Questions tagged [density-of-states]

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Density of states of Bogoliubov quasiparticles

For a simple fermionic system the formula for calculating the density of states (DOS) is $N(E) = \sum_{n}\delta(E-E_{n})$ where $\{E_{n}\}$ is the set of eigenvalues obtained after diagonalizing the ...
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1answer
127 views

What does it mean to divide by the degeneracy of the state in this textbook excerpt?

This section of Griffiths Introduction to Quantum Mechanics deals with Boltzmann, Fermi-Dirac, and Bose-Einstein distributions. I don't understand this line (highlighted in yellow): Let's talk only ...
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1answer
1k views

Photon density of states: Polarization/Helicity degree of freedom?

Sakurai's "Advanced Quantum Mechanics" states in Eq. (2.116) that the density of states of a single photon with $\vec k$ vector pointing into the solid angle $d\Omega$ is given by \begin{equation} \...
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1answer
693 views

Density of phonon states from dispersion relation

I have a dispersion relation $$ \omega( \textbf{q} ) = \omega_0 \sqrt{ \sum_{j=1}^{D} \sin^2{\frac{q_ja}{2}}. } $$ Where D is the dimension D=1,2,3. And my excersise is to calculate (numericaly on a ...
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75 views

Passing from a discrete summation to a continous integal

I'm having trouble understanding the math behind a step in an explanation of BCS theory. At one point the superconductor gap $\Delta$ is defined as \begin{equation} 1 = V \sum_q \frac{1}{\xi_q^2+\...
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1answer
224 views

Number of states $\Omega(E)$ derivation (Reif problem 2.4)

Problem 2.4 from Reif's "Fundamentals of Statistical and Thermal Physics": Consider an isolated system consisting of a large number $N$ of virtually non-interacting localized (not translating) ...
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478 views

Calculate density of states of particle in a 3D-harmonic oscillator

Problem: Calculate the density of states of a particle with mass $m$ in a 3D-harmonic oscillator with frequency $\omega$. $$ \rho(E) = \frac{m}{2\pi^2\hbar^3} \int d^3r \sqrt{2m(E-V(\vec{r}))}\Theta(...
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Density of States in 2D Tight Binding Model

Hello I am trying to find the density of states for the dispersion relation: $$E(k_x,k_y) =\cos(k_x a) -\cos(k_y a),$$ over an entire period, not simply around the minimum. For a crystal of length $L,$...
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2answers
1k views

Partition function for continuous energy

The partition function has usually two definitions: the first is for discrete microstates with energies $E_i$. In this case it is defined as $$Z = \sum_{i} e^{-\beta E_i},$$ where $i$ ranges over ...
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Landau quantization: degeneracy of first level

In some books the degeneracy of one Landau level in a two-dimensional gas of free electrons is calculated in the following way: Note: The electron spin is not considered. Number of states of a free ...
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1answer
85 views

Practical way of expressing the $\delta$-function [closed]

I have got a problem in using the $\delta$-function. As we know, this function is often used to define a 'density'-related quantity. Such as the density of states or some correlation function. Take ...
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1answer
121 views

Density of states from $k$ to $E$

Speaking about Quantum mechanics, considering the "particle in a box" condition as an approximation of the electrons condition in a semiconductor, let the material be represented by a volume $V$ with ...
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1answer
451 views

Fermi Dirac distribution and degenerate energy states

In Quantum Mechanics and in semiconductor materials, the number of electrons $N$ in conduction band is usually computed as follows: $$N = \int_{E_c}^{+\infty} g_c(E)f(E)dE$$ where $g_c(E)$ is the ...
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187 views

Density of States for a separable hamiltonian

There are $N$ non interacting electrons in a potential well: \begin{align} H&= -{1 \over 2 } \nabla^2 + U(x,y,z) \\ U(x,y,z)&={1\over2}\omega^2z^2 \; \mbox{for} \; (x,y) \in [0,L]\times [0,L]; ...
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392 views

Rigorous definition of density of states for continuous spectrum

For operators with pure point spectra it is clear how to count number of states corresponding to a given eigenvalue - one can just calculate the dimension of eigenspaces. I am wondering how to do it ...
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175 views

Fermi's golden rule and the DoS of scattering states

Can the Fermi's golden rule $$\Gamma_{fi} ~=~ \rho(E_f) \frac{2\pi}{\hbar} |M_{fi}|^2$$ be applied for transitions of discrete states to scattering states? If yes, then what should the density of ...
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1answer
175 views

What is the difference between these two expressions for the partition function, Z?

