Hamiltonian Definition and 833 Threads

In Rovelli's book, in chapter 7 it talks about the Hamiltonian operator for LQG. In manipulating the form for the Hamiltonian operator Rovelli makes the following expansions U(A,\gamma_{x,u})=1+\varepsilon u^a A_a(x)+\mathcal{O}(\varepsilon^2) where by fixing a point x and a tangent...

hi friends. i don't know how can i write a fortran code for expressing spins in Heisenberg model which have 3 dimension spin operator, sx,sy,sz? thanks for your help

Homework Statement The total energy may be given by the hamiltonian in terms of the coordinates and linear momenta in Cartesian coordinates (that is, the kinetic energy term is split into the familiar pi2/2m. When transformed to spherical coordinates, however, two terms are angular momentum...

Hi there, If the evolution operator is given as follows U(t) = \exp[-i (f(p, t) + g(x))/\hbar] where p is momentum, t is time. Can I conclude that the Hamiltonian is H(t) = f(p, t) + g(x) if no, why?

I'm currently reading Sakurai's 'Modern Quantum Mechanics' (Revised Edition) and at page 76 he introduces a spin half hamiltonian H = - (\frac{e}{mc}) \vec S \cdot \vec B. But what is c doing in this hamiltonian? Clasically the energy of a magnetic moment in a magnetic field is E = -...

Homework Statement "Show that if the Hamiltonian depends on time and [H(t_1),H(t_2)]=0, the time development operator is given by U(t)=\mathrm{exp}\left[-\frac{i}{\hbar}\int_0^t H(t')dt'\right]." Homework Equations i\hbar\frac{d}{dt}U=HU U(dt)=I-\frac{i}{\hbar}H(t)dt The Attempt at a...

Theorem: Prove that there exist $n$ edge disjoint Hamiltonian cycles in the complete graph $K_{2n+1}$. ---------------------------------------------------------------------------------- I have found two constructive proofs of this over the internet. But I would like to prove it...

Hi, I have a question about two different expressions of Jaynes-Cummings Hamiltonian H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} + g (a^{\dagger}\sigma_{-} +a\sigma_{+} ) and H=\Delta_c a^{\dagger}a+\Delta_a \sigma_{+} \sigma_{-} +i g (a^{\dagger}\sigma_{-} -a\sigma_{+}...

Hi I have a question regarding the pump-term in the Hamiltonian on page 9 (equation 2.9b) of this thesis: http://mediatum2.ub.tum.de/download/652711/652711.pdf. This term is not very intuitive to me. a is followed by a CCW phase, whereas a^\dagger is followed by a CW phase. How does this...

I don't understand this idea. For example we have cubic crystal which has a lot of unit cells. We define spin variable of center of cell like S_c. And spin variable of nearest neighbour cells with S_{c+r}. So the cell hamiltonian is...

Why is minus sine in definition of hamiltonian H=-\sum_{i,j}J_{i,j}(S_{i}^+S_{j}^-+S_i^zS_j^z) Why not? H=\sum_{i,j}J_{i,j}(S_{i}^+S_{j}^-+S_i^zS_j^z)

I have a Hamiltonian, consisting only of angular momentum components Lx,Ly,Lz. I need to go from it to some coordinate representation. But I don't have derivatives Lx' etc. in H. So, when I'll go to coordinates and momenta I'll have Hamiltonian equations like p_i=0, which doesn't have sense...

How can you tell if the Klein-Gordan Hamiltonian, H=\int d^3 x \frac{1}{2}(\partial_t \phi \partial_t \phi+\nabla^2\phi+m^2\phi^2) is time-independent? Don't you have to plug in the expression for the field to show this? But isn't the only way you know how the field evolves with time is...

I'm currently using David J. Griffiths 'Introduction to Quantum Mechanics' to teach myself quantum mechanics and I had a quick question about the way he factors the Hamiltonian into the raising and lowering operators for the potential V(x)=(1/2)kx² On page 42 he writes the Hamiltonian as...

In this paper called "Stepping out of Homogeneity in Loop quantum Cosmology" - http://arxiv.org/pdf/0805.4585.pdf. On page 4 they say "where the sum is over the couples of distinct faces at each tetrahedron, U_{ff'} = U_f U_{f_1} U_{f_2} \dots U^{-1}_{f'} where l_{ff'} = \{ f , f_1; f_2; \dots...

Generally, what sort of problems are handled better by Hamiltonian mechanics than by Lagrangian mechanics? Can anyone give a specific example?

