Hamiltonian Definition and 899 Threads

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. A

    Does the addition of a new vertex always guarantee a Hamiltonian path or cycle?

    Watching this lecture from nptel about NP completeness: http://www.youtube.com/watch?v=76n4BjlL1cs&feature=player_embedded -We have an algorithm, HC , which given a graph tells us whether or not it has a hamiltonian cycle. -We want to use it in order to create an algorithm that determines...
  2. andrewkirk

    Why ignore the potential term in the quantum Hamiltonian?

    I am reading about the recovery of some classical rules from quantum mechanics. My text (Shankar) considers a Hamiltonian operator in a one-dimensional space H = P^2 / 2m + V(X) where P and X are the momentum and position operators respectively. It then asserts that [X,H] = [X,P^2/2m]...
  3. T

    Solving Eigenvalues and Eigenfunctions of Hamiltonian

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  4. L

    Tight Binding Hamiltonian for Graphene

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  5. andrewkirk

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  6. A

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  7. V

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  8. M

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  9. W

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  10. M

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  11. T

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  12. L

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  13. fluidistic

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  14. fluidistic

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  15. C

    How do constraints affect the Hamiltonian in mean-field approximation?

    Please correct me if I make any mistakes along the way. Suppose we have a simple tight-binding Hamiltonian H=\sum_i \epsilon _i c_i^\dagger c_i - t\sum_{\langle i j\rangle} c_i^\dagger c_j + h.c., In half-filling systems, we tend to impose a constraint such that each site has only one...
  16. A

    Commutator of the Hamiltonian with Position and Hamiltonian with Momentum

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  17. J

    Why Does the Free-Field Dirac Hamiltonian Calculation Seem Incorrect?

    Homework Statement This is a simple problem I thought of and I'm get a nonsensical answer. I'm not sure where I'm going wrong in the calculation. Find the value of <-,p',v';+,q',r'|H|-,p,v;+,q,r> where H is the free-field Dirac Hamiltonian H =...
  18. B

    Maths of Hamiltonian / Lagrangian mechanics

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  19. 1

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  20. K

    Lagrangian, Hamiltonian and Legendre transform of Dirac field.

    In most of the physical systems, if we have a Lagrangian L(q,\dot{q}), we can define conjugate momentum p=\frac{\partial L}{\partial{\dot{q}}}, then we can obtain the Hamiltonian via Legendre transform H(p,q)=p\dot{q}-L. A important point is to write \dot{q} as a function of p. However, for the...
  21. M

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  22. K

    The Jaynes-Cummings Hamiltonian

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  23. D

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  24. R

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  25. Simfish

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  26. B

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  27. S

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  28. Y

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    I am having trouble typing the Hamiltonian symbol into latex. I found the symbol in the http://mirrors.med.harvard.edu/ctan/info/symbols/comprehensive/symbols-a4.pdf" , however, I had some difficulties in installing the font and stuff. I am using tex-live. I got the following error...
  29. A

    Difference between Hamiltonian operator and Total energy operator?

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  30. Z

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    When dealing with the Euler-Lagrange equation in a physical setting, one usually uses the Hamiltonian L=T-V as the value to be extremized. What is the physical interpretation of the extremizing of this value?
  31. Z

    What Is the Hamiltonian for a Wire Coil Under Reflection Transformation?

    When applying Noether's theorem to a coil of wire under reflection transformation invariance, what Hamiltonian would one use as as the extermized function? I realize that electromagnetism is not invariant over reflection transformations, that's what I am trying to prove.
  32. N

    How Does a Fluctuating Hamiltonian Affect the Expectation Value of Sx?

    Homework Statement Hi Say I have a Hamiltonian given by H = δSz acting on my system, where δ is a random variable controlled by some fluctuations in my environment. I have to show that if I start out with <Sx>=½, then the Hamiltonian will reduce <Sx> to <Sx> = ½<cos(δt)> where the <>...
  33. P

    Hamiltonian of Van der Pol Oscillator

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  34. K

    How do I know if the Hamiltonian is constant?

