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Questions tagged [dispersion]

Dispersion refers to the frequency dependence of the properties of a wave.

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Connection between dispersion relation and symmetries of the Hamiltonian

I am having trouble understanding intuitively the connection between the dispersion relation and the symmetries of the Hamiltonian. For example, suppose we have a lattice and there are four sub-...
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Deriving dispersion relation of hyperbolic material

I am trying to derive the following form of the dispersion relation for hyperbolic material: From the literature, this dispersion relation is achieved by setting the determinant of the below matrix ...
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Are there Dirac equations for different energy-momentum dispersion relations?

When I was introduced to the Dirac equation they wrote a PDE such that plane waves satisfy $E^2 = P^2 + m^2$. They went on to show that other options (ie Klein–Gordon) don't have spin. Are there Dirac ...
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Can an optical medium be lossless and dispersive?

Occasionally, I come across the phrase "lossless dispersive linear optical medium". How can such a medium be possible mathematically? I mean the real and imaginary parts of the electric ...
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Local Maxima in a Dispersion Curve

Hello I programmed a dispersion curve and plotted the phase velocity. I am noticing that for one of my modes (the S2 mode) my phase velocity reaches a maxima and then decreases. Is this a physical ...
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Factor for observed redshift - light dispersion (chromatic aberration) caused by a gravitational lensing

This thought tortured me for a while now, and I can't find if this has been hypothesized/discussed before, and if so, is there a consensus: Can redshift be to at least some degree a result of a ...
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Mathematical definition of a wave [duplicate]

I am aware that, in general, a wave is a type of energy transfer (though I admit I don't know if there are more complicated types of waves for which this isn't true). But I want to understand how a ...
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Trying to derive relativistic dispersion formula

If we define conserved quantities of motion as constants arising from continuous symmetries of the system (Lagrangian), why does the following argument not give the correct result? Let $\gamma: I \to ...
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Phonon dispersion and what wave it represents

To start things off, I'm doing a semestral project on phonons and I'm to find a dispersion relation of a crystal (with the use phonopy). I chose to do hexagonal boron nitride. From my understanding, ...
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Problem with calculating the variance of different sounds

Problem: In a layered crystal with weak connections between layers, the dispersion law of acoustic backgrounds has the form $$\omega _1=s_1q_{||}, \ \ \omega _2=s_1q_{||}, \ \ \omega _3=Sq^2_{||},$$ ...
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Dispersion Relations in a Cold, Unmagnetized, Collision-less Plasma

Currently I am trying to understand the physics of waves in cold plasmas. So in a cold, collison-less, unmagnetized plasma you can derive two different different dispersion relations: one for ...
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Is the relativistic energy-momentum relation fundamental? [duplicate]

The following relativistic energy-momentum relationship is taken to be fundamental: $$E^2=p^2c^2+m_0^2c^4.\tag{1}$$ Let us specialize to massless particles ($m_0=0$) so that we have: $$E=p\ c.\tag{2}$$...
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How to determine the effective mass for multivariate dispersion relation? [duplicate]

Consider the 1D dispersion obtained by the tight-binding method: $ E(k)=\epsilon_0-t\cos(ka) $ $\epsilon_0$ is the on-site energy and $t$ is the hopping energy. To find the effective mass, you can ...
Anthony K.'s user avatar
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Dispersion relation for non-harmonic waves

This question is related to my previous one. The entire linear theory of waves is built on dispersion relations, which represent the algebraic dependence of frequency on wave number. That is we ...
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Derivation in a Landau-Lifshitz ferromagnetism paper

I was reading the following paper by Landau and Lifshitz on ferromagnetism: https://www.sciencedirect.com/science/article/abs/pii/B9780080363646500089 In the paper, the following expression is used ...
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Exploring Phonon Dispersion in Two-Dimensional Materials: Analytical Techniques and Mode Identification

How can one analyze the phonon dispersion of 2D materials like graphene to discern specific curves corresponding to particular phonon modes, such as acoustic or optical, as well as longitudinal and ...
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Can anyone explain the relationship between the refractive index, the speed, wavelength and angle of a wave?

