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Principle of Least Action Question

Let's say we have a particle with no forces on it. The path that this classical particle takes is the one that minimizes the integral $$\frac{1}{2}m\int_{t_i}^{t_f}v^2dt.$$ So if we graph this for ...
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2answers
147 views

Eigenvalues of Hermitian operators are real and the dependence/independence of boundary conditions

Without reproducing proofs: Eigenvalues of a Hermitian operator are real (proof does not rely on the boundary conditions). The momentum operator is Hermitian (proof does not rely on the boundary ...
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0answers
39 views

Refraction of the electric field lines, at the interface of separation between two conductive media

Suppose we have 2 media with electrical parameters ${\varepsilon _1},\,{\sigma _1}$, respectively ${\varepsilon _2},\,{\sigma _2}$, separated by the plane surface $\Sigma $; electrical charge surface ...
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1answer
49 views

Ehrenfest Theorem and boundary Conditions

In what cases does Ehrenfests Theorem hold? If I look at the wavefunction of electrons in a squared box of length $L$ (with periodic boundary-conditions, $\Psi(0) = \Psi(L)$), then the solution to ...
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0answers
17 views

What is the Free Boundary Conditions?

I'm studying a paper given here click for the paper. This is a paper about Lattice Field Theory in curved spacetime. In Lattice simulations one usually uses periodic boundary conditions in every ...
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1answer
49 views

Infinite square well - periodic boundaries

If we have an infinite square well, I can follow the usual solution in Griffiths but I now want to impose periodic boundary conditions. I have $\psi(x) = A\sin(kx) + B\cos(kx)$ with boundary ...
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0answers
39 views

Bound states in two and three dimensional delta potential in non relativistic QM

I would like to find bound state energies in let's say 2D delta function potential. So my eigenvalue equation is: $$(-\frac{1}{2}\Delta - g\delta(r)) \psi = -B \psi$$ and by the means of Fourier ...
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0answers
53 views

Why is there a state which is annihilated by two different operators with same absolute Fourier index?

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposed a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
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1answer
28 views

Boundary value condition used during Jordan-Wigner transformation for a $1 D$ Ising chain

In the section 4.1 of Quantum Computation by Adiabatic Evolution, Farhi et al proposes a quantum adiabatic algorithm to solve the $2$-SAT problem on a ring. To compute the complexity of the algorithm ...
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0answers
12 views

Energy/density fluid/solid boundary condition

Working on solid/fluid couplings (for atmospheric purposes) I try to better understand how things work at the interface between an elastic medium and a compressible fluid. What I understood from ...
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0answers
29 views

Implementing fixed-temperature, solid-wall boundary conditions

I wish to simulate the behaviour of a fluid along one dimension, where the right boundary is transmissive and the left boundary is a solid wall at a fixed temperature. Temperature is not one of my ...
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1answer
41 views

On the boundary condition of solid boundaries for an enclosed, non moving fluid

First and foremost, I might say horribly wrong things. Feel free to correct any inconsistencies in the following post. Let's assume an incompressible, viscous, newtonian fluid that is not moving and ...
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0answers
47 views

How are boundary consitions implemented correctly in time dependent hydrodynamics?

I posted this question more than one year ago and got an answer recently. This answer looks good to me, but indicates that something is wrong in my original approach to the problem. Can someone tell ...
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1answer
49 views

What is the physics with examples behind the boundary conditions for heat, wave and Laplace equations?

Mainly there are three types of boundary conditions, TYPE I, TYPE II, TYPE III. TYPE I is also known as Dirichlet conditions, i.e, $u(0,t)=f(t),\,\,\mbox{and}\,\,u(L,t)=g(t)$ TYPE II is also known ...
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0answers
25 views

Mixed Neumann/Dirichlet BCs in Poisson-Schrodinger self-consistent solver

I'm currently in the process of writing a self-consistent Schrodinger-Poisson solver for a device heterstructure (High Mobility Electron Transistor). I am having a few difficulties however. It seems ...
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0answers
68 views

How can I prove that antinodes are present at both open ends of organ pipe mathematically?

I know that for anti node to be formed the magnitude of displacement should be maximum at there. For standing waves in an organ pipe, the boundary conditions are such that anti nodes are formed at ...
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2answers
143 views

Pressure standing wave nodes at the end of the open side of a tube

I do not understand why standing sound waves can be formed in a one-side or two-side open tube. Consider a one-side open tube. In particular how does the reflection of the wave at the open end occur? ...
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1answer
14 views

Conditions imposed in wave reflection and transmission in one dimension

In the study of trasmission and reflection of waves in one dimension I do not understand completely the meaning of the conditions imposed. Consider an impulse $\xi(x,t)$ moving on a rope linked with ...
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2answers
198 views

Standing wave on a rope fixed at both sides: minus sign in the reflected wave

I'm studying stationary waves on a rope fixed at both sides. In some books I find that the wave function studied is the sum of incident wave $\xi_1(x,t)$ and of the reflected wave $\xi_2(x,t)$. $$\xi(...
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0answers
21 views

Reversibility of uniqueness theorem

If we use uniqueness theorem then we know that if the boundary conditions and the charge distribution is same in 2 setups in a certain region then the electric field and potential in that region in ...
1
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3answers
163 views

Is it true that $\frac{d}{dt}\int_S \mathbf{B} \cdot d \mathbf{a}$ goes to zero if the amperian loop delimiting $S$ contracts indefinitely?

