1
vote
0answers
83 views

Energy of an Electron in a One Dimensional Periodic Potential

First, we consider the time independent Schrodinger equation of the form: $$\bigg(-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+u(x) \bigg)\phi_A(x)=E_A\phi_A(x)$$ Where $u(x)$ is a potential created by a ...
2
votes
1answer
92 views

How to calculate energy in two-band Hubbard model

It might be a very easy question for you, but I am confused and I need helps. In the simplest Hubbard model at one-dimensional lattice, I ignore the $U$ term and only remain the hopping term. ...
0
votes
1answer
81 views

Two particles state of a 1D massive scalar field

Perfectly localized states are not normalized so do not belong to the Fock space (they belong to the rigged version). Suppose we approximate localized states with gaussians, what is the mathematical ...
2
votes
1answer
154 views

Derive non-linear $\sigma$ model from a theory of SU(2) matirx

It's said in Chapter VI.4 of A. Zee's book Quantum Field Theory in a Nutshell, a theory defined as $L(U(x))=\frac{f^2}{4}Tr(\partial_{\mu}U^{\dagger}\cdot\partial^{\mu}U)$, can be write in the form of ...
2
votes
2answers
413 views

Probability of Different States - Canonical Ensemble - Partition Function

Consider a canonical ensemble of $N$ ideal gas atoms, which could have spin up or spin down. Why is it that the probability of finding the particle in a spin up state generally only involves the ...
1
vote
0answers
162 views

Average number of spin up particles

In a paramagnetic system, where $N = N_\uparrow + N_\downarrow$ is fixed, how does one calculate the average number of spin-up particles $\langle N_\uparrow \rangle$? You can assume we have the ...
2
votes
1answer
726 views

Partition Function for Two Level System

I have a system with $N_s$ sites and $N$ particles, such that $N_s >> N >> 1$. If a site has no particle, then there is zero energy associated with that site. The $N$ particles occupy the ...
1
vote
2answers
155 views

Paramagnetism and large N

In a paramagnetic system, we have: $$N = N_\uparrow + N_\downarrow$$. If we have a large system, with $N >> 1$, is it generally okay to assume $N_\uparrow \approx \frac{N}{2}$ and ...
1
vote
3answers
232 views

Distinguishable, Indistinguishable Paramagnetic Ideal Gas

In the canonical ensemble, the partition function for an ideal gas is given by: $$\frac{Z}{N!}$$ The factor $N!$ accounts for the indistinguishability of the particles of the ideal gas. What ...
4
votes
1answer
483 views

Paramagnetism Spin-1/2 Particles - Partition Function

I'm trying to come up with an expression for the partition function of a system of spin-1/2 ideal gas particles on a line of length $L$. The total number of particles $N$ is fixed, with $N = ...
1
vote
2answers
290 views

Anticommutatorrelation in Bogoliubov-de Gennes Hamiltonian

I almost solved the problem Equivalence of Bogoliubov-de Gennes Hamiltonian for nanowire. In the next steps I used the notation by arXiv:0707.1692: $$ \Psi^{\dagger} = ...
2
votes
1answer
107 views

Is this 2D structure triclinic?

The only rotation axis obvious to me is rotation by 360 degrees, the identity. Vertical mirror planes I've been dicing and cutting it through several planes and I still see none. Yet, the structure ...
2
votes
1answer
1k views

Specific heat capacity of a 2D free electron gas

I have got so far the 2D density of states as $g(\epsilon)=\frac{Am}{\pi\hbar^2}$ where $A$ is the area of the "square" and $m$ is the the electron mass. Then I have found an expression for the the ...
0
votes
1answer
1k views

Calculation of Number Density

Number density equation is given by $ n= \dfrac{(N_A)\rho}{M} $ where $ N_A =6.023\times10^{23} mol^{-1} $ $ \rho=8.02\ g/cm^3 $(at 1500 degree celsius.) $M=63.546*1.6605\times10^{-24} g$ Whats ...
7
votes
2answers
323 views

Why does the quantum Heisenberg model become the classical one when $S\to\infty$?

The Hamiltonian of the spin $S$ quantum Heisenberg model is $$H = J\sum_{<i,j>}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$$ I have read that when the spin quantum number $S\to\infty$, quantum fluctuation ...
1
vote
2answers
311 views

Momentum change in collisions (Drude Model)

A particle suffers elastic collisions with scattering centers with a probability of collision per unit time $\lambda$. After a collision the particle is in a direction caracterized by a solid angle ...
1
vote
2answers
200 views

Eigenfunctions in periodic potential

For Hamiltonian $\operatorname H$ and lattice translation operator $\operatorname T$, if $$\operatorname H\psi=E\psi, \qquad \operatorname T\psi=e^{ik\cdot R}\psi,$$ and $$\operatorname ...
2
votes
1answer
218 views

Inhomogeneous Effective Mass in a 2D Lattice

Consider a tight-binding square lattice in 2D. This lattice has two different nearest neighbor tunneling rates along the x and y directions; call them $J_{x}$ and $J_{y}$. All longer range tunneling ...
0
votes
1answer
108 views

1D Acoustical Relations beyond nearest neighbor couplings

Consider some 1D Lattice of atoms with nth neighbor coupling of strength k_{n}. I'm looking for the dispersion relation for acoustical phonons under these conditions. I start with the Lagrangian, ...
3
votes
0answers
471 views

Goldstone modes and Heisenberg model

The ideia is to show that, because of Goldstone modes, 2d systems are quite different from 3d ones. So, considering the Heisenberg model, I'll post here what I'm asked to and my current thoughts on ...
5
votes
1answer
968 views

Tight Binding Model in Graphene

I'm following a calculation done by a guy who's done it a bit different than what I've done before (used nearest neighbour vectors and a DFT instead of what I will show below), I'm not quite sure how ...
2
votes
1answer
217 views

True Ground State Population of Ideal Bose-Einstein Condensate at Critical Temperature

I'm supposed to demonstrate that although we make the assumption in an ideal BEC that the ground state population follows $N_0 = N\left[1-\left(\frac{T}{T_c}\right)^{3/2}\right]$ in reality the true ...
1
vote
0answers
285 views

Electron Fermi gas

My question is about 2-dimensional Fermi gas of electrons. What is magnetic susceptibility when $T<<T_F$ (where $T_F$ is Fermi Temperature) and, What is the ratio between Pauli and Curie ...