What is Hamiltonian: Definition and 895 Discussions
In the mathematical field of graph theory, a Hamiltonian path (or traceable path) is a path in an undirected or directed graph that visits each vertex exactly once. A Hamiltonian cycle (or Hamiltonian circuit) is a Hamiltonian path that is a cycle. Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem, which is NP-complete.
Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron. Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). This solution does not generalize to arbitrary graphs.
Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman, who, in particular, gave an example of a polyhedron without Hamiltonian cycles. Even earlier, Hamiltonian cycles and paths in the knight's graph of the chessboard, the knight's tour, had been studied in the 9th century in Indian mathematics by Rudrata, and around the same time in Islamic mathematics by al-Adli ar-Rumi. In 18th century Europe, knight's tours were published by Abraham de Moivre and Leonhard Euler.
Good evening,
unfortunately I can't get to the solution of my task
I wrote for the system:
## \frac{dz}{dt} = \nabla_p H ; \\
\frac{dp}{dt} = - \nabla_z H ##
Then the solution would be (as ## \nabla_p H = 0) ##:
## \frac{dz}{dt} = 0 \Rightarrow z = const. ## and ## p = zt + p_0 ##.
But...
I have read that if one measures the Hamiltonian and receives a value of h2, then the quantum state will be in ##|h2\rangle##. Finding the probability of a1 is done by projecting ##|a1\rangle## upon ##|h2\rangle## divided by ##\langle h2|h2\rangle##. In other words: $$\frac{|\langle...
Hello, I try to solve this problem, and I think a) wasn't too hard, I have the following solution:
##H = \lambda (\frac{\vec{S^2-(\vec{S_1}^2+\vec{S_2}^2)}{2})##.
I struggle with 2. I find it very abstract. When I have H as a matrix I know how to calculate eigenvalues, but I don't know how...
The Hamiltonian for a scalar field contains the term
$$\int d^3x m^2 \phi(x) \phi(x)$$, does it changs to the following form?
$$\int d^3x' {m'}^2 \phi'(x') \phi'(x')=\int d^3x' \gamma^2{m}^2 \phi(x) \phi(x)$$? As it is well known for a scalar field: $$\phi'(x')=\phi(x)$$ .
Homework Statement: Practical examples of Hamiltonian Mechanics sought
Relevant Equations: Hamilton Jacobi Equations, MTW
Hi,
I'm currently a bit stuck on Box 24.2 in MTW. I really need to get a better understanding of Hamiltonian Mechanics to be able to work my way through this and I...
So first I derived the expressions for the dynamics of the spin operators and got:
$$ \frac{d\hat{S}_y}{dt} = w\hat{S}_x^H $$
$$ \frac{d\hat{S}_x}{dt} = w\hat{S}_y^H $$
$$ \frac{d\hat{S}_z}{dt} = 0 $$
Now I want to calculate the time-dependence of the expectation values of the spin operators...
Consider the Heisenberg picture Hamiltonian $$H(t) = \int_{\textbf{r}}\psi^{\dagger}(\textbf{r},t)\frac{(-i\hbar\nabla+e\textbf{A})^{2}}{2m}\psi(\textbf{r},t)$$ where ##\psi(\textbf{r},t)## is a fermion field operator. To find the equations of motion that ##\psi,\psi^{\dagger}## obey. I would...
Hello, I am fairly new to the world of molecular spectroscopy, so I apologize for any ignorance on my part. For the last few months, I've been working on a diatomic spectral simulation tool and have reached a point where I want to incorporate more advanced theory to model complex interactions in...
We studied about the time translation operator and that its generator is the hamiltonian the question is could there be a time rotation operator in analogy with rotations in space and what would be it's relation to relativity?
So first I rewrote H as a matrix:
$$ H =
\begin{pmatrix}
a & b \\
b & c
\end{pmatrix} $$
And tried to find the eigenvalues/energies of H, so I solved
$$ det (H - \lambda I ) =
\begin{vmatrix}
a-\lambda & b \\
b & c-\lambda
\end{vmatrix} = (a-\lambda)(c-\lambda) - b^2 = ac - a\lambda -...
