Hamiltonian Definition and 833 Threads

Hello! I have recently bought the book The Principle of Relativity by Einstein (Along with Minkowski, Lorentz and Weyl). This book is simply a collection of papers published by Einstein (along with the other three scientists mentioned) concerning the development of Special and General...

Homework Statement Show that for the one-dimensional linear harmonic oscillator the Hamiltonian is: [; H = \frac{1}{2}[P^2+\omega ^2 X^2]-\frac{1}{2}\omega \hbar ;] [; =\frac{1}{2}[P+i\omega X][P-i\omega X]+\frac{1}{2} \omega \hbar ;] where P, X are the momentum and position operators...

Hi. I am working through a QFT book and it gives the relativistic Lagrangian for a free particle as L = -mc2/γ. This doesn't seem consistent with the classical equation L = T - V as it gives a negative kinetic energy ? If L = T - V doesn't apply relativistically then why does the Hamiltonian H =...

Hi all, This is the problem I want to share with you. We have the hamiltonian H=aP+bm, which we are commuting with the position x and take: [x,H]=ia, (ħ=1) Ok. Now if we take, instead of x, the operator X=Π+ x Π+ +Π-xΠ- where Π± projects on states of positive or negative energy the...

Homework Statement Given H=\frac{1}{2m}\left[ \vec{P}-q\vec{A}\right] ^{2}+qU+\frac{q\hbar }{2m}\vec{\sigma}.\vec{B} ..(1) show that it can be written in this form; H=\frac{1}{2m}\left\{ \vec{\sigma}.\left[ \vec{P}-q\vec{A}\right] \right\}^{2}+qU ...(2) Homework Equations [/B] In my...

This is what I have so far: I'm trying to show that the matrix D has to be unitary. It is the matrix that transforms the wavefunction.

I am reading Frohlic's paper on electron-phonon interaction. Frohlic.http://rspa.royalsocietypublishing.org/content/royprsa/215/1122/291.full.pdf Here author has introduced the quantization for complex B field in this paper and claimed to have arrived at the diagonalized form of the...

Homework Statement I am asked to find the Hamiltonian of a system with the following Lagrangian: ##L=\frac{m}{2}[l^2\dot\theta^2+\dot{\tilde{y}}^2+2l\dot{\tilde{y}}\dot{\theta}\sin{\theta}]-mg[\tilde{y}-l\cos{\theta}]## Homework Equations ##H = \dot{q_i}\frac{\partial L}{\partial...

Homework Statement Given a non-hermitian hamiltonian with V = (Re)V -i(Im)V. By deriving the conservation of probability, it can be shown that the total probability of finding a system/particle decreases exponentially as e(-2*ImV*t)/ħ Homework Equations Schrodinger Eqn, conservation of...

Using the Feshbach-Villars transformation, its possible to write the KG equation as two coupled equations in terms of two fields as below: ## i\partial_t \phi_1=-\frac{1}{2m} \nabla^2(\phi_1+\phi_2)+m\phi_1## ## i\partial_t \phi_2=\frac{1}{2m} \nabla^2(\phi_1+\phi_2)-m\phi_2## Then we can...

Hello. I'm studying quantization of electromagnetic field (to see photon!) and on the way to reach harmonic oscillator Hamiltonian as a final stage, sudden transition that the Fourier components of vector potential A become quantum operators is observed. (See...

Hi guys, I consider the qm-derivation of the electronic states of hydrogen. There are two different derivations (I consider only the coulomb-force): 1) the proton is very heavy, so one can neglect the movement 2) the proton moves a little bit, so one uses the relative mass ##\mu## The...

Hello everyone! I'm trying to follow a solution to a problem from the book "Problems and Solutions on Quantum Mechanics", it's problem 1017. There's a step where they go on too fast, and I can't follow. I've posted the solution and where my problem is down below. Homework Statement The dynamics...

Hello everyone, I m currently working on a problem that is freaking me out a bit, suppose I have a second quantized hamiltonian: \begin{eqnarray} H=H_{0}+ \epsilon d^{\dagger}d + V(d{\dagger}c_{0} + h.c) \end{eqnarray} In terms of some new operators, I would like to rotate the hamiltonian, so...

I came across a technique called "multiple-scale analysis" https://en.wikipedia.org/wiki/Multiple-scale_analysis where the equation of motion involves a small parameter and it is possible to obtain an approximate solution in the time scale of $$\epsilon t$$. I am wondering if it is possible to...

hi, i have studied the annihilation and creation operators and number operator N in relation with the simple harmonic oscillator that is governed by: H = hw(N+1/2) i don't understand the relation between the harmonic oscillator and for example, this hamiltonian: H = hw1a+a+hw2a+a+aa that i...

