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0
votes
1answer
21 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 ...
3
votes
1answer
80 views

Electric potential and field due to a continuous charge distribution

(1) The electric potential due to a continuous charge distribution is: $$\psi=\int_V \dfrac{\rho}{r}\ dV$$ To calculate this integral $\rho$ must be continuous over $V$. But $\rho$ is discontinuous ...
3
votes
1answer
89 views

Schrödinger Equation for a freely falling body near the surface of Earth

Near Earth's surface the Schrödinger equation of a freely falling particle takes the form, $$ \frac {-\hbar^2}{2m} \frac {d^2 \psi (y)}{dy^2} + mgy\psi (y) = E \psi (y). $$ Putting $k=\frac {\sqrt {...
0
votes
1answer
64 views

What is meant when we say “any solution is *the* solution” due to the uniqueness theorem?

I understand the proofs for the uniqueness theorems in electrostatics, but I'm having trouble understanding their application to problem solving. A classic example is a system of concentric shells of ...
1
vote
1answer
28 views

How does the image of $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfy the boundary conditions for the infinite square well?

I understand mathematically how $\sqrt{2/L} \ \sin\left({k_n x}\right)$ satisfies the boundary conditions for the infinite square well in terms of the fact that $\psi(0) = \psi(a) = 0$, and excuse the ...
1
vote
1answer
109 views

Boundary Condition for Dirac comb potential in solving independant Schrodinger Equation

The Periodic potential is And, the general solution is: Then, boundary condition at $x=a$ is: Where does $2\Omega u(a)$ comes from? I know that boundary condition is just 1) $U(x<a)(a)=U(x>...
2
votes
1answer
200 views

Infinite annular potential well. Trouble with solving Bessel equation to get eigenstates and energy

I have infinite annular potential well (scheme in the picture). Schrodinger equtation in the anullus (for $R_1 <r<R_2$ is $V=0$) with polar coordinates is \begin{equation} - \frac{ \hbar }{...
0
votes
2answers
122 views

Behavior of potential by infinite charge distribution

The picture is a question from the book Intro to Electrodynamics by Griffiths. In question as you can see we want to find potential due to an infinite strip maintained at constant potential in the ...
2
votes
1answer
550 views

Linear, homogenous and isotropic dielectric in electrostatic field. Why do I consider two potentials (inside & outside sphere)?

Presentation of the problem : We have a uniform homogenous isotropic dielectric sphere in an electrostatic field. To solve this problem, we remark that we have an azimuthal symmetry. So the ...
3
votes
3answers
461 views

$\sin$ and $\cos$ components in symmetric infinite potential well problem

Consider an infinite potential well in one dimension with boundaries at $\pm a/2$. Can $\psi(x) = A \sin(kx) + B \cos(kx)$ for this system? The way it was answered was "mathematically acceptable but ...
1
vote
1answer
54 views

Charge on a conductor's surface

I take a charged conductor completely insulated. The charge is distributed over the surface, maintaining the surface at a given potential. The charge distribution that gives this potential is unique?
-1
votes
1answer
367 views

A point charge near an infinite conducting plane

I want to calculate (with Poisson's equation) the electric field in the region containing a point charge near an infinite conducting plane with 0 potential. My textbook uses V(x,y,z)= 0 for every x,y,...
2
votes
2answers
414 views

How do I show that the Laplace equation has a unique solution under the Dirichlet closed-surface boundary condition?

In other words, when the the potential is specified at a finite boundary, how can I show the solution to $\nabla ^{2} V = 0$ exists and is unique? It is fine to show it for two dimensional Cartesian ...
1
vote
0answers
50 views

Trouble applying boundary conditions to Laplace equation

I'm having trouble finding which conditions to apply to Laplace equation in order to find the electrostatic potential of a specific configuration: There are 4 electrodes, given by the equations (each ...
0
votes
2answers
240 views

Solving the TISE for Infinite square well mathematical question

Consider the infinite square well situation where the potential is infinite at positions $|x| > a$ and $0$ otherwise. When solving the Time independent Schrodinger Equation (TISE) we can come to ...
1
vote
1answer
134 views

Energy spectrum for a step potential

Most of the books tend to give this explanation that for a bound physical system, the energy and momentum eigen values have discrete spectrum and otherwise, they have a continuous spectrum, which I ...
0
votes
1answer
26 views

An electrostatic problem for two disks in $\mathbb{R}^2$ - how can the solution be represented?