    Lets say H = \frac{m}{2} (\dot{Q}^2 - \omega^2 Q^2 ) where Q is the generalized coordinate. It doesn't explicitly depend on time, but the Q and the \dot{Q} does. If i differentiate it with respect to time it should be zero if it's constant, right? So i guess my question is should i treat the Q's...
  35. C

    Clarifying Open String Hamiltonian for Witten's Book

    trying to get the open string hamiltonian I use H=\int\,d\sigma(\dot{X}.P_{\tau}-L)=\frac{T}{2}\int(\dot{X}^{2}+X'^{2})d\sigma as in Witten´s book, but we are integrating the Virasoro constraint equal to zero. So, Is not the Hamiltonian zero? Please, clarifyme this equation.
  36. J

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  37. P

    Definition of Heisenberg Hamiltonian

    I have a question. What is the definition of Heisenberg hamiltonian? \hat{H}=-\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j or \hat{H}=-2\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j or \hat{H}=\sum_{i,j}J_{i,j}\hat{\bfs{S}}_i\cdot \hat{\bfs{S}}_j or...
  38. U

    What is a Hamiltonian vector field in General Relativity?

    I'm researching General Relativity and have stumbled upon a bit of Hamiltonian mechanics. I roughly understand the idea behind the Hamiltonian of a system, but I'm utterly confused as to what the hell a Hamiltonian vector field is. I've taken ODE's, PDE's, Linear Algebra, and I'm just being...
  39. M

    Understanding Hamiltonian Field Equations and Their Applications in Field Theory

    Hi there, physics lovers. I'm studying field theory. So far, so well. I got it with the lagrangian density. I understood it. But then I DIDN'T FIND stuff about the Hamiltonian density. I couldn't find anything in Landau-Lifgarbagez series, and that makes me worry. I've been looking in the...
  40. S

    Converging the Hamiltonian in Atomic Units?

    Homework Statement So the question is I have to use some trial function of the form \sum c_if_i to approximate the energy of hydrogen atom where f_i=e^{-ar} for some number a (positive real number). Note that r is in atomic unit. Homework Equations Because r is in atomic unit, I think I should...
  41. G

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    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...
  42. V

    Find Hamiltonian Value of 4x4 Matrix

    could you help me how to find the value of the attached 4x4 matrix.Could you give me the idea or which method i have to follow to get the value of that matrix.
  43. U

    What Are the Eigenvalues of a Hamiltonian with a Linear Term?

    Homework Statement Consider the Hamiltonian \hat{}H = \hat{}p2/2m + (1/2)mω2\hat{}x2 + F\hat{}x where F is a constant. Find the possible eigenvalues for H. Can you give a physical interpretation for this Hamiltonian? Homework Equations The Attempt at a Solution I don't think...
  44. D

    Definition of Hamiltonian Density & Deriving Energy Current

    Hi. In elementary quantum mechanics the continuity equation is used to derive the electron current, i.e. \frac{\partial \rho(\mathbf r,t)}{\partial t}+\nabla\cdot\mathbf j(\mathbf r,t)=0 and one then puts \rho(\mathbf r,t)=\psi^*(r,\mathbf t)\psi(\mathbf r,t). Now if I want to derive...
  45. R

    What is the Mistake in Deriving the EM Hamiltonian?

    For some reason I can't derive the Hamiltonian from the Lagrangian for the E&M field. Here's what I have (using +--- metric): \begin{equation*} \begin{split} \mathcal L=\frac{-1}{4}F_{ \mu \nu}F^{ \mu \nu} \\ \Pi^\mu=\frac{\delta \mathcal L}{\delta \dot{A_\mu}}=-F^{0 \mu} \\...
  46. M

    Constructing Eigenstate for a given hamiltonian

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  47. N

    Hamiltonian does NOT equal energy?

    Hello, Just to be sure: is the following correct? Imagine a long rod rotating at a constant angular speed (driven by a little motor). Now say there's a small ring on the rod that can move on the rod without friction. The ring is then held onto the rod by an ideal binding force (I don't...
  48. J

    About the Schrodinger equation, the Hamiltonian, time evolution?

    Forgive me if this is a poorly asked question but I am not yet completely fluent in quantum mechanics and was just looking at the energy eigenvalue equation H|\Psi\rangle = i\hbar \frac{\partial}{\partial t}|\Psi\rangle = E|\Psi\rangle . We've got the Hamiltonian operator H acting on the state...
  49. Peeter

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  50. B

    Restrictions on Numbers in Hermitian Hamiltonian Equation

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