Hamiltonian Definition and 901 Threads

I have constructed GNR(graphene nanoribbon Hamiltonian) which is of 18 by 18 matrix,i want to add magnetic field term how i can do that ,since earlier B was taken to be zero. Thanks

Can someone help me understand the Hamiltonian on the attached picture. What does the notation with the annihilation and creating operators written in a row vector exactly mean? Does it mean I should just take the dot product as written on the picture? Evidently it doesn't since this just gives...

Hello Seniors, I have done BSc in Physics but couldn't take lectures of Classical Mechanics. I am Almost blind in this subject. Since it's a core course in Physics, so i need your help to understand the basics in this course. If anyone of you have any helping material/notes/slides etc which...

hello, how to derive the hamiltonian for a free electron in electromagnetic field mathematically ? for a first step what is the lagrangian for a free electron in the EM field in classical mechanics ? the physics textbook always like to give the results directly.

Say I have a wavefunction that's a superposition of two-particle states: \Psi = \int dk ~f(k) c^{\dagger}_k c^{\dagger}_{-k} | 0 \rangle Here, ##|0\rangle## is the vacuum and ##c^{\dagger}_k c^{\dagger}_{-k} | 0 \rangle## represents a pair of fermions with momenta ##k,-k##. My goal is to solve...

Hi. Say we have found a hamiltonian ##H## for some system. So I know that if ##\frac{\partial H }{\partial t} \neq 0## then obviously the energy of the system is not conserved. But if ##\frac{\partial H }{\partial t} = 0##, is the energy always conserved? Or do we need to find that ##\frac{d H...

Hey guys, So here's the deal. Consider the Lagrangian \mathcal{L}=\bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi where \bar{\psi}=\psi^{\dagger}\gamma^{0} . I need to find the Hamiltonian density from this, using \mathcal{H}=\pi_{i}(\partial_{0}\psi_{i})-\mathcal{L} So I get the following...

I know the definition of the Poisson bracket and how to derive elementary results from it, but I'm struggling to understand intuitively what they are describing physically? For example, the Poisson bracket between position q_{i} and momentum coordinates p_{j} is given by \lbrace...

Please refer to p. 99 and 100 of Rovelli’s Quantum Gravity book (here). I wonder what is the signification of the “naturalness” of the definition of ##\theta_0=p_idq^i##? If I take ##\theta_0'=q^idp_i## inverting the roles of the canonical variables and have the symplectic 2-forms of the...

Given the half harmonic potential: \begin{equation}V=\begin{cases}1/2\omega^2mx^2 & x > 0\\\infty & x < 0\end{cases}\end{equation}What will be the Hamiltonian of the half oscillator?I understand that for x>0 the Hamiltonian will be...

Hello. Do I need to grasp well the Lagrangian and Hamiltonian formulations in classical mechanics to go through quantum mechanics without struggles?

Homework Statement I'm struggling to perform a symplectic reduction and don't really understand the process in general. I have a fairly solid understanding of differential equations but am just starting to explore differential geometry. Hopefully somebody will be able to walk me through this...

I might have learned what I am going to ask during my electrodynamics class long time ago but just that do not remember it now. I always wonder why does an electron moving in space with EM radiation have Hamiltonian of the form ## H = \left( \mathbf{p}-e\mathbf{A}/c \right)^2/2m +e\phi## where...

What would be the effects on the system for different values of the Hamiltonian preferred basis in Decoherence? Would it for example make the electrons higher in orbital or bands? Or what would be the exact effects?

"The hamiltonian runs over the time axis while the lagrangian runs over the trajectory of the moving particle, the t'-axis." What does the above statement means? Isnt hamiltonian just an operator that corresponds to total energy of a system? How is hamiltonian related to lagrangian intuitively...

Hello! (Smile) Longest path We have a graph $G=(V,E)$, lengths $l(e) \in \mathbb{Z}^{+}$ for each $e \in E$, a positive integer $K$ and two nodes $s,t \in V$. The question is if there is a simple path in $G$ from $s$ to $t$ of length at least $K$. Show that the problem Longest Path belongs...

Hi there, I'm reading on the hamiltonian method and it says we can ignore constraints? Is this true, or am I missing something here, so if we have a constraint in the system we do not have to include it in the final calculation for the equation of motion? Hope someone could clear this up, thanks!

Hey, I was hoping someone could clear this up for me. When using this method, how do you get the final equation of motion, that's where I am confused. So I know I start off using Lagrangian (T - U) -> momentum (partial L/ partial q dot) -> Hamiltonian T+U, and then using the hamiltonian...

