Questions tagged [boundary-conditions]
This tag is for questions regarding to the boundary conditions (b.c.) which expresses the behaviour of a function on the boundary (border) of its area of definition. The choice of the b.c. is fundamental for the resolution of the computational problem: a bad imposition of b.c. may lead to the divergence of the solution or to the convergence to a wrong solution.
1,240 questions
1 vote
0 answers
57 views
Path integral in quantum mechanics and expectation values
How to calculate any expectation value from path integral in quantum mechanics? In QM path integral the initial and final points are fixed and points between them are varied. But as far as i ...
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2 votes
1 answer
49 views
Trouble understanding the boundary conditions for the Blasius boundary layer equation
In Section 7.2.1 of Bergman's Fundamentals of Heat and Mass Transfer, there is a derivation of the Blasius equation $$ 2 \frac{\mathrm d^3 f}{\mathrm d \eta^3} + f \frac{\mathrm d^2 f}{\mathrm d \eta^...
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0 votes
1 answer
73 views
Mass lifting off a harmonic oscillator
Consider this setup: A classic, harmonic oscillator made of a spring with spring constant $k$ and a mass $m_1$ that oscillates vertically. $m_1$ is formed like a horizontal plate, and on the plate ...
6 votes
4 answers
330 views
Is there a version of Hamilton's Principle of Stationary Action when only initial conditions are known and the final end state is unknown? [duplicate]
Consider a dynamical system with Lagrangian $L$ and configuration space $X$, we are interested in trajectories of this system over a time interval $q:[t_0,t_1]\rightarrow X$. When one has the ...
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8 votes
5 answers
634 views
Fixed vs free ends in 1D standing waves
In introductory physics courses one often discusses standing waves on a string with two fixed ends. A standard experimental demonstration of this is given here. My problem is that in such a ...
- 2,018
0 votes
2 answers
82 views
Reasoning for fixed endpoints when constructing the action from the EL equations
Given a particle in space, the EL equations give us a differential equation for determining how the partilce will move as time evolves. I am comfortable deriving a least action principle which ...
- 21
3 votes
1 answer
91 views
Effect of non-self-adjointness on correlation functions of momentum for the particle in a box
I would like to calculate the optical conductivity of a particle in a box, which means calculating a correlation function of the momentum operator. The momentum operator appears in the calculation ...
- 613
2 votes
1 answer
78 views
Edge singularity and finite energy in a conducting wedge
Solving Laplace in a conducting wedge of opening angle $a$ with Dirichlet data on $\phi = 0, a$ gives $$ \Phi(r, \phi) = \sum_{n=1}^\infty A_nr^{n\pi / a} \sin \frac{n\pi \phi}{a}. $$ Near the apex $r ...
6 votes
1 answer
598 views
Why isn’t the magnetic field discontinuous at the surface of a current-carrying wire if the permeability inside and outside are different?
Question: Why isn’t the magnetic field $B$ discontinuous at the surface of a current-carrying wire if the permeability inside ($\mu_m$) and outside ($\mu_0$) are different? When deriving the magnetic ...
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1 vote
0 answers
129 views
EM Fields inside a wire
Imagine that we have a (copper) wire with radius $a$ and length $L$ oriented along the $z$ axis. Maxwell's equations inside the wire are: $$ \nabla \cdot \mathbf{E} = \rho/\varepsilon \\ \; \\ \nabla \...
- 2,069
0 votes
0 answers
55 views
How to incorporate boundary conditions in mean field descriptions while deriving macroscopic equations from microscopic stochastic processes?
The question is linked to this question. The microscopic stochastic processes are defined using homogeneous jump probabilities between sites. The assumption will be broken when we have physical ...
- 291
2 votes
1 answer
149 views
Why does the boundary behavior of Lagrange multipliers matter for gauge vs global symmetries?
In Henneaux & Teitelboim (Quantization of Gauge Systems, p. 30), they discuss the variation of a dynamical variable $$ \delta F = \int d^nx\, u(x)\,\{F, C(x)\}_{PB},\tag{1.62} $$ where $C(x)$ is a ...
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2 votes
1 answer
174 views
Associated Legendre equation solution for the angular equation
So in Griffith's, after doing separation of variables on the angular equation $$ \sin\theta \frac{\partial}{\partial\theta}(\sin\theta \frac{\partial Y}{\partial\theta}) + \frac{\partial^2 Y}{\partial ...
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1 vote
0 answers
88 views
Black Strings and $p$-Branes
I was trying to work out the details of the paper by Horowitz and Strominger titled "Black Strings and $p$-Branes". This can be accessed via https://inspirehep.net/literature/29494. I am ...
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4 votes
1 answer
151 views
Why can we use a non-symmetric propagator if the currents symmetrize the integral?
Imagine a simple QFT with action principle \begin{equation} S[\phi,J]=\int d^4x \Big\{-\frac{1}{2}\phi\Box \phi+J\phi\Big\}. \end{equation} As usual, we solve the field equations $\Box \phi=J$ to get \...
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