What is Hamiltonian matrix: Definition and 26 Discussions

In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix










{\displaystyle J={\begin{bmatrix}0&I_{n}\\-I_{n}&0\\\end{bmatrix}}}
and In is the n-by-n identity matrix. In other words, A is Hamiltonian if and only if (JA)T = JA where ()T denotes the transpose.

View More On Wikipedia.org
  1. AzAlomar

    A Reference for empirical Tight-binding Hamiltonian of spds* vs sps*

    Is there a clear reference article/note for the 20X20 Hamiltonian matrix of the spds* Zinc-Blende system similar to the sps* reference in [1] Table (A) of Vogl P, Hjalmarson HP, Dow JD. A Semi-empirical tight-binding theory of the electronic structure of semiconductors†. J Phys Chem Solids...
  2. M

    Spin-orbit coupling and the Zeemann effect

    Homework Statement Consider an electron in a hydrogen atom in the presence of a constant magnetic field ##B##, which we take to be parallel to the ##z##-axis. Without the magnetic field and ignoring the spin-orbit coupling, the eigenfunctions are labelled by ##\vert n, l, m, m_s \rangle##...
  3. CharlieCW

    Eigenvalues dependent on choice of $\vec{A}$?

    Homework Statement A particle with spin s=1/2 moves under the influence of a magnetic field given by: $$\vec{A}=B(-y,0,0)$$ Find the eigenvalues of the corresponding Pauli hamiltonian. Repeat the same process for: $$\vec{A}=\frac{B}{2}(-y,x,0)$$ Explain your result by relating the...
  4. F

    Calculating eigenvectors/values from Hamiltonian

    Homework Statement I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point. For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube. I'm then constructing the Hamiltonian...
  5. DeathbyGreen

    Mathematica Eigenvectors 4x4 Matrix in Mathematica

    Hi, I'm trying to calculate the eigenvectors of a 4x4 matrix, but I don't want the actual eigenvalues included in the solution, I simply want them listed as a variable. For example, I have the matrix: H_F = \left[ \begin{array}{cccc} \hbar\Omega&\hbar v_fk_- &0&0\\ \hbar...
  6. DeathbyGreen

    A Eigenvectors of a Floquet Hamiltonian

    I'm trying to recreate some results from a paper: https://arxiv.org/pdf/1406.1711.pdf Basically they take the Hamiltonian of graphene near the Dirac point (upon irradiation by a time periodic external field) and use Floquet formalism to rewrite it in an extended Hilbert space incorporating...
  7. U

    Hamiltonian operator affecting observable

    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...
  8. P

    I Calculating Hamiltonian matrix elements in a chaotic system

    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...
  9. Konte

    I Hamiltonian matrix - Eigenvectors

    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 \\...
  10. L

    Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

    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...
  11. M

    How to Find Ψ(x,t) for a Given Hamiltonian Matrix and Initial State?

    Homework Statement I have the matrix form of the Hamiltonian: H = ( 1 2-i 2+i 3) If in the t=0, system is in the state a = (1 0)T, what is Ψ(x,t)? Homework Equations Eigenvalue equation The Attempt at a Solution So, I have diagonalized given matrix and got...
  12. N

    Hamiltonian matrix for two electrons in a 1D infinite well

    Hi everyone, I need help for preparing a Hamiltonian matrix. What will be the elements of the hamiltonian matrix of the following Schrodinger equation (for two electrons in a 1D infinite well): -\frac{ħ^{2}}{2m}(\frac{d^{2}ψ(x_1,x_2)}{dx_1^{2}}+\frac{d^{2}ψ(x_1,x_2)}{dx_2^{2}}) +...
  13. B

    How Potts model hamiltonian is equal to hamiltonian matrix

    /How can I show that Potts model hamiltonian is equal to this matrix hamiltonian? Potts have these situations : { 1 or 1 or 1 or 0 or 0 or 0} but the matrix hamiltonian : { 1 or 1 or 1 or -1/2 or -1/2 or -1/2} I take some example and couldn't find how they can be equal.
  14. T

    Hamiltonian matrix off diagonal elements?