Can anyone explain the relationship between the refractive index, the speed, wavelength and angle of a wave? in my book is states that $$n = \frac{v_1}{v_2} = \frac{\sin θ_1}{\sin θ_2} = \frac{λ_1}{...
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"Derivation" of group velocity and "sharply" peaked wave functions

I was reading BH Bransden's chapter on wave packets and he derives the group velocity expression taking into account that: $$ \psi (x,t)=\int e^{i[p_{x}x-E(p_{x})t]/ \hbar} \phi (p_{x}) dp_{x} $$ ...
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Wave packet at $t<0$

If there's wave packet $\psi(x,t)$ in zero potential and $\psi(x,0)$ is given, what happens when $t<0$? I know that for $t>0$ the wave as a group moves at group velocity $\frac{d\omega}{dk}|_{...
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Why acoustic phonon dispersion cross $\omega=0$ at $k=0$?

When I am learning about phonons, it is taught that acoustic phonons necessarily have $\omega=0$ at $k=0$. while optical phonons have a finite $\omega$ at $k=0$. But I am confused about two things: ...
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Wave packet for particle in a potential

In the book Quantum Mechanics by Cohen-Tannoudji, the author explains that the solutions of the Schrodinger equation of a free particle in one dimension are plane waves: $$\psi (x, t) = A e^{i(kx-\...
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Wave dispersion relation as a differntial equation

I have seen the following dispersion relation as a differential equation: $$ \frac{ \partial \mathbf{k} }{ \partial t } + \left( \mathbf{V}_{g} \cdot \nabla \right) \mathbf{k} = 0 \tag{1} $$ in the ...
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Dispersion of light without deviation

Why do we have dispersive angle in dispersion without deviation if there is no deviation from the original path?
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The speed of light in medium with different frequency

We know that the speed of light in vaccuum can be expressed as $c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$ and thus the speed of light in vaccuum is thus $$v=\frac{1}{\sqrt{\mu\epsilon}}=\frac{c}{\sqrt{\...
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Periodic Zone Scheme - Bloch Theorem in Lattices

I am quite confused about the different representations of the dispersion relation in a lattice. This image makes a lot of sense to me, since it only represents one dispersion curve and transforms it ...
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How to Calculate frequencies, $C$ and Group $Cg$ from a dispersion relation?

Some people requested what was the context of the question and this is where it comes from: "Given the dispersion relation: $w^2=w_0^2+A^2k^2$ ( $w_0$ and $A$ are constants) Which frequencies ...
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Quantum mechanics, wave solutions + dispersion relations

So, I'm trying to write down the passages required to create a wave function (from first to second quantization, to just wave functions). From the Wikipedia page, basic behaviour, non-dispersive wave ...
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What does "non-dispersive" mean in terms of waves and group velocity?

I'm confused about the term wave group velocity: It is usually explained in terms of a superposition of harmonic waves with very closely spaced wave vectors and frequencies. It is then easily shown, ...
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Let's say I have manufactured a prism from a non-dispersive medium, then light coming from air wouldn't split into colours right?

Let's say I have manufactured a prism from a non-dispersive medium, then light coming from air incident on the prism wouldn't split into colours, right? I mean light still changes direction, but all ...
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Generally speaking, the speed of a wave depends on the medium and the wave type, so waves of the same type in the same medium attain the same speed [duplicate]

My question is, given the above statement, why electromagnetic waves attain different speeds in media other than vacuum even though they are of the same type and propagate in the same medium?
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Group Velocity at the Boundary of the Brillouin Zone

I'm having trouble understanding why the dispersion relation $\omega(\vec{q})$ is extremal at the boundaries of the first Brillouin zone. I understand why the dispersion relation is symmetric around ...
Tomas Noguera's user avatar
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Electron dynamics in a periodic potential: Why wavepackets? Why $\vec{v}=(1/\hbar)\nabla_{\vec{k}}E_n$ is its group velocity?

Electrons in a periodic potential, $V(\vec{ r})$, satisfying $V(\vec{r}+\vec{R}) = V(\vec{ r})$, are described by Bloch states $$\psi_{n,\vec{k}}(\vec{r})=e^{i\vec{k}\cdot\vec{r}}u_{n,\vec{k}}(\vec{r}...
Solidification's user avatar
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Deriving $\mathrm d\omega/\mathrm dk = v$

I am reading Matter and Light by de Broglie and a little bit confused about how he deduced the equation of $v_g := \frac{\mathrm d\omega}{\mathrm dk} = v$ I understand that a wave that is static in ...
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What are soft phonon modes?

