Hamiltonian Definition and 833 Threads

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##...

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...

Homework Statement calculate the Hamiltonian chain I for Te122-128[/B] hello I have to calculate The Hamiltonian and the parameters for U(5) for the Te122-128 this is the equation which i have to use but this is the first time i use such an equation. i searched in google but i don't know how...

Hi guys, I was wondering if anyone knows of a good source/paper that I can find online that details how to derive the MSW Hamiltonian from a field-theoretic approach, or from an effective field theory approach or anything like that? Every time I study the MSW effect, the Hamiltonian...

I'm having some trouble grasping the meaning of the exchange term in the Hamiltonian Heisenberg gives in his classic 1932 paper (the one typically given as the first to describe nucleons via a spin-like degree of freedom; NOTE: I realize this isn't the same as what is today called isospin, but...

Homework Statement H = \frac{2e^2}{\hbar^2 C} \hat{p^2} - \frac{\hbar}{2e} I_c cos\hat\theta , where [\hat\theta , \hat{p}] = i \hbar How can we write the expression for the Hamiltonian in the basis |\theta> Homework EquationsThe Attempt at a Solution I have already solved most part of...

Could someone please explain Hψ = Eψ? I understand that H = Hamiltonian operator and ψ is a wavefunction, but how is H different from E? I am confused. I am trying to understand "Hψ = Eψ" approach

Homework Statement Prove the relationship $$\left(\frac{\partial H}{\partial\lambda}\right)_{nn} = \frac{\partial E_{nn}}{\partial\lambda},$$ where ##\lambda## is a parameter in the Hamiltonian. Using this relationship, show that the average force exerted by a particle in an infinitely deep...

If the hamiltonian is defined as ##\mathcal{H}\equiv \sum _{ i }^{ all }{ p_{ i }\dot { q } _{ i } } - \mathcal{L} ##, how is it considered a function of ##p## and ##q## instead of ##p## and ##\dot{q}##? Chris

So, I should prove that: - \frac{\partial H}{\partial q_i} = \dot{p_i} And it is shown that: - \frac{\partial H}{\partial q_i} = - p_j \frac{\partial \dot{q_j}}{\partial q_i} + \frac{\partial \dot{q_j}}{\partial q_i} \frac{\partial L}{\partial \dot{q_j}} + \frac{\partial L}{\partial q_i} =...

I've read a couple of places that a hamiltonian can be a tool used in classical mechanics and that it's eigenvalues are useful pieces of information. I've tried finding info on the subject matter, as I want to see something that actually requires linear algebra, or at least makes good use of it...

In the lagrangian formalism, we treat the position ##q## and the velocity ##\dot q## as dependent variables and talk about configuration space, which is just the space of positions. In the hamiltonian formalism we talk about canonical positions and momenta, and we consider them independent. Is...

Homework Statement Prove that for any stationary state the average of the commutator of any operator with the Hamiltonian is zero: \langle\left[\hat{A},\hat{H}\right]\rangle = 0. Substitute for \hat{A} the (virial) operator:\hat{A} = \frac{1}{2}\sum\limits_i\left(\hat{p}_ix_i...

So, I was reading about the exchange interaction, and trying to work out what it referred to, and came across something strange in the treatment of the hydrogen molecule (I think it was on wikipedia): The hamiltonian given for the system included a term e2/(4πε0 * Rab) for the repulsion between...

Homework Statement A magnetic field pointing in ##\hat{x}##. The Hamiltonian for this is: ##H= \frac{eB}{mc}\begin{pmatrix} 0 & \frac{1}{2}\\ \frac{1}{2} & 0 \end{pmatrix}## where the columns and rows represent ##{|u_z\rangle, |d_z\rangle}##. (a) Write this out in Dirac...

According to my book, and wiki http://en.wikipedia.org/wiki/Hamiltonian_mechanics#As_a_reformulation_of_Lagrangian_mechanics, ##\frac{\partial{H}}{\partial{t}} = - \frac{\partial{L}}{\partial{t}}##, where ##L## is the Lagrangian. But how can this be? This assumes the generalized...

Is there any difference between Hamiltonian operator and E? Or do we describe H as an operation that is performed over (psi) to give us E as a function of (psi)??

Homework Statement So I just learned how to derive the equation of motion under the Lagrangian formulation which involves finding the euler-lagrange equation when setting the change in action to zero, chain rule, integration by parts etc.. Then I learned how to find the equations of motion...

Hello (and sorry for this stupid question), Could someone tell me the name of this Hamiltonian H = \left(\dfrac{p^2+q^2}{2}\right)^2 Thanks in advance

[SIZE="4"]Definition/Summary The Hamiltonian, like the Lagrangian, is a function that summarizes equations of motion. It has the additional interpretation of giving the total energy of a system. Though originally stated for classical mechanics, it is also an important part of quantum...

I've got a problem which is asking for the eigenvalues and eigenstates of the Hamiltonian H_0=-B_0(a_1 \sigma_z^{(1)}+a_2 \sigma_z^{(2)}) for a system consisting of two spin half particles in the magnetic field \vec{B}=B_0 \hat z . But I think the problem is wrong and no eigenstate and...

This is related to classical Hamiltonian mechanics. There is something wrong in the following argument but I cannot pinpoint where exactly the pitfall is: Consider an arbitrary (smooth) Hamiltonian (let us assume conservative) and 2n phase space coordinates (q,p). The Hamiltonian flow gives...

Hello, The question says that we can write the Hamiltonian of the harmonic oscilator like this: H=0.5*[P^2/m + (4*h^2*x^2)/(m*σ^4)] where h is h-bar I need to calculate the expectation value of energy of the oscilator with the next function: ψ(x)=A*exp{-[(x-bi)^2]/σ^2}. I tried to the...

I am working through Leonard Susskinds 'the theoretical minimum' and one of the exercises is to show that H=ω/2(p^2+q^2). The given equations are H=1/2mq(dot)^2 + k/2q^2, mq(dot)=p and ω^2=k/m. q is a generalisation of the space variable x, and (dot) is the time derivative if this helps...