Questions tagged [boundary-conditions]

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About the boundary conditions in the finite potential wall

Suppose we have a finite potential wall with a "depth" $V=-V_0$ and a radius of $a=1.5$ fm. The exercise now states that $\Psi(r) = u_0R(r) = u_0\frac{u(r)}{r}$, where $$u(r) = \begin{cases} A\cdot \...
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Why do we require that the gauge condition $\alpha(x)$ falls off at infinity?

Let's say we are working in QED. The lagrangian is \begin{equation} \mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\big(i\not{D}-m\big)\psi \end{equation} where $F_{\mu\nu}=\partial_{\mu}A_{\...
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Feynman's path integral and uncertainty principle?

The Feynman's path integral representation gives the quantum amplitude to go from point $x$ to point $y$ as an integral over all paths. How is that idea consistent with the uncertainty principle that ...
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12 views

Dirichlet/Neuman conditions for the concentration of oxygen at the interface air-water

I'd like to compute (with a simulation) the variation of the concentration $c$ ($kg/m^3$) of oxygen in water with a simple diffusion equation : $$\partial_t c = D\nabla^2 c $$ $$c(z=0)=c_0 $$ $$c(...
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Initial in time conditions and Lagrangian approach in classical mechanics

When we derive Euler-Lagrange equations in classical mechanics following the Lagrangian approach we introduce Boundary conditions at the starting- and end-points of the path in the configuration space....
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Polchinski's String theory Green's function on $RP_2$; eq. (6.2.38) p. 176

Is there a sign error in Polchinski's String Theory Equation (6.2.38) p 176 that is not on the Errata's page https://www.kitp.ucsb.edu/joep/links/joes-big-book-string/errata ? He write for the Green'...
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42 views

Showing that the $A$-$j$ coupling in classical electromagnetism is gauge invariant

I am attempting an early exercise from Altland's Condensed Matter Field Theory. The electromagnetic field's action is given as: $$S[A]=\int d^4x(c_1F_{\mu\nu}F^{\mu\nu}+c_2A_\mu j^\mu),$$ and I wish ...
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15 views

Method of Images - Point Charge with Semi-infinite Dielectric

To calculate the electric field for the system where a point charge is placed at a distance d from a semi-infinite dielectric medium (eg. http://farside.ph.utexas.edu/teaching/jk1/lectures/node42.html)...
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10 views

Time scale for surface reconstruction

I've been reading about surface reconstruction in condensed matter physics, and it got me wondering. Is there any way to assign a time scale to this phenomenon? For example, suppose we were to take a ...
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115 views

Heat Equation… with Newton Cooling?

I have the following differential equation that is purported to represent the equilibrium temperature at a point $x\in [0,L]$ on an uninsulated rod of length $L$, whose end points are kept constant $T(...
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Are the boundary conditions purely a consequence of Maxwell's equations?

The boundary conditions, namely were all these, realized only by looking at Maxwell's equations? Or is there a physical reasoning behind them? For example, Why does the component of the electric ...
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37 views

Jackson Green function expansion in spherical coordinates

In Jackson's "Classical Electrodynamics" third edition in section 3.9 page 121 while explaining a Green function expansion the following equation is attained: $ \frac{1}{r}\frac{d^2}{dr^2}(rg_l(r,r'))...
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Delta Potential Boundary Conditions on the wavefunction

I'm reading over how the delta function potential problems are solved and I can't really understand the origin of these boundary conditions: $(1) \,\,\psi \,\,$ is always continuous $(2) \,\, \dfrac{...
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92 views

Free particle on a Möbius strip: boundary condition

I've just see this video ( https://youtu.be/np6_1k99oRA ) and this guy tried to calculate the eigenvalues of a free particles on a Möbius strip using the boundary condition on the rectangle $[-\infty,\...
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No boundary conditions approach in cosmology

I have the following questions about James Hartle and Stephen Hawking's 'No-boundary' proposal: In their approach multiple histories would exist. These histories could yield universes with different ...
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1answer
43 views

Potential of hydrogen atom solution of the Laplacian: Missing boundary condition to fix integration constant $c_1$

I have following problem, I want to calculate the classical potential $\phi(r)$ of the hydrogen atom in its ground state. The charge density is known: $$\rho(r)=\frac{-e_{0}}{\pi a^3}e^{-\frac{...
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199 views

What are the energy eigenstates for a modified quantum harmonic oscillator?

Imagine a particle obeying Schrodinger's Equation with an harmonic oscillator potential modified with an additional linear potential and cut off with an infinite potential barrier at $x=0$. That is, $$...
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1answer
26 views

Why are Cauchy boundary conditions an over-specification of boundary conditions for solving Poisson’s equation?

I was referred to Physics.SE by the following content published in Jackson’s Classical Electrodynamics: This rather surprising result [the fact that the potential within a charge-free volume is ...
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65 views

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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48 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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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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134 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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100 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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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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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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42 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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50 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
70 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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52 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
52 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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55 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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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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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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55 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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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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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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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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28 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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39 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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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
34 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
45 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
148 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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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
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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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55 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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208 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 ...