Questions tagged [dispersion]
Dispersion refers to the frequency dependence of the properties of a wave.
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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
$...
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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 ...
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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 ...
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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 ...
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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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My phone sees the NIR from a TV remote. Why does it not see the NIR in rainbows?
My Pixel phone's camera shows the IR light from the LED of a TV remote as a magenta/lavender color. That wavelength is likely around 940 nm.
Now, when I take a picture of a rainbow in the sky, it does ...
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Refraction of light with different wavelengths [closed]
A red light and blue light enter a rectangular glass block normal to its surface at the same time. Strictly speaking, after passing through the block, which pulse exits first?
Should I consider their ...
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Can we use hybridised orbitals in Tight-Binding Method for calculation energy dispersion?
I am working on a model of a Fe square planar complex with nitrogen and oxygen given below is a monomer of that polymer. While constructing the hamiltonian matrix of this monomer I'm confused as to if ...
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Analyticity in the upper half plane and causality
Can you, please, help me to understand the following
How is the analyticity of a complex-valued function in the upper half plane related to causality and Kramers-Kronig relations? Namely, why is it ...
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Confused about the Kelvin wake pattern
What are the $v_{\text{phase}}$ and the $v_{\text{group}}$ in Kelvin wakes? Additionally there are tons of water waves combined to form the kelvin wake. How is it possible for the $v_{\text{group}}$ ...
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Fermi-Energy for different cases
I Have been working on the fermi energy recently and understand where the fermi-energy equation comes from for the non-relativistic case but does the fermi energy change for the relativistic and ultra-...
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Conformal field theory: How can we infer linear dispersion relation?
It is well known that conformal field theories (CFTs) in general have a gapless linear dispersion (up to $E_0$ shift) $E(\vec{k}) = c|\vec{k}|$ for $E\approx 0$ with $c$ a constant velocity.
How can ...
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Can I obtain the dispersion relation boosting the rest frame solution?
I have some complicated relativistic coupled wave equation for a scalar field $\psi$ and a 2-tensor field $h^{\mu\nu}$. I'm not able to find the dispersion relation for a general momentum $p^\mu$, so ...
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Diffusion versus dispersion
What is the difference between diffusion and dispersión in porous media?
In my model mathematical,
Advection-Diffusion-Reaction Equation
Or
Advection-Dispersion-reaction Equation ?
Help please
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Understanding radial velocity dispersion
I have to work with the radial velocity dispersion. Could someone help me to understand what this actually is and how I work with velocity dispersion? What happens if you multiply the radial velocity ...
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Dispersion relation for Rayleigh equation given unbounded vortex sheet
Reference: Drazin and Reid
Consider Rayleigh's equation away from the continuous part of the spectrum, which is governed by $$\phi''-\alpha^2\phi=0.$$ Given the shear flow $U(z)=\text{sgn } z$, one ...
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How to differentiate longitudinal and transverse phonons from the phonon dispersion curve?
I know that there are two kinds of phonons: acoustic and optical. Acoustic phonons have zero frequency at the $\Gamma$ point, whereas optical phonons have a non-zero frequency at the $\Gamma$ point.
I ...
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Why does Debye model predict that, for big wavelenghts, the relationship between the frequency and the wave number of phonons is linear?
I'm studying Debye model for the specific heat capacity of solids.
In class we found first of all that the number of modes per frequency is proportional to the frequency squared.
We found that there ...
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Spatial spread of a relativistic particle [closed]
I am trying to figure out a picture of the wave properties of relativistic particle, e.g., particle produced at collider. It seems that the particle detected at collider have well defined momentum and ...
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I'm struggling with a collisionless plasma textbook problem. Calculating the longitudinal dielectric function from the distribution
The problem is:
Given the distribution of the electrons
$$
f_0(\mathbf{p}) = \frac{1}{\pi^2}\frac{mun_e}{(p^2 + m^2u^2)^2}
$$
find longitudinal dielectric function and the dispersion equation for ...
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Rectangular lattice and periodic energy/$k$ dispersion
Given is a rectangular lattice (a=0.44 nm; b=0.34 nm), Fermi energy of $E_F = 6.4$eV and the energy/k dispersion along $k_a$ with $E(k)=E_0 - E_0 \cos(k\cdot a)$; $E_0 = 4$eV analogous in $k_b$ ...
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Why is the definition of energy density in auxiliary field method consistent with the common one?
In PRL 104, 087401 (2010) (PDF), the authors provide an alternative way to formulate dispersive photonic media.
Here I would like to take lossless dielectrics with Lorentz dispersion $\epsilon\left(\...
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Why varying the frequency of a propagation wave in a non-dispersive media doesn’t change it’s propagation velocity?
Why varying the frequency of a propagation wave in a non-dispersive media doesn’t change it’s propagation velocity?
Before anything, I want to note that I’ve seen already similar questions on this ...
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Dispersion Relation and Eigenvectors of SSH Model in Tight Binding
Consider a one-dimensional chain of atoms as shown in the figure. Let the spacing between the atoms be $a$. Assume that the onsite energy is the same at each point and is equal to
$0$ (without any ...
