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
J
=
[
0
I
n
−
I
n
0
]
{\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.
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...
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##...
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...
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...
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...
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...
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...
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...
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 \\...
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...
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...
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}}) +...
/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.
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...
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> =...
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...
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...
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 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...
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...
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"...
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...
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...
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...
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...
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 ?