the velocity of the fluid at the fluid-solid interface must be equal to that of the solid? What ...
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1answer
473 views

flat plate isoflux convection - how to calculate temperature

I am an engineer trying to design a simple 1D program for evaluating the temperatures of a multi-layer heatsink that includes convection and radiation heat transfer from the external surface. For a ...
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1answer
240 views

Boundary conditions and uniqueness of heat equation solution

I have some confusion about the uniqueness of solution in an unstable heat transfer problem. The domain of this problem is shown in the figure below, which is infinite in the left-right direction, ...
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2answers
148 views

Principle of least action and greedy algorithm

Is the principle of least action sort of a greedy algorithm that all mechanical systems follow?, sometimes to minimise and sometimes to maximise the quantity we call action, at each individual step.
5
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1answer
374 views

Derivation of open boundary conditions for Non-Equilibrium Green's Function (NEGF)

It is widely claimed that the Non-Equilibrium Green's Function (NEGF) equations for the study of quantum transport have been derived from the many-body perturbation theory (MBPT). Yet the bridge ...
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2answers
1k views

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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2answers
1k views

Difference for boundary condition, particle in a box

When solving the simple problem of a free particle in a box of volume $V = L^3$, we can impose either periodic boundary conditions $\psi(0) = \psi(L)$ and $\psi '(0)= \psi'(L)$ either strict boundary ...
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2answers
5k views

Vibrating string, free end boundary condition

When discussing the vibrating string problem with one end (or both) free to move in the vertical direction but constrained in the longitudinal direction (achieved by placing the "free" end in a ...
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2answers
5k views

Direction of H and B inside and outside a bar magnet

I seem to have encountered a contradiction when thinking about the directions of $\textbf{H}$ and $\textbf{B}$ inside and outside a bar magnet. Suppose that a bar magnet has a roughly constant ...
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3answers
811 views

What boundary conditions in a wave simulation would avoid reflections?

In simulating an elastic medium as a series of mobile points connected by ideal springs, it's straightforward to model conditions corresponding to a fixed endpoint, which results in an incoming wave ...
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90 views

How to solve numerically or analytically this Partial Differential Equation?

:D I'm modeling a problem of ecology with PDEs, So I gotta solve numerically this Reaction-Diffusion Partial Differential Equation $$ \frac{\partial u(t,x,y)}{\partial t}=D\Big( \frac{\partial^{2}u(...
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0answers
250 views

QFT with fixed boundary conditions

I am looking for references on the formulation of QFTs with fixed boundary conditions for the fields (typically $\phi(0)=\phi(L)=0$), and especially how to construct the corresponding perturbative ...
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1answer
473 views

Periodic boundary and differential equation in Quantum Mechanics

Consider 3d box of size $L$ with periodic boundary. Then the Schrodinger equation gives \begin{align} \frac{d^2 \Psi}{d x_i^2} = -k_i^2 \Psi \end{align} thus we can set the solution in the following ...
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2answers
679 views

Multipole expansion in cylindrical coordinates

I am seeking the general solution for the Laplace equation in cylindrical coordinates or $$\nabla^2 \omega = 0. $$ In several texts, the general solution can be found via separation of variables ...
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1answer
151 views

Which waveguide mode will be excited by circularly polarized laser mode $\mathrm{TEM}_{00}$ entering hollow fiber?

Linearly polarized Gaussian mode $\mathrm{TEM}_{00}$ mode, would couple into HE modes of circular hollow fiber. Which modes would circular polarized beam excite entering hollow circular fiber ?
5
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1answer
944 views

How to derive end-correction value relationship for open-ended air columns?