I suppose to have an ordinary magnetic field: in the answer I'm not interested to involve Dirac delta: the integral goes to zero. I want to focus on another point: an infinitesimal physical quantity ...
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2answers
56 views

Particle in a box, quantization of energy

I'm learning about how the energy of matter is quantized like how the energy of light is. My textbook illustrates the concept of quantization with the particle in a box: "A particle of mass $m$ ...
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2answers
80 views

Derivation of Euler-Lagrange equation from principle of least action

When deriving the Euler-Lagrange equation for a field $\phi$ the term $$ \int\textrm{d}x^{\mu}~\partial_{\mu}\left( \dfrac{\partial \mathcal{L} }{\partial(\partial_{\mu}\phi)}\right)\delta\phi $$ is ...
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2answers
955 views

Question about the apparent loophole in principle of least action

In Lagrangian formalism, given two points $(x_1,t_1)$ and $(x_2,t_2)$, we ask the question which paths $x(t)$ make the action $S=\displaystyle \int_{t_1}^{t_2}L\ \mathrm dt$ stationary and satisfy the ...
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0answers
11 views

When light reflects from a medium of lower index of reflection to a medium of higher index of refraction, why does the light undergo a phase shift? [duplicate]

I learned in my physics class that there is a phase shift when light reflects off a low $n$ from a higher $n$, but never got the explanation.
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1answer
31 views

Can you have a problem with a Dirichlet boundary condition but with waves that reflect off the boundary?

Say we are looking for a solution to the Helmholtz equation $$(\Delta + k^2) u = 0,$$ in in the upper half space ($y > 0$) in 2D with a Dirichlet boundary condition on the $x$-axis, that is, $u(x, ...
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0answers
72 views

Is it possible to do a path integral between two boundaries analytically on a quantum lattice?

I have been trying to perform some path integral between two boundaries for a massless scalar field $$\int_{\varphi(t_a, \vec{x})}^{\varphi(t_b, \vec{x})} \mathcal{D}\varphi(x)e^{iS[\varphi(x)]}$$ ...
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3answers
80 views

Why don't E&M fields change orientation after hitting a surface?

In essentially every derivation of the Fresnel equations, the general problem of radiation hitting a surface at a certain angle is broken into two parts (out of which we hope the solution any general ...
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0answers
32 views

Problem with understanding boundary conditions in electromagnetism

In some books on electrodynamics they stress that electric current won't radiate if it is placed on a perfect electrical conductor (PEC), citing image theory: exactly opposite current will appear and ...
5
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1answer
83 views

Why are periodic boundary conditions used for the derivation of phonons? [duplicate]

I am currently reading "Quantum Field Theory for the Gifted Amateur". In chapter 2 Phonons are introduced as solutions (in k-space) of a coupled harmonic oscillator. In real space the oscillator is ...
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3answers
498 views

Interpretation of boundary conditions in time-independent Schrödinger equation

The time-independent Schrödinger equation: $$\ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V\psi = E\psi$$ is second order, so we should expect the solution to have two "degrees of freedom" which can ...
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0answers
105 views

Why can we set the coefficient $c_- = 0$ in the solution of the quantum particle on a ring?

In the quantum particle in a ring problem, the general solution for the wavefunction, with $k = R \sqrt{2 m E / \hbar^2}$, $R$ being the ring radius, $c_{+, -}$ being constants, $E$ the energy, and $m$...
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1answer
42 views

Correlating two definitions of bound states in quantum mechanics

In Griffiths, he defines a bound state to be that stationary state for which the total energy E is such that $E<V(\pm\infty)$. Let $\psi(x)$ is a stationary state satisfying $E<V(\pm\infty)$ and ...
3
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1answer
25 views

Suitable boundary conditions magnetic field paradox

Consider a point charge $q$ situated at the origin, and a uniform magnetic field, covering all of space, pointing in the $z$ direction $\mathbf{B}=B_0\hat{\mathbf{k}}$. What happens when you turn off ...
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1answer
31 views

Boundary Conditions For Strings? [closed]

There seems to be two main boundary conditions for Strings. 1. Neumann Condition: Ends of Strings are free to move up or down. 2. Dirichlet Condition: Ends of Strings are fixed. What other ...
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0answers
18 views

When is the condition $V(x_i\rightarrow \infty)=0$ needed when solving the Laplace equation?

Im currently working on the solution of some Laplace Problems with Griffiths (Page 180). Often the condition $V(x_i\rightarrow \infty)=0$ appears when discussing problems of conducting plates, for ...
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0answers
25 views

Computing the value of an Action given some boundary conditions

Having being dealing with Actions for a while I have come across a question in which I am required to calculate the value for $S$ an action in the form of a function for some given boundary conditions....
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1answer
125 views

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
155 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
137 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
109 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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0answers
32 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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0answers
16 views

C.o.m. position and total momentum of open DD string

I'm trying to calculate c.o.m. position $q_\text{DD}^\mu$ and total momentum $p_\text{DD}^\mu$ of the open string with Dirichlet boudnary conditions on both ends. Something is going very wrong though. ...
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1answer
31 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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0answers
24 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
45 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 ...
3
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1answer
107 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
53 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
690 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
98 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 $ψ(x) → 0$ ...