Homework Statement Consider the quantum mechanical Hamiltonian ##H##. Using the commutation relations of the fields and conjugate momenta , show that if ##F## is a polynomial of the fields##\Phi## and ##\Pi## then ##[H,F]-i \partial_0 F## Homework Equations For KG we have: ##H=\frac{1}{2} \int...

Hi, I am learning classical mechanics right now, Particularly Noether's theorem. What I understood was that those kinds of transformations under which the the Hamiltonian framework remains unchanged, were the key to finding constants of motion. But here are my Questions: 1. What is...

As far as I know, the Hamiltonian of graphene in the Bloch's sums |A\rangle and |B\rangle near the points K or K' is a 2 \times 2 matrix with the components: \langle A|H|A\rangle, \langle A|H|B\rangle, \langle B|H|A\rangle,\langle B|H|B\rangle which all are parameters (and not variables). But in...

Homework Statement How can I solve this hamiltonian equation? H= p2/2m +α/ħ (σxpy - σypx) + γ/ħ (σxpx - σypy) + βxσx + βyσy Homework Equations Rashba coupling effect equation: HR=α(σykx - σxky) The Attempt at a Solution H0=ħ2k2/2m + HR+ HD

Homework Statement H=\sum^N_{i=1}(\frac{p_i^2}{2m}+\frac{1}{2}(x_{i+1}-x_i)^2+(1-\cos(2\pi x_i)) Homework Equations Hamilton equation of motion I suppose ##\dot{q}=\frac{\partial H}{\partial p}## ##\dot{p}=-\frac{\partial H}{\partial q}##[/B]The Attempt at a Solution If particles are...

is there a generally accepted candidate Hamiltonian for LQG? i've seen marcus post these papers recently http://arxiv.org/abs/1507.00986 New Hamiltonian constraint operator for loop quantum gravity Jinsong Yang, Yongge Ma (Submitted on 3 Jul 2015) A new symmetric Hamiltonian constraint...

V Chernyak, Wei Min Zhang, S Mukamel, J Chem Phys Vol. 109, 9587 (can download here http://mukamel.ps.uci.edu/publications/pdfs/347.pdf ) Eq.(2.2), Eq. (B1) Eq.(B4)-(B6). When I substitue Eq.(B4)-(B6) into Eq.(2.2), I can not recover Eq.(B1). Who can give me a reference or hint on how to write...

Pretty straightforward question. The Einstein-Hilbert Action says that the Lagrangian for Gravity is ##L=R(-g)^{1/2}## where ##g## is the determinant of the Metric Tensor and ##R## is the Ricci Scalar (Actually I am not sure if the determinant of the metric should be included there). From this...

I am trying to teach myself DFT (yet again) from books and my maths is only improving at a modest pace to understand how people calculate using QM. So a very basic question now. When a Hamiltonian for a many body system is written as given in page 8 on this presentation...

According to <x|H|x\prime>=(-\hbar ^2 /2m \frac{\partial^2 }{\partial x^2}+v(x)) \delta (x-x\prime) can one draw the conclusion that the Hamiltonian is always diagonal in the position basis?

Homework Statement A particle of mass m moves in a "central potential" , V(r), where r denotes the radial displacement of the particle from a fixed origin. From Hamilton´s equations, obtain a "one-dimensional" equation for {\dot p_r}, in the form {{\dot p}_r} = - \frac{\partial }{{\partial...

All I know is that Lagrangian is kinetic energy- potential energy and Hamiltonian is kinetic energy + Potential energy. Why do we calculate the lagrangian or hamiltonian?

Homework Statement Derive the Hamiltonian equation in terms of momentum and position ( p and r) for the given system whose lagrangian is stated as L=ř^2/(2w) - wr^2/2 Homework Equations L=ř^2/(2w) - wr^2/2 and H=př-L The Attempt at a Solution Notice here ř means first derivative of r. As i...

Why is Hamiltonian defined as 1st derivative with respect to time ? From the units of energy (kgm2s-2) I would expect it to be defined as 2nd derivative with respect to time. (I'm reading http://feynmanlectures.caltech.edu/III_11.html#Ch11-S2)

In weak field regime i know that it is possible to quantize the gravitational field obtaining a quantum theory of free particles, called gravitons, which is very similar to the one for the electtromagnetic field. Do you know some book in wiich i can study this theory? In anycase what is the...

Homework Statement The e-states of H^0 are phi_1 = (1, 0, 0) , phi_2 = (0,1,0), phi_3 = (0,0,1) *all columns with e-values E_1, E_2 and E_3 respectively. Each are subject to the perturbation H' = beta (0 1 0 1 0 1 0 1 0) where beta is a positive constant...

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...

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>...