Hi there. I need help to work this out. A particle with mass m is studied over a rotating reference frame, which rotates along the OZ axis with angular velocity \dot\phi=\omega, directed along OZ. It is possible to prove that the potential (due to inertial forces) can be written as: V=\omega...

Homework Statement A particle moves in a one dimensional potential : V(x) = 1/2(mω2x Show that the function ψ(x) = a0exp(-βx2) is an eigenfunction for the Hamiltonian for a suitable value of β and calculate the value of energy E1 Homework Equations The Attempt at a Solution...

Not sure I am posting this in the right subforum, if this is not the case, please feel free to move it. Anyway, the title about sums it up - I need to find a good source which offers a thourough treatment of Hamiltonian formalism for the explicitly time-dependent case - could someone possibly...

Homework Statement The problem is from Ashcroft&Mermin, Ch32, #2(a). (This is for self-study, not coursework.) The mean energy of a two-electron system with Hamiltonian \mathcal{H} = -\frac{\hbar^2}{2m}(∇_1^2 + ∇_2^2) + V(r_1, r_2) in the state ψ can be written (after an integration by...

Consider a ball of mass m rotating around an axis Oz (vertical). This ball is on a circle whose center is the same O. Given: Angular velocity of ring is d∅/dt = ω. Mind explaining it so we can prove that Hamiltonian here is different from Energy?!

Hi all, There is a Hamiltonian in terms of "a" and "a^{dagger}"bosonic operators H=ω*(a^{dagger}a+1/2)+alpha*a^2+β*a^{dagger}^2 and ω, alpha and β are real constants and its energy is E=(n+1/2)*epsilon where epsilon is ω^2-4*alpha*β. Now, I tried to find this energy but I couldn't. Would you...

Alright. So the Dirac Eq is (i \gamma^{\mu} \partial_{\mu} - m) \psi = 0 or putting the time part on one side with everything else on the other and multiplying by \gamma^0 , i \partial_t \psi = (i \gamma^0 \vec{\gamma} \cdot \nabla + \gamma^0 m) \psi I would think that this is the...

Homework Statement a light, inextensible string passes over a small pulley and carries a mass of 2m on one end. on the other end is a mass m, and beneath it, supported by a spring w/ spring constant k, is a second mass m. using the distance x, of the first mass beneath the pulley, and the...

i'm just ready to start QM and I looked at the text and I turned to Shro eq to see if I could understand it and they mentioned Hamiltonian operator. It looked like the book assumed knowledge of H and L mechanics. Do I need to know this stuff? I wasn't told by others that I needed this. I was...

I have two somewhat related questions. First, why would we care about the Lagrangian L = T - V (or K - U)? I understand with the Hamiltonian H = T +V, the total energy is conserved. But with the Lagrangian, what does it actually mean? Mathematically, it only inverts the potential energy...

Hi, A fundamental aspect in the Hamiltonian framework of mechanics is that the q's and p's are independent. I feel like I understand the steps in the Legendre transform from Lagrangian to Hamiltonian mechanics, but I don't see how you can go from a system where only the q's are independent...

This isn't a homework problem - I can't understand a particular statement in my professor's notes. As such, I hope it's in the correct forum. Homework Statement The Hamiltonian for a charged particle in a potential field A is \hat{H} = (1/2m) ( -i \hbar \nabla - q A)^{2} The square...

Homework Statement A two state system has the following hamiltonian H=E \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) The state at t = 0 is given to be \psi(0)=\left( \begin{array}{cc} 0 \\ 1 \end{array} \right) • Find Ψ(t). • What is...

Homework Statement I have to find the hamiltonian for a diatomic molecule, where the molecule can only rotate and translate and we supose that potencial energy doesn't change.Homework Equations The Attempt at a Solution Okey so I used Spherical coordinate system such as the kinetic energy of...

Dear all, I have a fundamental question about breaking the Hamiltonian. Here is the description: Suppose a particle, \lambda^{0}, is produced in a high energy nuclear collision with proton beam. It is produced by strong interaction, and it has fixed energy (can be obtained from its...

Hi guys, I'm reading Shankar and he's talking about the Variational method for approximating wave functions and energy levels. At one point he's using the example V(x) = λx^4, which is obviously an even function. He says "because H is parity invariant, the states will occur with alternating...

Homework Statement I have a question given to me by my prof that is a time dependent Hamiltonian H(q,p,t) = g(t)(p2/(2m) + kq2/2) where f(t) has 2 different forms i need to solve 1) eat 2) cos(gt) problem is goldstein only covers conserved hamiltonians in chapter 10 for the H-J...