What is the difference between these two expressions given for the partition function, Z? $$Z = \sum_{i}e^{-\varepsilon_i/kT}$$ $$Z = \sum_{j} g_je^{-\varepsilon_j/kT}$$ where each energy level has ...
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186 views

Density of states from the Retarded Green's function for a rotating black hole

I have been studying the scattering of a scalar field around a rotating black hole in the near-horizon extremal limit. The radial solution provides the retarded Green's function, just by taking the ...
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272 views

Calculate 2D Effective mass from bulk effective mass

I am trying to create a self consistent Shrodinger Poisson Solver for various semiconductors. There is already one done by Professor Hu from UC Berkeley - QM CV Simulator. Looking at the code, they ...
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1answer
3k views

Density of states for arbitrary dispersion relation

If I have a 3D dispersion relation $E=E(k_x, k_y, k_z)$ I have an equation for the density of states, which is $D(E)=\frac{1}{\nabla_k E}\int\frac{dS}{(2\pi)^3}$ 1) I am confused about the ...
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1answer
4k views

Relation between band structure, dispersion, density of states, and the Fermi energy and Fermi level

Despite the long title, this question is mostly qualitative (although I am interested in quantitative results if possible). Say you have an electronic band structure (energy as a function of "k") for ...
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1answer
653 views

Density of States in NOT Free Electron Gas

I think that I understand how the density of states works for a free electron gas. It is effectively just a conversion factor between summing over values of k and integrating over values of E. If you ...
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1answer
629 views

Density of states of classical harmonic oscillator in phase space

Since all classical harmonic oscillators are ellipses in phase (position-momentum) space, and since the entire phase trajectory of a given system (with a fixed rigidity and mass factor) can be ...
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1answer
750 views

How to prove that sum converges to integral using density of states?

Essentially, I would like to prove $$ \sum_k f(k) \to \int f(k) \rho dE \tag{1}$$ where $$ \rho = \frac{dk}{dE} \tag{2}$$ is the density of states and $k \to \infty$. The model is that there is a ...
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1answer
2k views

How to calcualte density of state from computed phonon dispersion?

I have numerically computed phonon dispersion using small displacement method by solving dynamical matrix along high symmetry direction [100], [110] and [111] for f.c.c lattice. Now I want to ...
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1answer
84 views

How does phonon scattering change the distribution function?

For a one-dimensional structure, we know that the modified distribution function has the following energy dependency in equilibrium: \begin{equation} Z(\varepsilon)\,f(\varepsilon) = \dfrac{N_\text{1D}...
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1answer
120 views

Density of states (DOS) integral when surface is not closed

According to the density of states (DOS) formula $$\rho(\varepsilon)\propto \int_{\varepsilon=\text{const}}\frac{dS}{|\nabla_k \varepsilon_k|}.$$ Since there is an integral on the constant energy ...
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591 views

DOS behavior of Van Hove singularity in a line

When there are some points in momentum space give $|\nabla_k \varepsilon_k|=0$, they are called Van Hove points and give singularity in the desity of states (DOS). But what if $|\nabla_k \...
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1k views

DOS of Van Hove singularity in 2D square lattice tight binding model

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\mathbf{k}=(0,\pi)$ and related momentum points are ...
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83 views

How to get the asymptotic expression of DOS near Van Hove singularity [duplicate]

For the simplest example, 2D square lattice tight binding model gives the energy band as $$\varepsilon_k=-2t(\cos k_x+\cos k_y) \, .$$ We know that $\vec{k}=(0,\pi)$ and related momentum points are ...
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2answers
818 views

Direct definition of density of states

I've been studying statistical mechanics and in the book there's something the author calls density of states which he introduced in a kind of indirect way. Basically, the author argues that if we ...
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0answers
478 views

Autocorrelation function corresponding to density of states with significant rotational motion

Most statistical physics textbooks (at least the ones I've found) state simply that the density of states of a system can be found as the temporal Fourier transform of the velocity autocorrelation ...
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0answers
83 views

Density of states for the diffusion

For the wave equation, the propagator in the Fourier domain is written as $$G(\mathbf{k},\omega)=-\frac{1}{\frac{\omega^2}{c^2}-\mathbf{k}^2+\mathrm i\epsilon}.$$ When $\omega/c$ is close to $\|\...
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1answer
5k views

What is the physical origin of van Hove singularity?