In chapter 4 of Charles Kittel's Introduction to Solid State Physics, a problem titled "soft phonon modes" asks us to derive the dispersion relation for the following setup: a 1d crystal ...
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Basics regarding $k$-space axis and Energy graph in direct semiconductor

We can see from the graphic axes, that energy increases along the y-axis, and k space increases along the y-axis. Considering a direct or indirect semiconductor whichever it doesn't matter, but why ...
Fredrick's user avatar
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Understanding the use of Group Velocity Dispersion Optics

I am a bit confused on the topic of GVD. Most information I read on the topic tells me something like this: "GVD means that the group velocity will be different for different wavelengths in the ...
fordanjen's user avatar
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Is it possible to witness a rainbow while facing the sun?

We know that a rainbow occurs due to the refraction and reflection of light inside raindrops. We also know that inside spherical raindrops total internal reflection is not possible. So some light is ...
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Superfluid dispersion relation

So I have derived that the action of the superfluid is $$ S_{\text{eff}} = \int d \tau d^d r \left ( \frac{(\partial_{\tau} \theta)^2}{2g} + \frac{1}{2m} \rho_0 (\nabla \theta)^2 \right ) $$ And we ...
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Why does the transmission oscillate when using a numerical dispersion?

I understand that the transmission probability oscillates when the electron is considered to be a plane wave - this is because of the interference effects of these waves. However, when a different ...
star123's user avatar
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How did Cohen-Tannoudji (QM) manipulated this? [closed]

In the first chapter of Cohen-Tannoudji’s Quantum Mechanics Book, there's this part of the chapter called "Time evolution of a free wave-packet". So one starts with the wave packet given by $...
Gean Araujo's user avatar
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Dispersion Relation for one dimensional monoatomic lattice in Kronig-Penny model and in Tight-Binding Approximation model

In the tight binding approximation model, we have the dispersion relation for a one-dimensional atomic lattice given as: $$E(k) = E_0 - \alpha - 2\beta \cos(ka)$$ Here, $\alpha$ and $\beta$ are ...
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What are the exact conditions for the constancy of the speed of a wave?

Do two mechanical transverse waves traveling in the same medium have the same speed whatever the source might be? If the answer to this question is yes, can I generalize and say "all waves of the ...
Jack's user avatar
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How to obtain the high frequency dispersion relation for transverse waves in the Drude model?

Context I am studying electrodynamics using Zangwill [1]. I have a narrow question regarding a proposition found there in. In the section on the Drude model of conducting matter that applies to ...
Michael Levy's user avatar
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Why does the green light change to red in the prism? [duplicate]

I know that white light, upon entering another medium from air/ vacuum, disperses into its constituent colours. Essentially when travelling in the air, all of the constituent colours have the same ...
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Does refraction depend on the colour of light?

Consider passing red and yellow lights separately through the same interfaces and with the same angles of incidence. Light of which colour will be refracted more? This question was given to me. All I ...
Sahaj's user avatar
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Discontinuities in the $u$ channel

if we consider a 2-to-2 scattering, we have normally an $s$ channel a $t$ channel and $u$ channel. In CMS frame $s$ is positive and $t$ and $u$ negative, by crossing symmetry there are kinematics ...
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A Rainbow Paradox

I was studying the phenomena of the formation of a rainbow. In my book, the following diagram is given: So, the rays at the red end of the spectrum make a larger angle with the incident ray than the ...
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How the $-2t\cos(k)$ term appears in the dispersion of the $1D$ tight binding model?

I am trying to derive the tight binding dispersion relation with periodic boundary condition with $N$ lattice sites of the simplest Hamiltonian: \begin{equation} \label{ham} \tag{1} H = -t\sum\...
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Complex values for the dispersion relation obtained through an $s$-band only tight binding model for diamond cubic crystal

Any given atom in a diamond cubic lattice (Like Si or Ge) has four nearest neighbours at at a distance $\sqrt{3}a/4$, being $a$ the lattice constant. The translation vectors to these neighbours can be ...
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Why do fluorescence microscopes use long-pass dichroics

Superresolution fluorescence microscopy is very sensitive to chromatic aberration. Dichroic mirrors, while being biplanar, are not normal to the optical axis and so will cause some dispersion in one ...
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