Hamiltonian Definition and 901 Threads

Homework Statement Let T = \frac{1}{2}T_{ij}\dot{q_{i}}\dot{q_{j}} and V = \frac{1}{2}V_{ij}q_{i}q_{j}. Verify that the equation of motion T_{ij}\ddot{q_{j}} + V_{ij}q_{j} = 0 imply that the energy T + V is conserved. Can the constancy of T+V be used to deduce the equations of motion...

Homework Statement I need to find the 2x2 Hamiltonian matrix for the Hamiltonian, which is written in second-quantized form as below for a system consisting of the electrons and photons. H = h/ωb†b + E1a†1a1 + E2a†2a2 + Ca†1a2b† + Ca†2a1b, a's are creation and annihilation operator for...

In his lectures on Quantum Physics, Richard Feynman derives the Hamiltonian matrix as an instantaneous amplitude transition matrix for the operator that does nothing except wait a little while for time to pass.(Chapter 8 book3) The instantaneous rate of change of the amplitude that the wave...

Homework Statement The Hamiltonian of a system has the matrix representation H=Vo*(1-e , 0 , 0 0 , 1 , e 0 , e , 2) Write down the eigenvalues and eigenvectors of the unperturbed Hamiltonian (e=0) Homework Equations when unperturbed the Hamiltonian will...

In my first Physics class (in high school by the way, a huge shame that i had so little before college), the first thing we talked about was the physics of elastically colliding bodies that have no interaction between them at all. However, I've only ever analyzed such systems with force...

Could anyone please help a lowly 2nd year undergrad understand what the hamiltonian function of action means! W = \int_{t_0}^t \mathcal{L}\,dt Apparently Schrodinger used it along with the Hamilton-Jacobi equation to derive the Schrodinger equation so it's a pretty important part of...

The Hamiltonian for a Hydrogen atom in Cartesian Coordinates (is this right?): \hat{H} = - \frac{\bar{h}^2}{2m_p}\nabla ^2_p - \frac{\bar{h}^2}{2m_e}\nabla ^2_e - \frac{e^2}{4\pi\epsilon _0r} In Spherical Coordinates do I just use: x=r sin θ cos φ, y = r sin θ sin φ, and z = r cos θ?

http://books.google.ca/books?id=Pd8-s6rOt_cC&pg=PA70&lpg=PA70&dq=%22show+that+in+a+hamiltonian+system+it+is+impossible+to+have+asymptotically%22&source=bl&ots=uJplwJKBPB&sig=4HBlny5uGifky2zfdNtWxDi-Uuo&hl=en&ei=yk6tS47hJJTssQOPvJmEDA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAkQ6AEwAA...

Hi, I was looking for a book that would explain classical field theory in a Hamiltonian setting. What I mean by this is that there be no *actions* around, no *Lagrangians* and *Legendre transforms* to define the Hamiltonian and so on. What I'm looking fo is an exposition of (classical) field...

Homework Statement A system with only one degree of freedom is described by the following Hamiltonian: H = \frac{p^2}{2A} + Bqpe^{-\alpha t} + \frac{AB}{2}q^2 e^{-\alpha t}(\alpha + Be^{-\alpha t}) + \frac{kq^2}{2} with A, B, alpha and k constants. a) Find a Lagrangian...

Homework Statement Show that the time-independent Schrödinger equation becomes -\frac{h^{2}}{2(m_{1}+m_{2})}\nabla^{2}_{R}\psi-\frac{h^{2}}{2\mu}\nabla^{2}_{r}\psi+V(r)\psi = E\psi Homework Equations -\frac{h^{2}}{2m_{1}}\nabla^{2}_{1}\psi-\frac{h^{2}}{2m_{2}}\nabla^{2}_{2}\psi+V(r)\psi =...

hey guys, this may be a little naive but, I can someone explain to me the physical aspect of the Hamiltonian? In the sense that if had to physically interpret its function, could I do it and if so how? Thanks

Hi there, My objective is to study Hamiltonian systems, integrable and non integrable systems, where there will be chaos, etc. I have a general idea of everything.. the destroyed tori, the symplectic structure of hamilton's equations, etc. But nothing is very clear to me! And the most...

Hello, I need help with a couple of questions. (The answers haven't come up properly and are too cryptic and I'm having some difficulty). I'm trying really hard to learn this, so could you explain it as fully and clearly as possible - that would be of great help. (I find there are some...

Homework Statement The hamiltonian of a spin in a magnetic field is given by: \hat{H} = \alpha\left( B_{x}\hat{S_{x}} + B_{y}\hat{S_{y}} + B_{z}\hat{S_{z}}\right) where \alpha and the three components of B all are constants. Question: Compute the energies and eigenstates of the...

