I am looking at page 2 of this document.https://ocw.mit.edu/courses/chemistry/5-04-principles-of-inorganic-chemistry-ii-fall-2008/lecture-notes/Lecture_3.pdf
How is the transformation matrix, ν, obtained? I am familiar with diagonalization of a matrix, M, where D = S-1MS and the columns of S...
I want to exactly diagonalize the following Hamiltonian for ##10## number of sites and ##5## number of spinless fermions
$$H = -t\sum_i^{L-1} \big[c_i^\dagger c_{i+1} - c_i c_{i+1}^\dagger\big] + V\sum_i^{L-1} n_i n_{i+1}$$
here ##L## is total number of sites, creation (##c^\dagger##) and...
Hi guys! I am starting to study Hubbard model with application in DFT and I have some doubts how to solve the Hubbard Hamiltonian. I have the DFT modeled to Hubbard, where the homogeneous Hamiltonian is
$$ H = -t\sum_{\langle i,j \rangle}\sigma (\hat{c}_{i\sigma}^{\dagger}\hat{c}_{j\sigma} +...
Homework Statement
Given the following quadric surfaces:
1. Classify the quadric surface.
2. Find its reduced equation.
3. Find the equation of the axes on which it takes its reduced form.
Homework Equations
The quadric surfaces are:
(1) ##3x^2 + 3y^2 + 3z^2 - 2xz + 2\sqrt{2}(x+z)-2 = 0...
I have the following matrix given by a basis \left|1\right\rangle and \left|2\right\rangle:
\begin{bmatrix}
E_0 &-A \\
-A & E_0
\end{bmatrix}
Eventually I found the matrix eigenvalues E_I=E_0-A and E_{II}=E_0+A and eigenvectors \left|I\right\rangle = \begin{bmatrix}
\frac{1}{\sqrt{2}}\\...
Firstly, thanks to everyone who participated in my last thread. It helped a lot! This will be the only other topic I can think of posting in physics forums, because, honestly, I don't know very much.
I remember sitting down one time and thinking I was quite brilliant when I started to make a...
Hi there,
This is a question about numerical analysis used particularly in the computational condensed matter or anywhere where one needs to DIAGONALIZE GIGANTIC DENSE HERMITIAN MATRICES.
In order to diagonalize dense Hermitian matrices size of 25k-by-25k and more (e.g. 1e6-by-1e6) it is not...
Hi!
I'm trying to understand how to diagonalize a Hamiltonian numerically. Basically I have a problem with a Hamiltonian such as
H = \frac{1}{2}c^{\dagger}\textbf{H}c
where c = (c_1,c_2,...c_N)^T
The dimensions of the total Hamiltonian are 2N, because each c_i is a 2 spinor. I need to...
Homework Statement
Not sure if this is advanced, so move it wherever.
A certain rigid body may be represented by three point masses:
m_1 = 1 at (1,-1,-2)
m_2 = 2 at (-1,1,0)
m_3 = 1 at (1,1,-2)
a) find the moment of inertia tensor
b) diagonalize the matrix obtaining the eigenvalues and the...
Given a system of linear differential equations $$x_{1}'=a_{11}x_{1}+a_{12}x_{2}+\cdots a_{1n}x_{n}\\ x_{2}'=a_{21}x_{1}+a_{22}x_{2}+\cdots a_{2n}x_{n}\\ \ldots\\ x_{n}'= a_{n1}x_{1}+a_{n2}x_{2}+\cdots a_{nn}x_{n}$$ this can be rewritten in the form of a matrix equation...
My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...
The problem asks for the diagonalization of (a(p^2)+b(x^2))^n, where x and p are position and momentum operators with the commutation relation [x,p]=ihbar. a and b are real on-zero numbers and n is a positive non-zero integer.
I know that it is not a good way to use the matrix diagonalization...
I don't really feel that I understand what it means for two matrices to be similar.
Of course, I understand the need to understand ideas on their own terms, and that in math analogies are very much frowned upon. In asking if you know of any "reasonable" analogies for what it means for two...
So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered
e^{At} \vec{u}(0) = \vec{u}(t)
as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
MIT OCW 18.06 Intro to Linear Algebra 4th edt Gilbert Strang
Ch6.2 - the textbook emphasized that "matrices that have repeated eigenvalues are not diagonalizable".
imgur: http://i.imgur.com/Q4pbi33.jpg
and
imgur: http://i.imgur.com/RSOmS2o.jpg
Upon rereading...I do see the possibility...
Mod note: I revised the code below slightly, changing the loop control variable i to either j or k. The reason for this is that the browser mistakes the letter i in brackets for the BBCode italics tag, which causes some array expressions to partially disappear.
Hello,
I am trying for the first...