Hamiltonian Definition and 833 Threads

I'm learning ray optics and feeling so confused by the definition of "Hamiltonian of light". What I learned was that the "Hamiltonian of light" defined by H = n-|\vec{p}| = 0 indicates the momentum conservation, where n is refractive index and \vec{p} here is the canonical momentum. The...

Hi all, I am working on DFT calculations using SIESTA. I have obtained the Hamiltonian file (.HSX) from the calculations. But, I cannot read the file it. How do I read the .HSX file .

Hi, The following contains two questions that I encountered in the books of Claude Cohen-Tannoudji, "Atom-Photon Interactions" and "Atoms and Photons: Introduction to Quantum Electrodynamics". The first one is about how to calculate two Fourier transforms, and the second one is a example of...

I completely do not understand How they work . Here is a link to my unanswered stack exchange question. Could anyone help me? Would you like any clarifications? http://physics.stackexchange.com/questions/112034/can-someone-explain-to-me-the-rocksar-kivelson-hamiltonian

Homework Statement a) The operators ##a## and ##a^{\dagger}## satisfy the commutation relation ##[a,a^{\dagger}] = 1##. Find the normalization of the state ##|\psi \rangle = C (a^{\dagger} )^2 |0\rangle##, where the vacuum state ##|0\rangle## is such that ##a|0\rangle = 0## b)A one...

Homework Statement Part (a): Show the Commutation relation [x, [H,x] ] Part (b): Show the expression by taking expectation value in kth state. Part (c): Show sum of oscillator strength is 1. What's the significance of radiative transition rates?Homework Equations The Attempt at a Solution...

1. Is the root(det(q)) term in the Hamiltonian Constraint what makes it non polynomial 2. Is the motivation for Ashtekar Variables to remove the non polynomial terms by replacing the Hamiltonian with a densitised Hamiltonian

Homework Statement I'm working (self-study) through Goldstein et al, Classical Mechanics, 3rd Edition, and I'm currently stuck on Problem 8.11: A particle is confined to a one-dimensional box. The ends of the box (let these be at \pm l(t)) move slowly towards the middle. By slowly we mean...

Homework Statement We want to get the time evolution of a wavefunction and the expectation value of the Hamiltonian, and from there we can show that it's the same as the time-independent result. So to be clear: given a wavefunction, get the time evolution of that function and the expectation...

Helloo, I don't understand how one arrives at the conclusion that the hamiltonian is a generating function. When you have an infinitesimal canonical transformation like: Q_{i}=q_{i}+ \delta q_{i} P_{i}=p_{i}+\delta p_{i} Then the generating function is: F_{2}=q_{i}P_{i}+ \epsilon...

Homework Statement The point of suspension of a simple pendulum of length l and mass m is constrained to move on a parabola z = ax^2 in the vertical plane. Derive a Hamiltonian governing the motion of the pendulum and its point of suspension. This is a two-dimensional problem...

Homework Statement A massless spring of length b and spring constant k connects two particles of masses m_1 and m_2 . The system rests on a smooth table and may oscillate and rotate. a) Determine the Lagrange's equations of motion. b) What are the generalized momenta associated with...

Homework Statement To try and relate the three ways of calculating motion, let's say you have a particle of some mass, completely at rest, then is acted on by some force, where F equals a constant, C, times time. (C*t). I want to find the equations of motion using Lagrangrian, but also Newton...

Hello, there are several papers on QCD in Hamiltonian formulation, especially in Coulomb gauge. Unfortunately the Hamiltonian H is rather formel and highly complex. Question: is there a paper discussing the contribution of individual terms of H to the nucleon mass?

How many ways can we change the Hamiltonian without affecting the wave functions (eigenvectors) of it. Like multiply all the elements in the matrix by a constant. I'm facing a very difficult Hamiltonian,:cry: I want to simplify it, so the wave function will be much easier to derive. Thanks in...

I know I've seen this point about roots discussed somewhere but I can't for the life of me remember where. I'm hoping someone can point me the right direction. Here's the situation:- The standard derivation of the quantum HO starts with the classic Hamiltonian in the form H = p2 + q2...

