Hamiltonian Definition and 833 Threads

1. In Classical Hamiltonian, it's equal to the kinetic energy plus potential energy.. but I read it that for a free particle, it doesn't even depend on position.. i thought the potential energy depends on position. If it doesn't depend on position, what does it depend on? 2. Since the...

To get the dynamics of particles in a box. You are supposed to get the Hamiltonian which is potential energy plus kinetic energy. But does the potential energy take into account the momentum of the particles in the box? What happens if you change the momentum of the particles.. do the potential...

I'm a beginner in QFT, starting out with D. J. Griffiths' book in this topic. I have a question on the operators used in QM. What are operators? What is the physical significance of operators? I can understand that ##\frac {d}{dt}## to be an operator, but how can there be a total energy...

The Hamiltonian in classical mechanics is not always equal to the total energy of the system. I believe this is only true if there is only a potential field and no vector potential. However, in quantum mechanics for a particle in an EM field, even if a vector potential is used the energy...

Hello! I read that if we apply the exactly same procedure for Dirac theory as we did for Klein Gordon, in quantizing the field, we obtain this hamiltonian: ##H=\int{\frac{d^3p}{(2\pi)^3}\sum(E_pa_p^{s\dagger}a_p^s-E_pb_p^{s\dagger}b_p^s)}## and this is wrong as by applying the creation operator...

The hamiltonian is not in the wave function but only exist when the amplitude is squared. But in the book "Deep Down Things". Why is the Schrodinger Equation composed of kinetic plus potential terms equal total energy. Is it not all about probability amplitude? How can probability amplitude have...

Hi! it's been a day since I have started this problem. I was wondering how I could arrive to this Hamiltonian? And I'm a bit at a lost on how exactly to derive this? I hope anyone can help me with this, even a suggestion of good starting point would be much appreciated. Basically the problem...

Hi! I am currently trying to derive the Fourier transform of a 2D HgTe Hamiltonian, with k_x PBC and vanishing boundary conditions in the y direction at 0 and L. Here is the Hamiltonian: H = \sum_{k}\tilde{c_k}^{\dagger}[A\sin{k_x}\sigma_x + A\sin{k_y}\sigma_y + (M-4B+2B[\cos{k_x} +...

How come a+a- ψn = nψn ? This is eq. 2.65 of Griffith, Introduction to Quantum Mechanics, 2e. I followed the previous operation from the following analysis but I cannot get anywhere with this statement. Kindly help me with it. Thank you for your time.

I am self studying the Book- Introduction to Quantum Mechanics , 2e. Griffith. Page 47. While the book has given a proof for eq. 2.64 but its not very ellaborate Integral(infinity,-infinity) [f*(a±g(x)).dx] = Integral(infinity,-infinity) [(a±f)* g(x).dx] . It would be great help if somebody...

Lagrangian is defined by ##L=L(q_i,\dot{q}_i,t)## and hamiltonian is defined by ##H=H(q_i,p_i,t)##. Why there is relation H=\sum_i p_i\dot{q}_i-L end no H=L-\sum_i p_i\dot{q}_i or why ##H## is Legendre transform of ##-L##?

Homework Statement The red box only Homework EquationsThe Attempt at a Solution I suppose we have to show L_3 (Π_1) | E,m> = λ (Π_1) | E,m> and H (Π_1) | E,m> = μ (Π_1) | E,m> And I guess there is something to do with the formula given? But they are in x_1 direction so what did they have...

Hi! I'm having some trouble on understanding how the Hamiltonian of the e-m field in the single mode field quantization is obtained in the formalism proposed by Gerry-Knight in the book "Introductory Quantum Optics". (see...

Homework Statement Finding eigenvalues of an hamiltonian Homework EquationsH = a S²z + b Sz (hbar = 1) what are the eigenvalues of H in |S,M> = |1,1>,|1,0>,|1,-1> The Attempt at a SolutionH|1,1> = (a + b) |1,1> H|1,0> = 0 H |1,-1> = (a-b) |1,-1> which gives directly the energy : a+b , 0 ...

Take a spin-1/2 particle of mass ##m## and charge ##e## and place it in a magnetic field in the ##z## direction so that ##\mathbf B=B\mathbf e_z##. The corresponding Hamiltonian is $$\hat H=\frac{eB}{mc}\hat S_z.$$ This must have units of joules overall, and since the eigenvalues of ##\hat S_z##...