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Calculating Fourier spectra of particle current autocorrelation functions
The particle current density can be defined as :
$$\textbf j(\textbf r,t)=\sum_{i=1}^{N} \textbf v_i\delta(\textbf r-\textbf r_i(t))$$
Its spatial Fourier transform :
$$\textbf j(\textbf k,t)=\sum_{i=...
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Is the statement that "It has dispersion relation with frequency(or energy)" equal to "It is a function of frequency(or energy)"?
I'm reading an article "RAMAN SPECTROSCOPY OF GRAPHENE AND RELATED MATERIALS" (https://www.physics.purdue.edu/quantum/files/Raman_Spectroscopy_of_Graphene_NOVA_Childres.pdf)
which describes ...
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Is there any physical explanation for negative extinction coefficient, $k(E)$, over a spectral range of energies?
Measured data for $n(E)$ of air were fitted to equations for $n(E)$ and $k(E)$. The measured data for n(E) spanned 0.734 to 6.702 eV. To obtain fits to the equations for n(E) and k(E) in the absence ...
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How should I plot a tight-binding dispersion when I use multiple gridpoints in real space?
Consider a simple tightbinding calculation with spacing $a$. I can write down a dispersion relation $$E(k) = 2t - te^{ika} - te^{-ika}$$
Say I am solving a simple tightbinding problem numerically, on ...
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Why Klein-Gordon and Dirac's waves have a phase velocity of $c$?
From De Broglie’s relations and the energy-momentum dispersion relation one finds:
$$v_p=\frac{w}{k}=\frac{E}{p}=\frac{\gamma mc^2}{\gamma mv}=\frac{c^2}{v} \tag{1}$$
Where, $v_p$ is the phase speed ...
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Optical and Acoustic Phonons: Choice of Unit Cell
According to my book, phonon dispersion relation for three dimensions, the number of acoustic phonons is 3 per unit cell while the number of optical phonons is 3(M-1) where M is the number of atoms in ...
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Wave propagation speed in non-linear differential equations
Could it happen than a solitary travelling wave (soliton) had a different propagation speed when seen from the usual wave equations from that in a non-linear equation. I mean, suppose a solution $F=f(...
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Deriving a dispersion relation in Hall MHD for inhomogeneous plasmas
I'm trying to derive the dispersion relation of a drift wave in Hall MHD. The derivation is done in this paper (https://aip.scitation.org/doi/10.1063/1.859752)
The governing equations are something ...
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About the graphene's Dirac points
I am studying Hasan & Kane's colloquim about topological insulators, and a question has emerged when I reahced the 4th page of this thesis. 1
Passages below the equation (3) show that near the ...
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Birefringince in 2D array of masses and springs
Assume a 2D square array of masses with mass $m$ connected by springs with constant $\kappa$. Then the equation of motion for small perturbations in the plane of the array to the mass at $(l,n)$ is
$$
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What is the $E$-$k$ relation of electron in valence band?
What is the $E$-$k$ relation of electron in valence band? $E$-$k$ relation for electron in conduction band is known but how to calculate energy of electron in valence band?
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What is meant by a dispersion relation when complex integration in the Cutkosky rules' derivation or the computation of form factors $F(q^2)$ is used?
In the context of calculations of loop diagrams with Cutkosky-rules very often the concept of dispersion relation is mentioned. For instance this technique is used in Vol. 4 of Landau-Lifshitz for the ...
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Why does light not disperse in air?
I was studying the prism and how and why it disperses light. My teacher told me that in a prism different wavelengths of light have different refractive indexes because the refractive index depends ...
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$E$-$k$ diagram under an applied Electric Field
We know from solid-state physics, if an electric field is applied, the band bends in the band diagram. How does this change the $E$-$k$ diagram?
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What happens to the optical branch solution in the limit of same masses in a diatomic chain?
For a monoatomic chain, we get the following dispersion relation:
$\omega(k) = \sqrt{\frac{4 \kappa}{m}\sin^2{\left(\frac{ka}{2}\right)}}$,
where $m$ is the atom mass, $\kappa$ is the spring constant, ...
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Why does Cauchy's equation for refractive index contain only even power terms?
Cauchy's equation is an empirical relation that defines the refractive index of a material in terms of wavelength:
$$ n(\lambda) = A + \frac{B}{\lambda^2} + \frac{C}{\lambda^4} + ...$$
While I ...
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One-dimensional diatomic chain phases
In my solid state book, when deriving the dispersion relation of a diatomic one-dimensional chain, the author makes the following Ansatz for the oscillation of the first (x) and seond (y) atom in the ...
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How to interpret dispersion curves?
I have trouble understanding the dispersion curves, especially the x-axis. As i understand it, the x-axis is the wave-vector k. Which would mean that waves of different wavelengths, move at different ...
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What causes phase walk-off reversal in inverted domains in Quasi-Phase-Matching?
Lets take for example a Periodically Poled Lithium Niobate crystal (PPLN) used in a nonlinear process such as Second Harmonic Generation (SHG). For this crystal, the purpose of periodic poling is to ...
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Why does high frequency have high energy?
The electromagnetic spectrum's wavelengths all travel at the same speed, $c$. Also, the wavelength $\lambda$ and frequency $\nu$ are related by $c = \lambda \cdot \nu$. Since all moving particles here ...
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