1. If I know that ##H(q_i,p_i,t)## is a valid Hamiltonian for which the hamilton equations hold. Now we are given that ##Q_j(q_i,p_i)## and ##P_j(q_i,p_i)## are canonical transformations. This means that there is a function ##K(Q_j,P_j)##, the new hamiltonian, for which the Hamilton equations...

We went over this concept quite fast in class and there is one thing that confused me: When transforming from a set of ##q_i## and ##p_i##to ##Q_i## and ##P_i##, if one checks that the transormations are canonical the new Hamiltonian ##K(Q_i, P_i)## obeys exactly the same equations.This has...

Homework Statement I have a hamiltonian: \begin{pmatrix} a &0 \\ 0&d \end{pmatrix} + \begin{pmatrix} 0 &ce^{i w t} \\ ce^{-iwt}&0 \end{pmatrix}=\begin{bmatrix} a & c e^{i w t} \\ c e^{-i w t}&d \\ \end{bmatrix} Where the first hamiltonian can be labeled with states |1> and |2>...

Homework Statement Consider the Hamiltonian: $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$ where $C$ is a constant and the magnetic field is given by $$\vec{B} = (0,B,0) $$ and the spin is $$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$ with$$\hat{S}_{x}...

Hello, In my problem I need to We are advised to create the Cooper pair box Hamiltonian in a matrix form in the charge basis for charge states from 0 to 5. Here is the Hamiltonian we are given H=E_C(n-n_g)^2 \left|n\right\rangle\left\langle...

Hello, I was just watching a youtube video deriving the equation for the Hamiltonian for the harmonic oscillator, and I am also following Griffiths explanation. I just got stuck at a part here, and was wondering if I could get some help understanding the next step (both the video and book...

Hey I have a tight binding Hamiltonian of a BCC lattice which is a 4x4 matrix in k space (the 4 elements correspond to 4 atoms that are in a unit cell) I want to expand it for small k's around the symmetry points P or Gamma or H. I'm looking at a paper by J. L. Ma˜nes, PHYSICAL REVIEW B 85...

Hello, I want to understand how bracket operations in general are related to symmetry and infinitesimal transformations (in hindsight of quantumfieldtheory), so I calculated an example with a particle that is moving on a circle with a generic potential. (I used simple polar coordinates in two...

Given a Hamiltonian ##H##, with a spectrum of eigenvalues ##\lambda##, you can define its zeta function as ##\zeta_H(s) = tr \frac{1}{H^s} = \sum_{\lambda}^{} \frac{1}{\lambda^s}##. Subsequently, the log determinant of ##H## with a spectral parameter ##m^2## acts as a generating function for...

Recommend an easy going introduction to lagrangian and hamiltonian mechanics (for self study)

Hi I am looking at the problem of optical absorption in direct gap semiconductors. It seems like the perturbing Hamiltonian is an oscillating perturbation , ie. an electromagnetic wave. Why can't the problem be treated as the absorption of a single particle , ie. a photon ?

The book calculates the commutator $[H,x_i]$ as $$[H,x_i] = \left[ \sum_j \frac{p_j^2}{2m}, x_i \right] = \frac{2}{2m} \sum_j p_j \frac{\hbar}{i} \delta_{ij} = - \frac{i \hbar p_i}{m},$$ where the hamiltonian operator $H$ is $$H = \sum_j \frac{{\mathbf p}_j^2}{2m_j} + V({\mathbf x}).$$ The book...

I found this in a Phd thesis consider a two level atom interacting with the electromagnetic field. The atom is described by ##H_{at} = \hbar ω_0 J_z## a monomode electric field is described by ##H_{em} = \hbar \omega (a^\dagger a + 1/2)## We have ##E = E_0(a^\dagger + a)## and the dipolar...

We are given the vectors la> = (1,0) and lb> = (0,1) and then a Hamiltonian H which is a 2x2 matrix with 2 on the diagonal entires and zero elsewhere. I am asked to now represent H in the basis of the vectors la'> = 1/sqrt(2)(1,1) and lb'> = 1/sqrt(2)(1,-1), which are also eigenvectors of H...

Homework Statement I am having too many troubles finding the eigenfunctions of a given Hamiltonian. I just never seem to know what exactly to do. My idea here is not for you to help me solve each problem below, but I would like to just set the equations. I know you guys don't like it when...