Homework Statement A canonical transformation is made from (p,q) to (P,Q) through a generating function F=a*cot(Q), where 'a' is a constant. Express p,q in terms of P,Q. Homework Equations The Attempt at a Solution A generating function is supposed to be a bridge between (p,q) and...

I am trying to understand how Hamiltonian gradient works. [SIZE="4"]H(q,p)=U(q)+K(p) U(q): potential energy K(p): kinetic energy q: position vector p: momentum vector both p and q are functions of time H(q,p): total energy [SIZE="4"]\frac{d{{q}_{i}}}{dt}=\frac{\partial H}{\partial {{p}_{i}}}...

If I consider the problem of for example the hydrogen atom. I.e. a central force problem with an effective potential V(r) that depends only of r, the distance between the positively charged nucleous and the negatively charged electron. In the Schrödinger's equation, one considers the...

Homework Statement Consider a harmonic oscillator with generalized coordinates q and p with a frequency omega and mass m. Let the transformation (p,q) -> (Q,P) be such that F_2(q,P,t)=\frac{qP}{\cos \theta }-\frac{m\omega }{2}(q^2+P^2)\tan \theta. 1)Find K(Q,P) where \theta is a function of...

Homework Statement a) the lagrangian for a system of one degree of freedom can be written as. L= (m/2) (dq/dt)2sin2(wt) +q(dq/dt)sin(2wt) +(qw)2 what is the hamiltonian? is it conserved? b) introduce a new coordinate defined by Q = qsin(wt) find the lagrangian and hamiltonian...

Consider a mass m confined to the x-axis and subject to a force Fx=kx where k>0. Write down and sketch the potential energy U(x) and describe the possible motions of the mass. (Distinguish between the cases that E>0 and E<0. It is the part in parenthesis that confuses me. I can't...

In QFT, Lagrangian is often mentioned. While in QM, it's the Hamiltonian, Is the direct answer because in QFT "position" of particle is focused on and Lagrangian is mostly about position and coordinate while in QM, potential is mostly focus on and Hamiltonian is mostly about potential and...

Homework Statement Particle of mass m constrained to move on the surface of a cylinder radius R, where R^2 = x^2 + y^2. Particle subject to force directed towards origin and related by F = -kx Homework Equations L = T - U H = T + U The Attempt at a Solution So I have the solution, but...

Homework Statement Show that the equation below is an eigenfunction for the Quantum Harmonic Oscillator Hamiltonian and find its corresponding eigenvalue. Homework Equations u1(q)=A*q*exp((-q^{2})/2) The Attempt at a Solution Ok, so I know that the Quantum Harmonic Oscillator...

Does the Hamiltonian is always equal to the energy of the system?? I have this doubt since a few weeks ago. For the Newtonian case we have that H=K+U, kinetical energy plus potential energy, but given that the definition of the Hamiltonian is H=\dot{q}P-L, my question is, Does exist a system or...

Homework Statement How do I obtain [H,P_x]? P_x is the polarization operator. Homework Equations H=-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial x^2}+V(x) P_x=2Re[c_+^*c_-] The Attempt at a Solution I know how to commute H and x. But somehow can't think of a way to...

It doesn't seem like a time dependent hamiltonian would have stationary states, am I wrong? I've run into conflicting information.

Hi. I am going to start my MSc in a couple of months majoring in nonlinear quantum optics. I have a good basic in quantum mechanics, but have never looked at quantum optics before. My topic will be to investigate quantum properties of nonlinear optical coupler but i have problem with the...

Homework Statement Two spin-half particles with spins S1 and S2 interact with a spin-dependent Hamiltonian H=λS1*S2 (the multiplication is a dot product and is a positive constant). Find the eigenstates and eigenvalues of H in terms of |m1,m2>, where (hbar)m1 and (hbar)m2 are the z-components...

Hi, I know how to prove the orthonormality of the hamiltonian when it is real but am struggling to work out how to prove it when the hamiltonian is not real. When proving for a real hamiltonian the lefthand side equals zero as H(mn)=H(nm)complexconjugate. but if the hamiltonian is not...

Hi, A general question.. In analytical mechanics, we take a given hamiltonian and re-write it in term of generalzed coordinates. In a way- we recode the hamiltonian to concern only the "essence" of the problem. However, it seems to me, that in QM we do the opposite- we look for operators that...

What is the main difference between Langrangian, Hamiltonian, and Netwonian Mechanics in physics, and what are the most important uses of them? I'm currently a high school senior, with knowledge in calculus based physics, what would the prerequisites be in order for me to begin Langrangian...