Hamiltonian Definition and 833 Threads

Homework Statement So the question is I have to use some trial function of the form \sum c_if_i to approximate the energy of hydrogen atom where f_i=e^{-ar} for some number a (positive real number). Note that r is in atomic unit. Homework Equations Because r is in atomic unit, I think I should...

Can somebody explain to me why, when we work with fermions, the tight binding Hamiltonian matrix has a form 0 0 -t -t 0 0 +t +t -t +t 0 0 -t +t 0 0 the basis are |\uparrow,\downarrow>, |\downarrow,\uparrow>, |\uparrow\downarrow,0>, |0,\uparrow\downarrow>, Why there is +t and -t? (I...

could you help me how to find the value of the attached 4x4 matrix.Could you give me the idea or which method i have to follow to get the value of that matrix.

Homework Statement Consider the Hamiltonian \hat{}H = \hat{}p2/2m + (1/2)mω2\hat{}x2 + F\hat{}x where F is a constant. Find the possible eigenvalues for H. Can you give a physical interpretation for this Hamiltonian? Homework Equations The Attempt at a Solution I don't think...

Hi. In elementary quantum mechanics the continuity equation is used to derive the electron current, i.e. \frac{\partial \rho(\mathbf r,t)}{\partial t}+\nabla\cdot\mathbf j(\mathbf r,t)=0 and one then puts \rho(\mathbf r,t)=\psi^*(r,\mathbf t)\psi(\mathbf r,t). Now if I want to derive...

For some reason I can't derive the Hamiltonian from the Lagrangian for the E&M field. Here's what I have (using +--- metric): \begin{equation*} \begin{split} \mathcal L=\frac{-1}{4}F_{ \mu \nu}F^{ \mu \nu} \\ \Pi^\mu=\frac{\delta \mathcal L}{\delta \dot{A_\mu}}=-F^{0 \mu} \\...

Homework Statement Hi; I am trying to construct eigenstate for the given hamiltonian. I have the energy eigenvalues and corresponding eigenvectors. But How can I construct eigenstates? Homework Equations The Attempt at a Solution I tried to use the H . Psi= E . Psi...

Hello, Just to be sure: is the following correct? Imagine a long rod rotating at a constant angular speed (driven by a little motor). Now say there's a small ring on the rod that can move on the rod without friction. The ring is then held onto the rod by an ideal binding force (I don't...

Forgive me if this is a poorly asked question but I am not yet completely fluent in quantum mechanics and was just looking at the energy eigenvalue equation H|\Psi\rangle = i\hbar \frac{\partial}{\partial t}|\Psi\rangle = E|\Psi\rangle . We've got the Hamiltonian operator H acting on the state...

An old QM exam question asked for consideration of a two-level quantum system, with a Hamiltonian of H = \frac{\hbar \Delta}{2} ( {\lvert {b} \rangle}{\langle {b} \rvert}- {\lvert {a} \rangle}{\langle {a} \rvert})+ i \frac{\hbar \Omega}{2} ( {\lvert {a} \rangle}{\langle {b} \rvert}- {\lvert...

Homework Statement In terms of the usual ladder operators A, A* (where A* is A dagger), a Hamiltonian can be written H = a A*A + b(A + A*) What restrictions on the values of the numbers a and b follow from the requirement that H has to be Hermitian? Show that for a suitably chosen...

The first part of QFT seems to be almost entirely mathematical formalism, without really requiring a whole lot of physical insight to proceed. For instance, we can start with the free-field scalar Lagrangian, minimize it using the Euler-Lagrange equation to arrive at the Klein-Gordon equation...

Homework Statement Find (classify) all graphs with n vertices who's Euler's path is the same as their Hamiltonian cycle. The Attempt at a Solution I'd say that any regular graphs with a degree greater than n/2 (Dirac's Theorem) with and n divisible by 2 has both and Euler's path and a...

The dimensions of action divided by the dimensions of electric field strength are distance x time x charge. Does this mean that distance x time x charge - whatever one might call that - is the "conjugate momentum" of an electric field? If so - is there any physical significance to this...

I was thinking about this. In every problem I have worked, we suppose a hamiltonian exists which can describe the system. There are obviously Hamiltonians which are not possible classically, such as in the 1-D ising model of paramagnetism, where the Hamiltonian contains terms of s_i. s_j where...

In equation 5.8 in this document http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf I am trying to derive this Hamiltonian. I find H= \pi \dot{\psi} - L = i \psi^\dagger \dot{\psi} - \bar{\psi} ( i \gamma^\mu \partial_\mu - m ) \psi = i \bar{\psi} \gamma^0 \partial_0 \psi - i \bar{\psi}...

