# 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.

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1. ### Solve this Hamiltonian System in Several Ways

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...

11. ### I Hamiltonian of the bead rotating on a horizontal stick

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...
12. ### A Confused about going from relativistic to non-relativistic Hamilonian

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...
13. ### I Interpreting Unitary Time Evolution

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) =...
14. ### I Quantum Circuit Confusion On Time Evolution

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...
15. ### Two-level Quantum System - initial state

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...
16. ### Classical Introductory books to Hamiltonian chaos

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...
17. ### I The Hamiltonian elements in Anderson dimer

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...
18. ### I The Hamiltonian and Galilean transformations

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...

35. ### B What is a non-local Hamiltonian?

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?
36. ### How can negative integers be used in deriving the Hamiltonian for open strings?

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...
37. ### I Numerical Calculation of Hamiltonian Overlaps

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...
38. ### I Hamiltonian formalism and partition function

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...
39. ### I Strange Hamiltonian of two particles on the surface of a sphere

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...
40. ### The total molecular Hamiltonian

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...
41. ### A Secular Approximation of Dipole-Dipole Hamiltonian

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...
42. ### Mathematica Piecewise Time-Dependent Hamiltonian in Mathematica Strategy

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...
43. ### I Separability of a Hamiltonian with spin

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...
44. ### I Separable Hamiltonian for central potential

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...
45. ### I How to get the energy eigenvalue of the Hamiltonian: H0+λp/m ?

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.
46. ### I Deriving the Commutator of Exchange Operator and Hamiltonian

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...
47. ### I Hamiltonian of a particle moving on the surface of a sphere

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...
48. ### 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...
49. ### I How to obtain Hamiltonian in a magnetic field from EM field?

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...
50. ### A SO(3) group, Heisenberg Hamiltonian

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...