I am trying to build physical intuition about van Hove singularities. The density of states for a system with energy dispersion $E_\mathbf{k}$ is defined as $$ D(E) = \int_{S(E)} \frac{dS}{4\pi^3} \...
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3answers
6k views

Density of states of 3D harmonic oscillator

Consider the following passage, via this image: 5.3.1 Density of states Almost all of the spin-polarized fermionic atoms that have been cooled to ultralow temperatures have been trapped by ...
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1answer
4k views

How to compute the density of state from the Green function?

I'd like to plot the density of state (DOS) for a specific system, say an s-wave BCS superconductor, the Green function of which is $$G\left(p,\omega\right)=\dfrac{\omega+\xi}{\omega^{2}-\xi^{2}-\...
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1answer
786 views

Density of states and fermi energy

I am trying to find the fermi energy and density of states. From the equation: K_F^3=3*(pi)^2*n and n=NAvagadrosdensity/m what is the big N here?
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1answer
926 views

What is the definition of 'relative population' in context of partition function?

In statistical mechanics, what is the definition (or mathematical definition) when authors refer to relative population in the case of a classical particle system?
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1answer
164 views

is it necessarily true that the partition function $Z$ (with degeneracies) $ =1$?

The partition function with degnerate energies is $$\text{Z}=\sum _ig_ie^{{-E_i}/{k_BT}}.$$ Because the partition function Z is defined as the normalisation constant, does Z always = 1?
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1answer
90 views

Summing over quantum states

For a system of $N$ identical particles we deal in quantum mechanics with wave functions $\langle \{\mathbf{r}_i \} \mid \Psi \rangle=\Psi(\mathbf{r}_1,\dots,\mathbf{r}_N)$ from which determine the ...
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1answer
648 views

How to understand Density of States with dispersion relation

I am having trouble understanding the Density of states concept. As I currently understand it, for the density of states $g(k)$ it is the number of microstates with wave number in the range $[k,k+\...
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1answer
569 views

Why is the density of states in $k$-space constant?

Why are the allowed states in $k$-space equidistant in every direction? As a consequence of this, the density of states for phonons in 3D is $$\frac{V}{(2\pi)^3}$$ while for electrons it is $$2 \frac{...
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1answer
341 views

Density of states and anisotropic distribution functions

We consider a $3D$ dynamical system. Its distribution function is given by the function ${ (\mathbf{x},\mathbf{v}) \mapsto f (\mathbf{x},\mathbf{v})}$, so that $$ \mathrm{d}^{3} \mathbf{x} \, \mathrm{...
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812 views

3D Density of states

I have the following dispersion relation: $$\epsilon(\vec{k})=\frac{\hbar}{2}\left(\frac{k_x^2}{m_1}+\frac{k_y^2}{m_2}-\frac{k_z^2}{m_3}\right)$$ (note the minus sign in the third term). And I am ...
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5answers
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Why do the high frequency waves have the most number of modes?

While reading the Wikipedia page of Ultraviolet Catastrophe, I came across how Rayleigh and Jeans applied the equipartition theorem. They told that each mode must have same energy. Now as the number ...
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2answers
343 views

How does the density of states for black-body radiation change with geometry?

If I have a hollow conducting cylinder with another conducting cylinder inside it (as with a coaxial cable), would the density of states of the photons/radiation between the two cylinders be any ...
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1answer
2k views

Density of States vs Dispersion

I have a rather naive question regarding DOS and dispersion. We showed the existence of a band gap in class for a small, periodic perturbation in class last week. When drawing this, the professor ...
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2answers
809 views

What methods exist to calculate the density of states in the continuum of a molecule?

Say I have an arbitrary molecule in the Born-Oppenheimer approximation, and furthermore say that I can approximate the molecule as having only one active electron. What methods exist to calculate the ...
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1answer
757 views

Density of classical states in quantum theory

Let's first treat electrons as classical objects. I can evaluate the classical energy of each state in a configurational space (3N real numbers and, say, spins) using just Coulomb's law. Then I ...