Hamiltonian Definition and 901 Threads

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...

Homework Statement Given an initial (t=-∞) Fock state , \left|n\right\rangle, and a function f(t), where f(±∞)=0, show that for a Harmonic Oscillator perturbed by f(t)\hat{x} the difference \left\langle H(+∞) \right\rangle - \left\langle H(-∞) \right\rangle is always positive.Homework Equations...

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. 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 \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 Basically, I'm given a Hamiltonian H = H(p,q) and asked to find a new Hamiltonian K = K(Q,P,t) using the generating functions method H = 1/2 (p^2 + q^2) Generating function f(q,P,t) = qp sec ( t ) - 1/2 (q^2 + P^2) tan ( t ) So, I have no problem finding the new...

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...

Homework Statement We need to find the Hamiltonian that corresponds to a given Lagrangian by finding the Legendre transform. The system is a rigid body pinned down in some point. This means the motion is described essentialy by SO(3). So the Lagrangian is given in terms of these matrices and...

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...

Homework Statement Let (V(x,t) , A(x,t)) be a 4-vector potential that constructs the electromagnetic field (in gaussian Units) by E(x,t) = -∇V(x,t) - (1/c)δtA(x,t) , B = ∇xA , (x,t) elements of R3xRt Consider the lagrangian L=.5mv2 - eV(x,t) + (ev/c)(dot)A(x,t) a) compute and interpret the...

Homework Statement Find the Hamiltonian and Hamilton's equations of motion for a system with two degrees of freedom with the following Lagrangian L = 1/2m1\dot{}xdot12 + 1/2m2\dot{}xdot22 + B12\dot{}xdot1x2 + B21\dot{}xdot1x1 - U(x1, x2) Explain why equations of motion do not depend...

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...

I am new to quantum physics. My question is how to write the Hamiltonian in dirac notation for 3 different states say a , b , c having same energy. I started with Eigenvaluee problem H|Psi> = E|psi> H = ? for state a? SO it means that indvdually H= E (|a><a|) for state a and for all three...

I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation} The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where L is angular momentum operator and S is spin operator. The eigenvalue for \begin{equation}L^2 ...

Homework Statement I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation} Homework Equations The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where L is angular momentum operator and S is spin operator. The...

Homework Statement Find the matrix elements of the Hamiltonian in the energy basis for the ISW. Is it diagonal? Do you expect it to be diagonal? Homework Equations H=\frac{p^2}{2m}+V \frac{d}{dt}\langle Q \rangle = \frac{i}{\hbar} \langle[\hat H, \hat Q] \rangle + \langle...

My question is simple is every classical mechanics problem which is solvable by Lagrangian & Hamiltonian methods also solvable by Newtonian methods of forces and torques? And why does it seem that LH make solutions to be a lot more easier than Newtonian methods, and is it always this way?

Homework Statement I have the state: |\psi>=cos(\theta)|0>+sin(\theta)|1> where \theta is an arbitrary real number and |\psi> is normalized. And |0> and |1> refer to the ground state and first excited state of the harmonic oscillator. Calculate the expectation value of the Hamiltonian...

A textbook gives the following interaction Hamiltonian describing the interaction of an atom (having transition dipole moment \mu) with a photon whose polarization can be \epsilon_{1} or \epsilon_{2}): H = g \Sigma^{2}_{s=1}\mu\bullet\epsilons\sigma^{-}a^{+}_{s} + h.c. where \sigma^{-} is...

Homework Statement Suppose the potential in a problem of one degree of freedom is linearly dependent on time such that the Hamiltonian has the form: H= p^2/2m - mAtq where m is the mass of the object and A is contant Using Hamilton's canonical equations that are give below. Find the...

Hi all, I've a hamiltonian that describes the coupling of electrons in a crystal (bloch electrons) to an EM field described by a vector potential A \begin{equation} \mathscr{H} = \frac{e}{mc}\left[\mathbf{p}(-\mathbf{k}) \cdot \mathbf{A}(\mathbf{k}, \omega)\right] \end{equation}...

Can anyone explain how the time evolution operator commutes with the Hamiltonian of a system ( given that the the Hamiltonian does not depend explicitly on t ) ?

Homework Statement Show that the Hamiltonian operator (\hat{H})=-((\hbar/2m) d2/dx2 + V(x)) is hermitian. Assume V(x) is real Homework Equations A Hermitian operator \hat{O}, satisfies the equation <\hat{O}>=<\hat{O}>* or ∫\Psi*(x,t)\hat{O}\Psi(x,t)dx =...

Hi there, just wondered if anyone could help me... If I am given a hamiltonian describing a particle in one dimension H=p^2/2m +1/2 (γ(x-a)^1/2) + K(x-b) how do I go about finding the eigenstates and eigenvalues of this hamiltonian? Many thanks

Hi , I can't understand the general formula for weyl ordering of the hamiltonian . It is written in Srednicki field theory book in page 68 . Can someone explain how to derive this formula ?