    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...
  15. Roodles01

    Hamiltonian matrix and eigenvalues

    OK. An example I have has me stumped temporarily. I'm tired. General spin matrix can be written as Sn(hat) = hbar/2 [cosθ e-i∅sinθ] ...... [[ei∅sinθ cosθ] giving 2 eigenvectors (note these are column matrices) I up arrow > = [cos (θ/2)] .....[ei∅sin(θ/2)] Idown arrow> =...
  16. S

    Feynman clock's Hamiltonian matrix reduction

    Homework Statement I have this 2^n*2^n matrix that represent the evolution of a system of $n$ spin. I know that I can have only one excited spin in my configuration a time. (eg: 0110 nor 0101 ar not permitted, but 0100 it is) s_+ is defined with s_x+is_y and s_- is defined with s_x-is_y...
  17. G

    Tight binding hamiltonian matrix

    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...
  18. N

    Hamiltonian matrix eigenvalue calculation

    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...
  19. R

    Finding 2x2 Hamiltonian Matrix for Second-Quantized Hamiltonian

    Homework Statement I need to find the 2x2 Hamiltonian matrix for the Hamiltonian, which is written in second-quantized form as below for a system consisting of the electrons and photons. H = h/ωb†b + E1a†1a1 + E2a†2a2 + Ca†1a2b† + Ca†2a1b, a's are creation and annihilation operator for...
  20. L

    A Starting with the Schrodinger equation, how do we find the Hamiltonian matrix?

    In his lectures on Quantum Physics, Richard Feynman derives the Hamiltonian matrix as an instantaneous amplitude transition matrix for the operator that does nothing except wait a little while for time to pass.(Chapter 8 book3) The instantaneous rate of change of the amplitude that the wave...
  21. M

    Computing Hamiltonian matrix for a 1-D spin chain.

    I'm not going to "follow the template provided" in the strictest sense - but I'm going to include all the same information expected - statement of the problem and a showing of how I've tried to do it, in the intended "spirit" of the template, since these different components are kind of "mixed"...
  22. W

    How to establish the hamiltonian matrix as soon as possible?

    in the bose-hubbard model, we need to enumerate all the possible basis usually, the basis vectors are taken as the fock states The problem is that, how to arrange the basis and how to establish the matrix of the hamiltonian as soon as possible It is apparent the the matrix will be very...
  23. R

    How to Label Spin Hamiltonian by Ms in EPR Experiments?

    dear members, My problem is... suppose take the spin Hamiltonian Hham=D[Sz2 -S(S+1)/3 +(E/D)(Sy2-Sy2)] +Hi\vec{S} (most often in EPR experiments, etc). here external magnetic field Hamiltonian Hi = \betagiBiext and i =x, y and z. Also gx=gy=gz=2 and the external magnetic field is...
  24. L

    Constructing Hamiltonian Matrix from Sz Basis States for Quantum Spin Chains

    Hallo! My question relates to the use of basis states to form operator matrices... In the context of quantum spin chains, where the Hamiltonian on a chain of N sites is defined periodically as:H = sumk=0N-1[ S(k) dot S(k+1) ] (apologies for the notation) so there is a sum over k=0 to N-1...
  25. H

    How to dertermine the Hamiltonian matrix

    Hi, guys, I do not know how to determine the Hamiltonian matrix of the following question with the basis of two stationary state. Pls give me some hints about it. Consider first a single Hydrogen atom, made up of a proton at some location A in space, and an electron. We assume that the...
  26. M

    Hamiltonian Matrix Eq. 8.43 Explained - Feynman III Quantum Mechanics

    In the volume III of R Feynman series which is on Quantum Mechanics , please explain to me the eq.8.43 given on page 1529, i know how we got the equation but the 2nd part of 1st equation (H12)C2, what does it mean ?