Hi Lets say I have a Hamiltonian which is invariant in e.g. the spin indices. Does this imply that spin is a conserved quantity? If yes, is there an easy way of seeing this? Niles.

The Hamiltonian operator in quantum field theory (of Klein Gordon Lagrangian) is H=\frac{1}{2} \int \frac{d^3p}{(2 \pi)^3} \omega_{\vec{p}} a_{\vec{p}}^\dagger a_{\vec{p}} after normal ordering Now we construct energy eigenstates by acting on the vacuum |0 \rangle with a_{\vec{p}}^\dagger...

Homework Statement Consider a two-state system. We denote the two orthonormal states by |1>and |2>. The Hamiltonian of the system is given by a 2 × 2 matrix: [omitted in this post, has 4 entries of course, not very interesting] Write the action of H on the states |1> and |2>. 2. The...

My answer would be "yes," and here's my argument: If we let H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + \frac 12 m \omega^2 x^2, it is a Hermitian operator with familiar normalized eigenfunctions \phi_n(x) (these are products of Hermite polynomials and gaussians) and associated...

Homework Statement I'm completing a question regarding position and momentum operators, however I'm stuck on one term. What does [\hat{H}, i\hbar] equal? Or what does it mean? Thanks.

Homework Statement see attached Homework Equations The Attempt at a Solution so i don't see why you need H to be time independent.. if you use TDSE to differentiate <Ek| psi> then you get d/dt of that = 0 regardless of whether H is time indep? surely? Also not sure how to...

Hi, I haven't posted this in the homework section, as I don't really see it as homework as such. I'm trying to derive the Heisenberg equations of motion for the Klein Gordon field (exercise 2.2 of Mandl and Shaw). I'm trying to derive the commutator of the Hamiltonian and canonical momentum...

I know the eigenvalues of a Hermitean operator are necessarily real, and we want energies to be real...but isn't it possible for non-Hermitean operators to have real eigenvalues? If that's so, shouldn't it be possible for at least some Hamiltonians to be non-Hermitean? Also, is it possible for...

Homework Statement A bead of mass, m is threaded on a frictionless, straight rod, which lies in the horizontal plane and is forced to spin with constant angular velocity, \omega, about a fixed vertical axis through the midpoint of the rod. Find the Hamiltonian for the bead and show that it...

Homework Statement A spin system with only 2 possible states H = (^{E1}_{0} ^{0}_{E2}) with eigenstates \vec{\varphi_{1}} = (^{1}_{0}) and \vec{\varphi_{2}} = (^{0}_{1}) and eigenvalues E1 and E2. Verify this & how do these eigenstates evolve in time? Homework Equations...

Homework Statement I am given the Hamiltonia operator of a system in two-dimensional Hilbert space: H = i\Delta(|w1><w2| + |w2><w1|) and am asked to find the corresponding eigenstates. I wrote this operator as a matrix, where H11 = 0, H22 = 0, and H12= i\Delta and H21= -i\Delta...

Homework Statement Homework Equations The Attempt at a Solution I think I have to use the fact that [a+ , a] = 1 but I don't know where to apply this.

When given a Hamiltonian operator (in this case a 3x3 matrix), how do you go about find the ground state, when this operator is all that is given? By the SE when have H\Psi=E\Psi. I can easily solve for Eigenvalues/vectors, but which correspond to the ground state, or am I missing something?

In The Road to Reality, § 20.2, Roger Penrose talks about a "vector field on the phase space T^*(C)", where C is a configuration space. He calls this vector field "the Hamiltonian flow", draws it as little arrows in Fig. 20.5 (that's his typical way of drawing tangent vectors, in contrast to the...

If anyone has time could they please answer this question. I was looking and concept of the the http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" , I was wonder is their a difference between the two terms? If so how are Hermitian and the Hamiltonian different? Can anyone give...

1. A particle of mass m is in the environment of a force field with components: Fz=-Kz, Fy=Fx=0 for some constant K. Write down the Hamiltonian of the particle in Cartesian coordinates .What are the constant of motion? 2. H=kinetic energy +potential energy [b]3. Is the Hamiltonian...

since the time derivative is second order, the KG equation can not be put in the form i \dot{\psi}= H \psi so there is no H in KG equation? and no Heisenberg picture for KG equation?

Apparently things like the Lorentz' force can't be handled as a hamiltonian system. I heard other people describe the hamiltonian mechanics as an equivalent characterization of classical mechanics, but this is wrong, then?