According to Young and Freedman's Physics textbook, in open-ended air columns like some woodwind instruments, the position of the displacement antinode extends a tiny amount beyond the end of the ...
2
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3answers
702 views

Particle in a box: value for wave function $u(x)$ when potential $V(x)$ is infinity

The time-independent Schrödinger equation (TISE) is: $$ -\frac{\hbar^2}{2m}\frac{d^2 u(x)}{dx^2}+V(x)u(x)=Eu(x) \hspace{15pt}$$ where $E$ is a constant. Imagine now a infinity potential well as ...
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1answer
3k views

Clarification on slip, no-slip and no-penetration for incompressible flow

This is my current understanding: The no-slip condition at a boundary means that there is no velocity relative to the boundary, this means that the individual components is zero. So we have (in two ...
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1answer
603 views

The boundary condition for delta function

Beginning with the Schr\"odinger equation for $N$ particles in one dimension interacting via a $\delta$-function potential $$(-\sum_{1}^{N}\frac{\partial^2}{\partial x_i^2}+2c\sum_{<i,j>}\delta(...
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2answers
212 views

What is the interpretation of a wave function of the Universe in Hawking's no boundary proposal?

In the path integral formalism we have an in state $\Psi_{in}[\phi]$ and and out state and we find the amplitude for going from one to the other: $$\Delta[\Psi_{in},\Psi_{out}] = \int \Psi_{in}[\phi]...
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163 views

Proof of reflected and refracted waves being in the same plane as that of the incident wave and its projection on a planar interface

Please give a proof of reflected and refracted waves being in the same plane as that of the incident wave and its projection on an ideal planar interface between two linear, homogeneous, isotropic, ...
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1answer
341 views

Boundary conditions in holomorphic path integral

Consider the holomorphic representation of the path integral (for a single degree of freedom): $$ U(a^{*}, a, t'', t') = \int e^{\alpha^{*}(t'') \alpha(t'')} \exp\left\{\intop_{t'}^{t''} dt \left( -a^...
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4answers
9k views

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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1answer
494 views

Number of states in a given Landau level

For an electron in a uniform magnetic field, in free space, we seek to find the number of allowed states in a given rectangle $L_x L_y$ (for some fixed Landau level). In effect we are tiling 2-D ...
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2answers
155 views

Mathematical confusion in quantum mechanics

During a class about Ehrenfest theorem, my teacher use an equation to proceed its derivation (to prove $\frac{d<r>}{dt}=\frac{<p>}{m}$ ) and that is: $$\int{x\psi\nabla^2\psi^*}d\tau=\int{\...
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1answer
524 views

Where exactly is the antinode of an air column with open-closed boundary conditions?

Suppose that I have an air column with closed-open boundary condition. The air pressure at the open end of the tube is constrained to match the atmospheric pressure of the surrounding air. Therefore,...
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2answers
1k views

Goldstein's derivation of the 'principle of least action'

I want make an punctual question ands it's about The derivation of the expression $$ \Delta\int_{t_1}^{t_2} Ldt=L(t_2)\Delta t_2-L(t_1)\Delta t_1 + \int_{t_1}^{t_2} \delta L dt. \tag{8.74}$$ You can ...
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2answers
3k views

Rectangular potential barrier

Take the usual rectangular potential barrier, that is: $$V(x)=0 \: \text{if} \: x<0 \: \text{or}\: \: x>a$$ $$V(x)=V_0 \: \text{if} \: 0\leq x \leq a.$$ I've looked at several notes and books ...
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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, ...
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1answer
137 views

Reflection of EM waves

In reflection of e m waves at the boundary, to show the reflected magnetic fields we put negative sign in the unit vector, example, if the B is along z direction we put (-k) in he reflected wave, ...
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2answers
2k views

Boundary conditions for Maxwell's equations at the interface between two media

Consider the following simple Maxwell's equations: $$ \nabla\cdot\mathrm{D}=\rho $$ $$ \nabla\times\mathrm{E}+i\omega\mathrm{B}=0 $$ $$ \nabla\cdot\mathrm{B}=0 $$ $$ \nabla\times\mathrm{H}=\mathrm{J}+...