Homework Statement Consider the free real scalar field \phi(x) satisfying the Klein-Gordon equation, write the Hamiltonian in terms of the creation/annihilation operators. Homework Equations Possibly the definition of the free real scalar field in terms of creation/annihilation operators...

I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...

In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that : In Hamiltonian formulation, there can be no constraint equations among the co-ordinates. Why is this necessary ? Any simple example which will elaborate this fact ? But in Lagrangian...

Homework Statement Consider an experiment on a system that can be described using two basis functions. We begin in the ground state of a Hamiltonian H0 at a time t1, then rapidly change the hamiltonian to H1 at the time t1. At a later time tD>t1 you preform a measurement of an observable D...

I'm working on this problem "Consider an experiment on a system that can be described using two basis functions. In this experiment, you begin in the ground state of Hamiltonian H0 at time t1. You have an apparatus that can change the Hamiltonian suddenly from H0 to H1. You turn this apparatus...

I am familiar with the usual solution of the hydrogen atom using the associated legendre functions and spherical harmonics, but my question is: is it possible to extend the hamiltonian of the hydrogen atom to naturally encompass half integer spin? My guess is that spin only pops in naturally in...

The system in which I tried to calculate the Hamiltonian matrix was a particle in a stadium (Billiard stadium). And I used the principle where we take a rectangle around the stadium in which the parts outside the stadium have a very high potential V0. We know the wave function of a rectangular...

Hi! I'm trying to understand how to diagonalize a Hamiltonian numerically. Basically I have a problem with a Hamiltonian such as H = \frac{1}{2}c^{\dagger}\textbf{H}c where c = (c_1,c_2,...c_N)^T The dimensions of the total Hamiltonian are 2N, because each c_i is a 2 spinor. I need to...

Hey, I just had a quick question about using hamiltonians to determine energy levels. I know that the eigenvalue of the hamiltonian applied to an eigenket is an energy level. H |a> = E |a> But my question is if I am given an equation for a specific Hamiltonian: H = (something arbitrary) And...

In classical mechanics, the Hamiltonian and the Lagrangian are Legendre transforms of each other. By analogy, in quantum mechanics and quantum field theory, the relationship between the Hamiltonian and the Lagrangian seems to be preserved. Where can I find a derivation of the Lagrangian...

Graphene's Hamiltonian contains first order derivatives (from the momentum operators) which aren't invariant under simple spatial rotations. So it initially appears to me that it isn't invariant under rotation. From reading around I see that we also have to perform a rotation on the Pauli...

If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way?

Homework Statement Given \hat{P}_r\psi=-i\hbar\frac{1}{r}\frac{\partial}{\partial{r}}(r\psi), show \hat{H}=\frac{1}{2m}(\hat{P}^2_r+\frac{\hat{L}^2}{r^2}) Homework EquationsThe Attempt at a Solution The solution starts out with...

I am reading Cohen-Tannoudji's Atom photon interactions (2004 version), in the Appendix he explains that for atom-light interaction, the electric dipole Hamiltonian (d.E form) is got from the original, "physical" (in line with his language) p.A form Hamiltonian by a time-independent unitary...

Homework Statement Consider a charge ##q##, with mass ##m##, moving in the ##x-y## plane under the influence of a uniform magnetic field ##\vec{B}=B\hat{z}##. Show that the Hamiltonian $$ H = \frac{(\vec{p}-q\vec{A})^2}{2m}$$ with $$\vec{A} = \frac{1}{2}(\vec{B}\times\vec{r})$$ reduces to $$...

Imagine I have a complicated second-order differential equation that I strongly suspect can be derived from a Hamiltonian (with additional momentum dependence beyond p2/2m, so the momentum is not simply mv, but I don't know what it is). Are there any ways to test whether or not the given...

I am given a hamiltonian for a two electron system $$\hat H_2 = \hat H_1 \otimes \mathbb {I} + \mathbb {I} \otimes \hat H_1$$ and I already know ##\hat H_1## which is my single electron Hamiltonian. Now I am applying this to my two electron system. I know very little about the tensor product...

Yesterday I was asking questions from someone and in between his explanations, he said that the Euclidean action in a QFT is actually equal to its Hamiltonian. He had to go so there was no time for me to ask more questions. So I ask here, is it true? I couldn't find anything on google. If its...