Hello. I'm studying for an ODE/PDE qualifier and I'm wondering how to do this problem. I feel like it should be pretty easy, but anyway.. Show that a Hamiltonian system in \mathbb{R}^{2n} has no asymptotically stable critical points. Any suggestions? Thanks..

Hey all, I've got a Hamiltonian of the form H = \omega (\sigma_z^1 - \sigma_z^2) + J \sigma_z^1 \sigma_z^2 where \omega is a frequency ( I think), J is the indirect dipole-dipole coupling, and \sigma_z^i is the Pauli Z operator on the ith particle. Does anybody know what this...

Hi, I'm trying to find the Hamiltonian for a system using cylindrical coordinates. I start of with the Lagrangian L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2)-U(r,\theta,z) From that, using H=\sum p\dot{q}-L...

I keep hearing jargon like "the algebra generated by the hamiltonian", and I'd like to get to the bottom of it. Given a set of hamiltonians, does the "algebra that they generate" refer to the unitaries that they generate and their subsequent combinations? Or does it refer to the hamiltonians...

HERE http://vixra.org/pdf/1007.0005v1.pdf is my proposed proof of an operator whose Eigenvalues would be the Imaginary part of the zeros for the Riemann Hypothesis the ideas are the following* for semiclassical WKB evaluation of energies the number of levels N(E) is related to the integral of...

Hi, there. It should be yes, but I'm very confused now. Consider a simple one-dimensional system with only one particle with mass of m. Let the potential field be 0, that's V(r) = 0. So the Hamiltonian operator of this system is: H = -hbar^2/(2m) * d^2/dx^2 \hat{H} =...

How would one find the eigenstates/values for the following Hamiltonian? H=A S_z + B S_x where A,B are just constants. Any help is appreciated. Thanks.

Sorry if this question is very general/vague, but I would prefer a general answer rather than a specific solution... I'll put more detail in if necessary though. So, say we have a Hamiltonian for a system (of fermions, spin 1/2); then we find its eigenvalues and hence eigenstates. These are...

Hello, I'm studying the Heisenberg Model. Given the Hamiltonian H = - 2 \frac{J}{\hbar^2} \vec{S}_1 \vec{S}_2 with \begin{equation} \vec{S} = \frac{\hbar}{2} \; \left( \begin{array}{ccc} \sigma_x \\ \sigma_y \\ \sigma_z \end{array} \right) \end{equation} \sigma_{x,y,z} \quad...

how is Hamiltonian energy of orbital differ from orbital in certain atom with another atom although the two orbital have the same no. of electrons... ex...Hamiltonian energy differ in lithium and hydrogen although both have one electron in last orbital please help me quickly

Homework Statement Given is the Hamiltonian for a particle in free fall: H(z,p) = P^2/(2m) + mgz At time t=0 there is an region given by the constrains: p1 less than or equal to p less than or equal to p2 E1 less than or equal to E less than or equal to E2 What is the area of...

L=\frac{1}{2}m(\dot{q}_1-\dot{q}_2)^2-V(q_1,q_2) Because if we put p_1=\frac{\partial L}{\partial \dot{q}_1} p_2=\frac{\partial L}{\partial \dot{q}_2} we get p_1=-p_2=m(\dot{q}_1-\dot{q}_2) We can't invert to get \dot{q_1} in terms of the two momenta. We can still write down a...

[QM] Hamiltonian and symmetries Homework Statement Let there be the hamiltonian: H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2(x^2+y^2+z^2)+kxyz+\frac{k^2}{\hbar \omega}x^2y^2z^2 Find the expectation value of the three components of \vec r in the ground state using ONLY the symmetry properties of...

The definition of a Legendre transformation given on the Wikipedia page http://en.wikipedia.org/wiki/Legendre_transformation is: given a function f(x), the Legendre transform f*(p) is f^*(p)=\max_x\left(xp-f(x)\right) Two questions: what does \max_x mean here? And why is it not...

the idea is can a Hamiltonian in 1-D of the form H=p^2 + V(x) for a certain function V(x) be unbounded and have NEGATIVE energies , for example a Hamiltonian whose spectra may be E_{n} = ...,-3,-2,-1,1,2,3,... and so on, so we have an UNBOUNDED Hamiltonian with positive and negative energies...

Hi, I have this article in which I saw that for a spin 1/2 particle confined to move along a ring positiond in a magnetic field with a z and \varphi The Hamiltonian is given by: (in second attacment) What I do not understand is how do you get the last term in the Hamiltonian. Any help...