The electrostatic Laplace problem for the exterior of a disk can be solved in a straightforward manner using separation of variables. Suppose we have a unit disk $\Omega$ with a charge density of $f$ ...
0
votes
1answer
645 views

Dielectric cylinder in uniform electric field: nonlong cylinder

Problem of dielectric cylinder in uniform electric field is well known. For example, Jackson textbook or Griffiths textbook or online solution here. Solution always given for case of long cylinder. ...
1
vote
2answers
601 views

Intuition in infinite grounded conducting plane with a point charge above it

In this problem, we use the Method of Images and get the resulting properties of the charge distribution on the plane. At the end of this process, Griffiths, in Introduction to Electrodynamics, ...
4
votes
2answers
213 views

Dipole field issue in particle-mesh Ewald method with periodic boundary conditions

I am working on a thesis that makes a great use of molecular-dynamics simulations, and I am trying to understand how the particle-mesh Ewald method works. The problem is, I have difficulties ...
2
votes
1answer
33 views

Choosing $A_l=0$ to guarantee bounded potential in infinity

I'm taking a course in Electrodynamics and quite often, when using the spherical approach $$\Phi=\sum\limits_{l~=~0}^{\infty}\left(A_lr^l+B_lr^{-(l+1)}\right)P_l(\cos\gamma),$$ there's the argument ...
1
vote
1answer
177 views

boundary condition for embedded dielectric sphere

Is the potential across the boundary continuous for a dielectric sphere embedded in a dielectric material, so that the potential inside the sphere can be set equal to the potential outside of it at $r=...
2
votes
1answer
109 views

Integrating Poisson Equation over two different regions knowing only two boundary conditions for the potentials

Poisson Equation for electric potential is: $$\nabla^2 V=-\frac{\rho}{\epsilon}$$ Solving the equation require two boundary condition. I'm confused about the use of these boundary conditions in some ...
1
vote
1answer
322 views

Why Does Electric Potential Approach Zero at Infinity: Boundary Conditions for Infinite Conducting Sheets

Imagine an infinitely long conducting "trough," as shown in the figure. The two sides are grounded, and the bottom strip is maintained at potential $V_0$. Suppose we want to know the electric ...
1
vote
1answer
182 views

Understanding the proof for uniqueness of solutions (Poisson's equation)

We assume there are two functions $V_1$ and $V_2$ satisfying Poisson's equation and the same boundary conditions, and we define $U=V_1-V_2$, then: $$\nabla^2U=\nabla^2V_1-\nabla^2V_2=0$$ Now, lets say ...
0
votes
1answer
321 views

Boundary condition of charge sheet in an external electric field

Please kindly refer to page 88 in the link below Click here For a sheet with surface charge $\sigma = \sigma_{f}+\sigma_{b}$, the electric flux through the gaussian pill box of area A can be ...
0
votes
1answer
273 views

Uniqueness theorem in uniform electric field

Consider the following: An uncharged metal sphere of radius $R$ placed in a uniform electric field $\vec{E} = E_0 \hat{z}$. The field will push positive charge to the northern surface of the sphere, ...
3
votes
1answer
127 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, ...
2
votes
2answers
625 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 ...
2
votes
3answers
614 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 ...
1
vote
1answer
141 views

Can Ampere's Circuital law be used on an infinite number of alternating Helmholtz coils?

I have the following surface current density $$ \bar{\sigma}_s = \hat{\phi} \sin(kz) |\bar{\sigma}_s| $$ to approximate an infinite number of alternating Helmholtz coils stacked along the z-axis with ...
5
votes
3answers
553 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 ...
1
vote
1answer
349 views

Eigenvalues of the radial Schrödinger equation on a finite integration interval

There are numerous ways to estimate the eigenvalues of a radial Schrödinger equation, see http://arxiv.org/abs/math-ph/0703040 as an example. Anyhow, the formulas only cover the Schrödinger equations ...
3
votes
1answer
183 views

Inconsistency in the delta potential

I encountered an inconsistency in the one-dimensional delta potential. Suppose we have a one-dimensional infinitely deep square well from $-L$ to $+L$. We know the eigenstates are sine and cosine ...
1
vote
1answer
488 views

Particle Outside the Box

What prohibits, mathematically, that a particle cannot be found outside the box ? Here, I am referring to particle in a box problem (infinite potential on both ends & zero potential along the ...
2
votes
1answer
694 views

Dielectric sphere placed in another dielectric medium with uniform external field: is there a surface charge density?

Consider a dielectric sphere placed within a dielctric medium. There is a uniform electric field $E_0$ present throughout in the medium. Would there be surface charge on the sphere?
2
votes
1answer
568 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 ...
7
votes
5answers
3k views

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 ...
10
votes
6answers
48k 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: ...
2
votes
3answers
6k views

Barrier in an infinite double well

I am stuck on a QM homework problem. The setup is this: (To be clear, the potential in the left and rightmost regions is $0$ while the potential in the center region is $V_0$, and the wavefunction ...
4
votes
2answers
659 views

The appearance of volume $V$ in the Fourier series representation of a periodic cubic system

In the textbook Understanding Molecular Simulation by Frenkel and Smit (Second Edition), the authors represent a function $f(\textbf{r})$ (which depends on the coordinates of a periodic system) as a ...
1
vote
2answers
1k views

Image charges, laplace equation and uniqueness theorem

Consider a well-known problem of the electric field generated by a system composed of a point charge in proximity of a large earthed conductor. It is said that the potential due to an image charge ...
1
vote
2answers
4k views

Does the wave function need to be zero at the boundaries? [closed]

Recently I had a quiz in my physics class and I feel like the professor made a mistake on the solution for it. Yes, I already have the answer to this question. I am not trying to get people's answers ...