Hi there, I'm working on getting a presentation together for a graduate course I'm taking and chose to give a brief introduction on spin polarizabilities. In the case of the nucleon, these 4 intrinsic quantities manifest themselves in a 3rd-order expansion of the Compton Scattering...

Hi guys Say I have a Hamiltonian given by H = \sum\limits_{i,j} {a_i^\dag H_{ij} a_j^{} } I wish to perform a transformation given by \gamma _i = \sum\limits_j {S_{ij} a_j }. Now, what my teacher did was to make the substituion \gamma_i \rightarrow a_i and a_i \rightarrow \gamma_i, so...

What's a good definition (or a practical definition) of this? It's actually with regards to electrodynamics but I just want to know in terms of a general system.

Homework Statement Write out: H_{SE}(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle) and exp(-iH_{SE}t)(\left|\right\beta,i_{\beta}\rangle\otimes\left|\right e_{j}\rangle) Where...

Hi guys I have the Hamiltonian, which describes my lattice of NxN metal atoms, and their mutual coupling. What I need is the density of states of this lattice, and I am quite sure that there is a way to find it from my Hamiltonian; I just need to find out how. What I thought was that I can...

The hermiticity of Hamiltonian comes up as a result of requiring real energy eigenvalues and well-defined inner-product for correlation amplitudes. In the corresponding Lagrangian picture (path-integral), I am not clear about the explicit restriction that the above hermiticity of Hamiltonian...

Homework Statement Question 1 on the following page: http://www.maths.tcd.ie/~frolovs/Mechanics/PS10.pdf It's the second part I'm stuck on ('Explain why the equations of motion do not...') Homework Equations The Attempt at a Solution I first found equations for x_1 'dot' and...

Hi, In the usual tight-binding Hamiltonian for semiconductor materials, say GaAs, the basis in which the Hamiltonian matrix elements are specified are the atomic wavefunctions for each atom in the basis. So for GaAs, including just the valence wavefunctions 2s,2px,2py,2pz, we have 8 basis...

If the Hamilton's operator H(t) depends on the time parameter, what is the definition for the time evolution of the wave function \Psi(t)? Is the equation i\hbar\partial_t\Psi(t) = H(t)\Psi(t)\quad\quad\quad (1) or the equation \Psi(t) =...

Evaluate the commutator [H,x], where H is Hamiltonian operator (including terms for kinetic and potential energy). How does it relate to p_x, momentum operator (-ih_bar d/dx)?

(Sorry for my poor English, Please, forgive mistakes, if any.) Dear Friends A system of second order, in normal form, differential equations can be rewritten as a similar first order one, in infinite way (usually that's done introducing simply auxiliaries variables, but it can also be...

Hello, Is it generally the case that [J, H] = dJ/dt? I saw this appear in a problem involving a spin 1/2 system interacting with a magnetic field. If so, why?This seems like a very basic relation but I'm having a bit of brain freeze and can't see the answer right now.

Hello! I'm looking for some introductory materials on the Hamiltonian Treatment of General Relativity; can anybody help with some references?

Homework Statement The title presents my problem. I know in principle how to find eigenvalues and eigenfunctions of the Hamiltonian if it depends only on orbital operators or in spin operators. On the other hand I have no clue how to solve it if there are both types of operators. The...

Homework Statement The title presents my problem. I know in principle how to find eigenvalues and eigenfunctions of the Hamiltonian if it depends only on orbital operators or in spin operators. On the other hand I have no clue how to solve it if there are both types of operators. The...

The Hamiltonian for particle in an EM field is H = 1/2m (p - qA)^2 + q phi If we take the cross-terms, which corresponds to the paramagnetic term, we have H para = -q/2m * (p.A + A.p ) = iqh/2m * (\nabla .A + A.\nabla) What I do not understand is how this simplifies...

How should the Hamiltonian look and what are the necessary forces?? Hi I have a problem which I have to solve. I have a wire which is suspended in two points like the figure below: http://img12.imageshack.us/img12/5660/59905008.png The figure denotes the value of the costate vector in...

Hello, I just have a quick question about Quantum Mechanics. It's probably a bit basic but I'm trying to get my head around the off-diagonal Hamiltonian elements of a perturbation. We can assume the unperturbed Hamiltonian to be degenerate. If I have a Hamiltonian H=H_{0}+H' where the...

how would you solve the hamilton - jacobi equation for something with a hamiltonian with mixed terms like 1/2(p1q2 + 2p1p2 + (q1)^2) well its quite trivial obtaining the HJ equation since there is no time dependence, 1/2( (ds/dq1)q2 + 2(ds/dq1)(ds/dq2) + (q1)^2 ) = E I can't see how...

