Questions tagged [boundary-conditions]

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34
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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 ...
5
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3answers
3k views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu A^\...
40
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5answers
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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-...
3
votes
2answers
661 views

What restrictions on time boundary conditions does it have to use Fourier transform to solve wave equation?

The wave equation can be solved using Fourier transform, by assuming a solution of the form of $$\mathbf{E}(x,y,z,t)~=~\mathbf{E}(x,y,z)e^{j\omega t}$$ and then reducing the equation to the Helmholtz ...
7
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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^...
14
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6answers
1k views

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, ...
4
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4answers
2k views

Is the momentum operator well-defined in the basis of standing waves?

Suppose I want to describe an arbitrary state of a quantum particle in a box of side $L$. The relevant eigenmodes are those of standing waves, namely $$ \left<x|n\right>=\sqrt{\frac{2}{L}}\cdot ...
4
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3answers
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Question on open organ pipe

Although open organ pipe is open on both ends, how standing waves are produced in a open organ pipe. Can someone explain with more clarity?
4
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2answers
16k views

Reflection of sound waves from the open end of an organ pipe & relationship b/w nodes & pressure [duplicate]

We know that standing waves are created when any wave traveling along the medium will reflect back when they reach the end. But in an open organ pipe, there is nothing to oppose the wave and reflect ...
31
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5answers
4k views

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 ...
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 ...
11
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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'...
1
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1answer
623 views

Boundary conditions of the radial Schrodinger equation

Consider the radial differential equation $$\bigg( - \frac{d^2}{dr^2} + \frac{(\ell+\frac{d-3}{2})(\ell+\frac{d-1}{2})}{r^2} + V(r) + m^2 \bigg) \phi_\ell (r) = \lambda\ \phi_\ell (r),$$ which I've ...
9
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3answers
12k views

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 ...
14
votes
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 ...
10
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6answers
49k views

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: ...
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 ...
5
votes
2answers
998 views

Why my 4-divergence term added to a Lagrangian modifies the equation of motion?

I take this Lagrangian: $$\mathcal{L}=\mathcal{L}_0+\partial_\alpha f(\phi, \partial_\mu \phi).$$ In this topic Does a four-divergence extra term in a Lagrangian density matter to the field ...
7
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2answers
2k views

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 ...
5
votes
3answers
559 views

Why do we not require higher derivatives to match at boundary when solving the Schrödinger equation in a given potential?

When solving the time independent Schrödinger equation for a given potential in 1D, the main part of the solving involves matching boundary conditions. Usually, we require the value and the first ...
4
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1answer
319 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^...
5
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3answers
745 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 ...
3
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5answers
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The discontinuity of Electric Field

''electric field always undergoes a discontinuity when you cross a surface charge $\sigma$'' GRIFFITHS In the derivation; Suppose we draw a wafer-thin Gaussian Pillbox, extended just barely over the ...
2
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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 ...
0
votes
1answer
214 views

How to show that $\lim_{r\to0} U(r)=0$ in the radial differential equation?

We have the following differential equation: $$\left({- \hbar ^2 \over 2 \mu } \frac{d^2}{dr^2} + {\hbar^2 \ell(\ell+1) \hbar^2 \over 2 \mu r^2} + V(r) \right)U(r)= EU(r)$$ in order to find the ...
32
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1answer
1k views

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 ...
16
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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 ...
14
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2answers
1k views

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 ...
9
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1answer
1k views

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 ...
5
votes
3answers
3k views

Comparison of 1D and 3D wave functions

When discussing the Schroedinger equation in spherical coordinates, it is standard practice in QM handbooks to point out that the radial part of the 3-dimensional wave equation bears a strong analogy ...
4
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2answers
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Dirichlet and Neumann Boundary condition: physical example

Can anybody tell me some practical/physical example where we use Dirichlet and Neumann Boundary condition. Is it possible to use both conditions together at the same region? If we have a cylindrical ...
2
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1answer
534 views

Physical intuition on the integral contained in D'Alembert's Formula for the wave equation

If $\phi(t,x)$ is a solution to the one dimensional wave equation and if the initial conditions $\phi(0,x)$ and $\phi_t(0,x)$ are given, then D'Alembert's Formula gives $$\phi(t,x)= \frac 12[ \phi(...
1
vote
2answers
889 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 ...
5
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2answers
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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 ...
3
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3answers
594 views

How does a photon “know” that it's left one charge and that it's going to another one?

How does it know the same charge it left will be the same charge it will return to? My understanding is photons are neutral and have no charge. i.e. Like charges repel, unlike attract. All charged ...
2
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1answer
466 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 ...
2
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1answer
570 views

An Electric Potential Glued to a Cube-Shaped Insulator to Replicate a Point Charge: Charge Distribution

I have been going back over this problem with a friend for the better part of a day: A potential is glued to a cube-shaped insulator so that outside of the insulator the field is the same as a point ...
5
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1answer
827 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 ...
1
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3answers
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Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box?

Why is $ \psi = A \cos(kx) $ not an acceptable wave function for a particle in a box with rigid walls at $x=0$ and $x=L$ where $$ k = \frac {(2mE)^{1/2}} {\hbar} \, ?$$ I had plugged the wave ...
0
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1answer
535 views

Lagrangian mechanics and initial conditions vs boundary conditions

It bothers me that many basic books on the classical mechanics don't discuss the following difference between "Newton's laws" and the "Principle of stationary action". Newton's laws can predict the ...
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2answers
349 views

An uniformly charged infinite surface plane

The electric field of an uniformly charged infinite surface plane is totally independent of how far away the test charge is. But isn't electric field depends upon distance by the relation $1/r^2$? I ...
17
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4answers
3k views

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{...
13
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1answer
954 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 ...
6
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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{\...
2
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2answers
794 views

Detailed conditions for symmetries of Lagrangian

Edit: To clarify the question, I am asking why we are justified in calling a continuous symmetry a symmetry of a system when it changes the Lagrangian by a total derivative of a function of $t, q(t)$ ...
10
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2answers
838 views

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: ...
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?
6
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2answers
3k views

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 ...
5
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3answers
1k views

How does Bloch's theorem generalize to a finite sized crystal?

I would be fine with a one dimensional lattice for the purpose of answering this question. I am trying to figure out what more general theorem (if any) gives Bloch's theorem as the number of unit ...
2
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1answer
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Higher-Order Derivatives in the Lagrangian

I am trying to derive the equations of motion for a Lagrangian which depends on $(q, \dot{q}, \ddot{q}).$ I proceed by the typical route via Hamilton's Principle, $\delta S = 0$ by effecting a ...