Hi,
In David Morin's "Introduction to classical mechanics", Problem 6.8, when deriving Hamiltonian of the bead rotating on a horizontal stick with constant angular speed, the Lagrangian derivative over angular speed isn't included.
Why is that?
Specifically, the Lagrangian takes form...
Hello! My question is related to going from Eq. 32 to Eq. 33 in this paper (however I have seen this in other papers, too). In summary, starting with:
$$H \propto \bar{e}\gamma_\mu\gamma_5 e \bar{q}\gamma^\mu q$$
where we have the gamma matrices, e is the electron field and q is the...
Hi all,
This should be a simple question but it has been bothering me for a bit:
Consider 2 Hamiltonian terms ##H_{1},H_{2}## that satisfy ##[H_{1},H_{2}] = 0##. Suppose we are working in the Heisenberg picture and we time evolve some operator ##A## according to ##A(t) =...
Hi all,
When working in the Heisenberg picture, we can represent implementing time evolution on an operator via a Hamiltonian H through a quantum circuit type picture like the following:
where time is on the vertical axis and increases going up and the block represents the unitary gate...
TL;DR Summary: Find the initial state of a two-level quantum system, given the probability of measurements for two observables and the expected value of an operator.
Dear PFer's,
I have been struggling with the following problem. It was assigned at an exam last year.
Problem Statement
For a...
I'm looking for books (or any other reference) to start studying the emergence of Hamiltonian chaos and KAM theory. You know, something that doesn't require a Ph.D in math to understand but is comprehensive enough to give a good understanding of the topic. Added bonus if it has a discussion on...
In a system with two orbitals ##c## and ##d## (each with two spin degrees of freedom), consider the Hamiltonian ##H=V(d^{\dagger}_{\uparrow} c_{\uparrow} + c^{\dagger}_{\uparrow}d_{\uparrow}+d^{\dagger}_{\downarrow} c_{\downarrow} + c^{\dagger}_{\downarrow}d_{\downarrow})##. Also suppose that...
In a classical example, for a system consisting of a mass attached to a spring mounted on a massless carriage which moves with uniform velocity U, as in the image below, the Hamiltonian, using coordinate q, has two terms with U in it.
But if we use coordinate Q, ##Q=q−Ut##, which moves with the...
Hey all,
I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian:
$$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$
where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p}...
Hi,
I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian.
However, I don't see how to find the eigenvectors.
To find the eigenvalues for the first excited state I build this matrix
##...
I'm reading through Hohenberg's seminal paper titled: "Inhomogeneous Electron Gas" that help lay the foundation for what we know of as Density Functional Theory (DFT) by proving the existence of a universal functional that exactly matches the ground-state energy of a system with a given...
I am trying to find the expected value of the variance of energy in coherent states. But since the lowering and raising operators are non-hermitian and non-commutative, I am not sure if I am doing it right. I'm pretty sure my <H>2 calculation is right, but I'm not sure about <H2> calculation...
Hi all
I was reading a certain paper that involves solving the Bohr-Mottelson Hamiltonian for a 5dimential square well potential, the B-M Hamiltoian reads:
my question is just how do I calculate the mass parameter "B" for a certain nuclei, and with a 5D infinite potential well how do I get the...
Hi,
I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it.
From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle##
Also, a linear combination of all states is also an solution.
If U commutes...
Hi,
In my book I have and expression that I don't really understand.
Using the definition of action ##\delta S = \frac{\partial L}{\partial \dot{q}} \delta q |_{t_1}^{t_2} + \int_{t_1}^{t_2} (\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}) \delta q dt##
Where L...
I am stuck on Question e and then how to proceed to f. I cannot seem to show this using the steps in the prior questions. My answers are:
a)
b)
c)
c) continued - and d) at the bottom of the page
d)I am not sure where I have gone wrong, as I am not sure how to apply the relevant...
A toy model of a QFT lattice (in 1 dimension) is given in [here] (at 5:55):
We assume that ##\Psi## is a vector set of four complex numbers having some values at every point on the grid, for instance:
$$\Psi_{100} =
\begin{bmatrix}
1+2i \\
3+4i \\
5+6i \\
7+8i
\end{bmatrix}$$
and...