Hi I have been looking at canonical transformation using generating functions. I am using the Goldstein book and it gives the following example - F1=qQ ⇔ Q=p and P=-q F2=qP ⇔ Q=q and P=p F3=pQ ⇔ Q=-q and P=-p F4=pP ⇔ Q=p and P=-q I'm confused ! Obviously functions 1 and 4 give the...

Hi All, Greetings! I have a 3d material and I use result from first principal for getting the potential (U(x,y,z)). I then find average U(x) from U(x,y,z). Now if I write one dimensional Hamiltonian in X direction and use this value of U(x), can I get bandgap of the original 3d material (I...

Hi All, Greetings ! Here is what I wish to know. Specifying a tight binding hamiltonian requires values of potential (U). Consider a 3d solid. If I have potential profile in x direction (U1, U2, U3...so on) can I directly plug in these U values into the tight binding hamiltonian or do I...

I am reading an article on the "energy surface" of a Hamiltonian. For a simple harmonic oscillator, I am assuming this "energy surface" has one (1) degree of freedom. For this case, the article states that the "dimensionality of phase space" = 2N = 2 and "dimensionality of the energy surface" =...

For two electron system ##\vec{S}_1##, ##\vec{S}_2## \mathcal{H}=J\vec{S}_1\cdot \vec{S}_2=J(\frac{1}{2}(S_{tot})^2-\frac{3}{4}) (1) How you get relation (1)?

Just a little doubt. When we are performing a canonical transformation on a Hamiltonian and we have the equations of the new coordinates in terms of the old ones we have to find the Kamiltonian/new Hamiltonian using K = H +\frac{\partial G}{\partial t}. My question is: do we have to derive the...

I have a question if you have an Hamiltonian given by H = \sum_{i,i+1} \sigma_i \cdot \sigma_{i+1} where i can even or odd bonds so in a 1D lattice so if you have 4 sites(1 2 3 4 1) then (12) and (34) are even bonds and (23) and (41) are odd bonds. and I was checking if [H_{x...

Homework Statement Hi, I want to diagonalize the Hamiltonian: Homework Equations H=\phi a^{\dagger}b + \phi^{*} b^{\dagger}a a and b are fermionic annihilation operators and \phi is some complex number. The Attempt at a Solution Should I use bogoliubov tranformations? I...

How to select the good basis for the special Hamiltonian?? For the Hamiltonian H=\frac{P^2}{2\mu} -\frac{Ze^2}{r}+ \frac{\alpha}{r^3} L.S (which we can use L.S=\frac{1}{2} (J^2-L^2-S^2)in the third term) how to realize that the third term,\frac{\alpha}{r^3} L.S, commutes with sum of the...

For an exercise I am given the attached Hamiltonian, but I don't understand it completely. We sum over spin -½ and ½ and the paulimatrices seem to be dependent on this since they are labeled by σσ'. What does this mean? I mean the pauli matrices are just operators for the spin in the...

Homework Statement The vector \psi =\psi_{n} is a normalized eigenvector for the energy level E=E_{n}=(n+\frac{1}{2})\hbar\omega of the harmonic oscillator with Hamiltonian H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}. Show that...

Hey, I consider a diagonalized Hamiltonian: H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const with fermionic creation and annihilation...

Homework Statement I have problem where I need to commutate my hamiltonian H with a fermionic anihillation operator. Had H been written in terms of fermionic operators I would know how to do this, but the problem is that it describes phonon oscillations, i.e. is written in terms of bosonic...

If a Hamiltonian is unbounded from below, say the hydrogen atom where the Hamiltonian is -∞ at r=0, is there a way to tell if the ground state is bounded (e.g. hydrogen is -13.6 eV and not -∞ eV)? It seems if the potential is 1/r^2 or less, then the energy will be finite as: \int d^3 r (1/r^2)...

##H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}m\omega^2x^2## Parity ##Px=-x## end ##e## neutral are group of symmetry of Hamiltonian. ## PH=H## ##eH=H## so I said it is group of symmetry because don't change Hamiltonian? And ##e## and ##P## form a group under multiplication. Is there...

Consider the attached exercise. I am having some trouble understanding exactly what time dependent hamiltonian it refers to. Because from the equation it refers to it seems that the hamiltonian is by definition time independent. Am I to assume that the H diagonal is a time independent...