I am currently in a computational physics course and am working on a final project involving carbon dimers. The reason this topic is applicable in my class is that once I figure out the physics involved, the problem involves using a lot of the numerical methods I learned in class. I am solid on...

Homework Statement The system described by the Hamiltonian H_0 has just two orthogonal energy eigenstates, |1> and |2> , with <1|1>=1 , <1|2> =0 and <2|2>=1 . The two eignestates have the same eigenvalue , E_0: H_0|i>=E_0|i>, for i=1 and 2. Now suppose the Hamiltonian for the...

Homework Statement The mechanics of a system are described by the Lagrangian: L = \frac{1}{2}\dot{x}^2 + \dot{x}t Homework Equations (a) Write the Energy (Jacobi function) for the system. (b) Show that \frac{dh}{dt} \neq \frac{\partial h}{\partial t} (c) Write an expression for...

Homework Statement Consider a physical system with a three-dimensional state space. In this space the Hamiltonian is represented by the matrix: H = hbar\omega \[ \left( \begin{array}{ccc} 0 & 0 & 2 \\ 0 & 1 & 0 \\ 2 & 0 & 0 \end{array} \right)\] The state of the system at t = 0 in...

Homework Statement A single spin-one-half system has Hamiltonian H=\alpha*s_x+\beta*s_y, where \alpha and \beta are real numbers, and s_x and s_y are the x and y components of spin . a) Using the representation of the spin components as Pauli spin matrices, find an expression for H^2...

In the article from Wikipedia called: Geodesics as Hamiltonian Flows at: http://en.wikipedia.org/wiki/Geodesics_as_Hamiltonian_flows" It states the following: It is frequently said that geodesics are "straight lines in curved space". By using the Hamilton-Jacobi approach to the...

Homework Statement Assuming that the Hamiltonian is invariant under time reversal, prove that the wave function for a spinless nondegnerate system at any given instant of time can always be chosen to be real. Homework Equations \psi(x,t)=<x|e^{-iHt/\hbar}|\psi_0> The Time-Reversal...

I'm not going to "follow the template provided" in the strictest sense - but I'm going to include all the same information expected - statement of the problem and a showing of how I've tried to do it, in the intended "spirit" of the template, since these different components are kind of "mixed"...

Dear all, could please give my some links or references to material that justifies the mathematical and physical reasons for introducing these two formalisms in mechanics? Thanks. Goldbeetle

I have a brilliantly engineered system of a bead-on-a-circular-loop (mass=m) rigidly attached to a massive block (mass=M) on one side and a spring on the other. The spring motion is constrained to be in x-direction only, while the bead is free to move on the wire anyway it wants to (no \phi...

Hi, I have a problem involving the Hamiltonian of a particle of mass m, charge q, position r, momentum p, in an external field defined by vector potential A and scalar potential X. Here's the Hamiltonian: H(r,p) = (1/2m)[p - qA(r,t)]2 + qX(r,t) = (1/2m)(pjpj - 2qpjAj + q2AjAj) + qX The...

Hello everybody! I'm currently attending lectures on Hamiltonian dynamics from a very mathematical viewpoint and I'm having trouble understanding two facts: 1. An inner product defined in every tangent space and a symplectic form both establish a natural isomorphism between tangent and...

Homework Statement Let ( Eo 0 A ) ( 0 E1 0 ) ( A 0 Eo ) be the matrix representation of the Hamiltonian for a three state system with basis states |1> |2> and |3> . If |ψ(0)> = |3> what is...

Homework Statement The Hamiltonian for a particle mass m, moving in a central force field is given as: H = 1/(2m) * |p^2| - V(r). Take the Hamiltonian to be invariant, such that it can be shown that L = r x p the angular momentum vector is a conserved quantity: dL/dt = {L,H} = 0. Homework...

Homework Statement A particle that moves in 3 dimensions has that Hamiltonian H=p^2/2m+\alpha*(x^2+y^2+z^2)+\gamma*z where \alpha and \gamma are real nonzero constant numbers. a) For each of the following observables , state whether or why the observable is conserved: parity , \Pi; energy...

\hat{H}=\sum_{\vec{p}}\frac{p^2}{2m}\hat{b}^+_{\vec{p}}\hat{b}_{\vec{p}}+\frac{1}{2V}\sum_{\vec{p}_1,\vec{p}_2,\vec{p}_3}W(\vec{p}_1-\vec{p}_3)\hat{b}^+_{\vec{p}_1}\hat{b}^+_{\vec{p}_2}\hat{b}_{\vec{p}_3}\hat{b}_{\vec{p}_1+\vec{p}_2-\vec{p}_3} Is this correct form or maybe...