Homework Statement A quantum system with a ##C^3## state space and a orthonormal base ##\{|1\rangle, |2\rangle, |3\rangle\}## over which the Hamiltonian operator acts as follows: ##H|1\rangle = E_0|1\rangle+A|3\rangle## ##H|2\rangle = E_1|2\rangle## ##H|3\rangle = E_0|3\rangle+A|1\rangle##...

Dear all, this is my first thread in the forum. I am trying to solve the following problem. it was given during a written exam at my university (many years ago) and I really would appreciate if someone will help me to solve it 1. Homework Statement Show that if the hamiltonian of the strong...

In my physical chemistry course, we are learning about the Schrödinger Equation and were introduced to the Hamiltonian Operator recently. We started out with the simple scenario of a particle in 1D space. Our professor's slide showed the following "derivation" to arrive at the expression for the...

I'm a bit puzzled over the structure of the Hamiltonian in the Schrodinger equation (SE). First we take the famous expression, Eψ=Hψ. From what I'm aware of, the classical Hamiltonian is H=Kinetic energy (KE) + Potential energy (PE). However, in the SE, there appears to be a negative sign...

As I understand it, the Hamiltonian is the kinetic plus the potential energy of the wave function. When a measurement is done what happens to the kinetic and potential energy? Does it dissipate? Is it conserved in the measured state? Does it decrease? Does the Hamilton or kinetic+Potential...

Homework Statement Been struggling with a particular problem that keeps coming up in one of my modules, so i thought i'd see if anyone here can enlighten me. A Hamiltonian H0 is represented by the matrix: top row: 3 0 -1 Middle row: 0 a 0 Bottom row: -1 0 3...

Hi everyoneI have been give a matrix operator and asked to find the eigen values, I have done so and then I was given a state |ψ> of some particle. The part I'm struggling with is it then asks for <H>, the expectation value of the matrix operator. It's a 3x3 matrix also. I've tried using the...

Hi all, I'm attempting to prove that i \frac{d \xi (t)}{dt}=[\xi(t),H(p,q ; t)] where the Hamiltonian is explicitly time-dependent, in general. We also have some unitary U(t) which generates time-evolution. I wrote up a quick proof but realized afterward that I had assumed that H and...

for a degnerate system it's in my notes that you can write: H^{(0)}\Psi _{1}=E_{0}\Psi _{1} H^{(0)}\Psi _{2}=E_{0}\Psi _{2} and (not related) we write the general Schrodinger equation H_{0}\Psi + V\Psi = E\Psi Please could someone tell me what both the upper and lower zeros on the H mean...

Given psi as function of x^2, and the potential energy as function of x, find the kinetic energy. My reasoning: KE=P^2/2m and use the momentum operator. My professor's reasoning: Calculate the hamiltonian operator and subtract the potential energy then divide by psi. Note: I talked to my...

Homework Statement Two identical spin-1/2 particles of mass m moving in one dimension have the Hamiltonian $$H=\frac{p_1^2}{2m} + \frac{p_2^2}{2m} + \frac{\lambda}{m}\delta(\mathbf r_1-\mathbf r_2)\mathbf s_1\cdot\mathbf s_2,$$ where (pi, ri, si) are the momentum, position, and spin operators...

Homework Statement Example Question: an electron with mass m is confined in a thin wire, with periodic boundary conditions applied in the x direction and harmonic potentials in the y and z direction. Write an expression for the wave functions in the ground state. Write down all the energy eigen...

Can anyone please explain to me what is the Ising model, Hilbert space, and Hamiltonian ? However, please explain it as simple as possible because I am a freshman. I have looked up all three things. I've tried my best to make some sense of it, but I am, honestly, still confused on what any of...

so I am taking a quantum mechanics course, we started taking about dispersion. so he the lecturer gave us an example about the fission of uranium by alpha ray... he said that we should place a detector in order to detect the alpha particlee , but the detector can only detect particlees with...

Hi guys, I'm having a hard time with that one from Cohen-Tannoudji, ##F_{VI}## # 6. I'm translating from french so sorry if some sentence are weird or doesn't use the right words. 1. Homework Statement We consider a system of angular momentum l = 1; A basis from it sub-space of states is...

Homework Statement charge e is within 2 dimensions in presence of magnetic field. H = 1/2m (p - e/c A)^2 A = 1/2* B x r p and r have two components Show: H in terms of B along z axis resembles 2D HO (with some extra term) express H in terms of x, y, p_x, L_y Homework Equations L = r x...

Hi everyone! I am trying to create the density matrix for a spin-1/2 particle that is in thermal equilibrium at temperature T, and in a constant magnetic field oriented in the x-direction. This is a fairly straightforward process, but I'm getting stuck on one little part. Before starting I...