Hello I'm reaaaaaally stuck here, if someone can explain me it'd be great because i just dn't know how to do all that Homework Statement Consider a system with the moment of inertia l=1 A base of the space of the states is constituted by the three eigenvectors of Lz |+1> , |0>, |-1> of...

Homework Statement Show that the two-body hamiltonianH_{\text{sys}}=\frac{\mathbf{p}_1^2}{2m_1}+\frac{\mathbf{p}_2^2}{2m_2}+V( \mathbf{r}_1,\mathbf{r}_2)can be separated into centre of mass and relative...

Homework Statement The simple form H=T+U is true only if your generalized coordinates are "natural"(relation between generalized and underlying Cartesian coordinates is independent of time). If the generalized coordinates are not natural, you must use the definition H=Ʃpq'-L. To illustrate...

Homework Statement (a) Consider a system with one degree of freedom and Hamiltonian H = H (q,p) and a new pair of coordinates Q and P defined so that q = \sqrt{2P} \sin Q and p = \sqrt{2P} \cos Q. Prove that if \frac{\partial H}{\partial q} = - \dot{p} and \frac{\partial H}{\partial p} =...

What would the Hamiltonian for a system of two classical point particles, with no interaction except for an elastic collision between them at a point look like? My gut says it's the usual T + V, with T = p12/2m1 + p22/2m2 and V = Kδ(r1-r2) With K approaching infinity -- each particle...

Homework Statement Find the eigenvalues of the following Hamiltonian. Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |RHomework Equations â|\phi_{n}>=\sqrt{n}|\phi_{n-1}> â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}> The Attempt at a Solution By applying the Hamiltonian to a random state n I...

I've dealt with both the Hamiltonian function for Hamiltonian mechanics, and the Hamiltonian operator for quantum mechanics. I have a kind of qualitative understanding of how they're similar, especially when the Hamiltonian function is just the total energy of the system, but I was wondering if...

I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I don't understand is: Why are the \mu B parts not diagonal? If the Hamiltonian is \vec{\mu} \cdot...

Homework Statement For the SHO, find these commutators to their simplest form: [a_{-}, a_{-}a_{+}] [a_{+},a_{-}a_{+}] [x,H] [p,H] Homework Equations The Attempt at a Solution I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first...

I've seen that the lagrangian for a relativistic free particle is -mc \sqrt{\eta_{\mu\nu} \dot{x^{\mu}}{\dot{x\nu}} but when I construct the hamiltonian as p_{\mu} \dot{x^{\mu}} - L I seem to get zero. I am not really sure what I'm doing wrong. I find that if in the first term of...

Dear All, To give a background about myself in Classical Mechanics, I know to solve problems using Newton's laws, momentum principle, etc. I din't have a exposure to Lagrangian and Hamiltonian until recently. So I tried to read about it and I found that I was pretty weak in coordinate...

I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures on quantum mechanics". My question is the following: I have momentum variables that depend on the...

This has bothered me for some time. In the ADM formulation, we foliate spacetime into 3+1 dimensions by creating 3 dimensional hypersurfaces via ##T = constant## along the worldline of some observer whose proper time is ##T##. This allows us to write dynamical equations for the evolution of some...

Hi everyone. Suppose I have an Hamiltonian which doesn't depend on the position (think for example to the free-particle one H=p^2/2m). I know that the classical partition function for the canonical ensemble is given by: $$ Z(\beta)=\int{dpdq e^{-\beta H(p,q)}}. $$ What does it happen to...

For example, I have a 3D particle that experiences a harmonic oscillator potential only in the X,Y plane for all Z ie. a free particle in the Z direction. This seems like cylindrical coordinates but I'm not sure how to express the Hilbert space if I want to be able to describe states and...

I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics. It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should...

Homework Statement Problem 7.15 from Aitchison and Hey, Volume I, 3rd Edition. Verify the forum (7.139) of the interaction Hamiltonian \mathcal{H_{S}^{'}}, in charged spin-0 electrodynamics. Equation 7.139 is \mathcal{H_{S}^{'}}= - \mathcal{L}_{int} - q^2 (A^0)^2 \phi^{\dagger} \phi...

I try diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor is all fine. However, with a superconductor I don't get the correct result for the energy spectrum of the Hamiltonian (arxiv:1302.5433) H=\eta(k)τz+Bσ_x+αkσ_yτ_z+Δτ_x...