Consider the interaction of a two level atom and an electric field (semiclassically, we treat the field as 'external' i.e. not influenced by the atom; the full quantum treats the change in the field as well)
Electric field in semiclassical Hamiltonian: plane wave
##H_{int,~semiclassical}=-\mu...
Hi,
I believe that I have an acceptable level of understanding where SRT, GRT, QM and QFT come from. This is not true for me regarding the "good old stuff".
Newton, okay, this is relatively (:wink:) clear to me but do you know something about the historical motivation for Lagrangian and...
Source: Scully and Zubairy, Quantum Optics, Section 1.1.2 Quantization
Questions:
1. Why are the destruction and creation operators considered a canonical transformations?
2. If these are canonical transformations, does it suggest that we are also canonically transforming the Hamiltonian...
I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation:
$$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$
where they...
I think that the quantum numbers are l=1 and ml=0, so I write the spherical harmonic Y=Squareroot(3/4pi)*cos(theta).
I would like to know how to compute the wave function at t=0, then I know it evolves with the time-evolution operator U(t), to answer the first request.
Suppose some quantum system has a Hamiltonian with explicit time dependence ##\hat{H} := \hat{H}(t)## that comes from a changing potential energy ##V(\mathbf{x},t)##. If the potential energy is changing slowly, i.e. ##\frac{\partial V}{\partial t}## is small for all ##\mathbf{x}## and ##t##...
Greentings,
I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand.
When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like
$$
\begin{align}
H_D S_F(x-y) &= (i \hbar...
If I understand it correctly, the Hamiltonian represents the total energy of the system. Can it be non-local? If yes, doesn't this contradict relativistic locality?
On ***page 38*** of Becker Becker Schwarz, we're given ***equation 2.69*** which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2})$$
Considering the open string we have...
Hi all,
I am currently reading through this paper: https://iopscience.iop.org/article/10.1088/1367-2630/10/4/045030
and would like to reproduce their results for N=5.
My roadblock is with (9), which models the classical motion of the system. Now symbolically finding the eigenstates of the matrix...
In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##.
For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##.
Can i consider ##\theta## or ##cos...
I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators...
Hello.
As an assignment, I have to explain the total molecular Hamiltonian. Problem is, I can't find it anywhere in my book (Atkins, Physical Chemistry: Quanta, Matter, and Change, 2nd Edition), even when I access the index for "Hamiltonian -> polyatomic molecules". They do give the electronic...
Hey folks,
I'm looking for a derivation of the secular approximation of the dipole-dipole Hamiltonian at high magnetic fields. Does anybody know a reference with a comprehensive derivation or can even provide it here?
Given we have the dipolar alphabet, I'd like to understand (in the best...
Hi all,
I'm doing some light simulations for an experiment I'm going to be running soon. I've ran through the math symbolically on paper but I'm not exactly eager for handling this large of matrices by hand so I'm trying to work through it and see if I can generate a simulated signal to compare...
I'd like to know if this Hamiltonian ##\hat{H}=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2+\frac{A}{\hbar^2}(J^2-L^2-S^2)## is separable into two parts ##H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2## and ##H_2=\frac{A}{\hbar^2}(J^2-L^2-S^2)## and ##[H_1,H_2]=0##. Here A is a constant. I did so...
In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...
Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.
In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...
In a quantum mechanical exercise, I found the following Hamiltonian:
Consider a particle of spin 1 constrained to move on the surface of a sphere of radius R with Hamiltonian ##H=\frac{\omega}{\hbar}L^2##. I knew that the Hamiltonian of a particle bound to move on the surface of a sphere was...
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...
To calculate the Hamiltonian of a charged particle immersed in an electromagnetic field, one calculates the Lagrangian with Euler's equation obtaining ##L=\frac{1}{2}mv^2-e\phi+e\vec{v}\cdot\vec{A}## where ##\phi## is the scalar potential and ##\vec{A}## the vector potential, and then we go to...
We have commutation relation ##[J_j,J_k]=i \epsilon_{jkl}J_l## satisfied for ##2x2##, ##3x3##, ##4x4## matrices. Are in all dimensions these matrices generate ##SO(3)## group? I am confused because I think that maybe for ##4x4## matrices they will generate ##SO(4)## group. For instance for...