Hello, I am stuck at the beginning of an exercise because I have some trouble to understand how are the energy level in this problem : In a crystal we have Ni2+ ions that we consider independent and they are submitted to an axial symmetry potential. Each ion acts as a free spin S=1. We have the...

Hello everybody, The general expression of molecular Hamiltonian operator for any molecule is: $$\hat{H} = \hat{T}_n+\hat{T}_e+\hat{V}_{ee}+\hat{V}_{nn}+\hat{V}_{en}+\hat{f}_{spin-orbit} $$ where: ##\hat{T}## correspond to kinetic energy operator ##\hat{V}## correspond to potential energy...

Homework Statement I'm currently working on a homework set for my intermediate QM class and for some reason I keep drawing a blank as to what to do on the first problem. I'm given three potentials, V(x), the first is of the form {A+Bexp(-Cx^2)}, the others I'll leave out. I'm asked to draw the...

basically, as far as I know we can derive 1/2mv2 from ∫F⋅ds=1/2mv2=(p2)/2m for wave equation we use Hamiltonian H=P2/2m+V where P and V are both operators However, I wonder how we can say that P2/2m is the term for kinetic energy because ∫F⋅ds=∫(dp/dt)⋅ds=1/2mv2 is saying that knowing F and...

Hi everyone, Currently, I am self-learning Renormalization and its application to PDEs, nonequilibrium statistical mechanics and also condensed matter. One particularly problem I face is on the conservation of symmetry of hamiltonian during renormalization. Normally renormalization of...

I am trying to figure out how to get the Hamiltonian for a mass on a fixed smooth hemisphere. Using Thorton from example 7.10 page 252 My main question is about the Potential energy= mgrcosineθ is the generalized momenta Pdotθ supposed to be equal to zero because θ is cyclic? Or is Pdotθ=...

Hello everybody, From a complete set of orthogonal basis vector ##|i\rangle## ##\in## Hilbert space (##i## = ##1## to ##n##), I construct and obtain a nondiagonal Hamiltonian matrix $$ \left( \begin{array}{cccccc} \langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\...

In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...

Hi, I am a physics student and i was asked to answer some questions about Hydrogen atom wavefunctions. I hope you can help me (sorry for my english, is not my motherlanguage, i will try to explain myself properly) 1. In order to find hamiltonian eigenfunctions of Hydrogen atom, we make then be...

The permutation operator commutes with the Hamiltonian when considering identical particles, which implies: $$ [\hat{P}_{21}, \hat{H}] = 0 \tag{1}$$ Now given a general eigenvector ##{\lvert} {\psi} {\rangle}##, where $$ \hat{P}_{21} (\hat{H}{\lvert}{\psi}){\rangle} = (\hat{P}_{21} \hat{H})...

Homework Statement How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x) The attempt at a solution H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x) I know how to find the matrix of the normal...

## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density for a real scalar field in 4-d, ##u=0,1,2,3 = t,x,y,z##, below ##i = 1,2,3 =x,y,z## In order to compute the Hamiltonian I first of all need to compute...

This isn't exactly homework, but something which you'd get as an assignment, so I'll still post it here in order to reach the right people.. I'm attempting to freshen up my knowledge on Hamiltonian systems, so I've tried to formulate the Kepler problem in Hamiltonian dynamics...

Homework Statement Working through problems in Mahan's 'Many Particle Physics' book, and at the end of the 1st chapter there's a question where we're asked to consider a fermion system with three energy states with eigenvalues E1, E2, E3, and matrix elements M12, M23, M13 which connect them and...

Hi folks, I am looking to learn the Lagrangian and Hamiltonian approach to celestial mechanics - I have previous experience in Newtonian numerical solutions for orbital motion but am looking to achieve similar things but through the use of Hamiltonian formulations. After having a poke around...

The path integral in quantum mechanics involves a factor ##e^{iS_{N}/\hbar}##, where ##S_{N}\equiv \sum\limits_{n=1}^{N+1}[p_{n}(x_{n}-x_{n-1})-\epsilon H(p_{n},x_{n},t_{n})].## In the limit ##N \rightarrow \infty##, ##S_{N}## becomes the usual action ##S## for a given path.When the...

In chapter 11, Lancaster takes us through the 5 steps for canonical quantization of fields, and in example 11.3 he derives a mode expansion of the Hamiltonian which ends in this: $$E=\int d^3 p E_p (a _p^{\dagger} a_p + \frac{1}{2} \delta^{(3)}(0)) $$ Which I have no problem with, but then...