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. J

    Help with Heisenberg's "isospin" Hamiltonian

    I'm having some trouble grasping the meaning of the exchange term in the Hamiltonian Heisenberg gives in his classic 1932 paper (the one typically given as the first to describe nucleons via a spin-like degree of freedom; NOTE: I realize this isn't the same as what is today called isospin, but...
  2. T

    How Do You Write the Hamiltonian in the Basis |\theta>?

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  3. T

    What is the difference between H and E in the equation Hψ = Eψ?

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  4. A

    Proof of relationship between Hamiltonian and Energy

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  5. kq6up

    Question About the Hamiltonian

    If the hamiltonian is defined as ##\mathcal{H}\equiv \sum _{ i }^{ all }{ p_{ i }\dot { q } _{ i } } - \mathcal{L} ##, how is it considered a function of ##p## and ##q## instead of ##p## and ##\dot{q}##? Chris
  6. Daaavde

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  7. BiGyElLoWhAt

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  8. C

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  9. A

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  10. M

    Does Adding a Constant to the Hamiltonian Affect Quantum System Dynamics?

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  11. B

    How Does a Magnetic Field Influence Quantum State Evolution?

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  12. N

    Hamiltonian mechanics: ∂H/∂t = ?

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  13. P

    Hamiltonian Operator: Difference vs. E?

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  14. C

    What are the main differences between Hamiltonian and Lagrangian mechanics?

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  15. K

    What is the name of this Hamiltonian?

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  16. Greg Bernhardt

    Hamiltonian: Definition, Equations & Explanation

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  17. ShayanJ

    Eigenvectors of this Hamiltonian

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  18. B

    Hamiltonian Noether's theorem in classical mechanics

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  19. L

    Is any Hamiltonian system integrable?

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  20. A

    Exceptation of energy by Hamiltonian

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  21. J

    Hamiltonian for classical harmonic oscillator

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  22. genxium

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  23. A

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  24. R

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  25. S

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  26. C

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  27. U

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  28. Q

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  29. W

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  30. E

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  31. JonnyMaddox

    Infinitesimal transformations and the Hamiltonian as generator

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  32. W

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  33. W

    Hamiltonian of two rotating and oscillating masses

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  34. B

    Lagrangian and Hamiltonian equations of motion

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  35. tom.stoer

    Hamiltonian formulation of QCD and nucleon mass

    Hello, there are several papers on QCD in Hamiltonian formulation, especially in Coulomb gauge. Unfortunately the Hamiltonian H is rather formel and highly complex. Question: is there a paper discussing the contribution of individual terms of H to the nucleon mass?
  36. I

    Changing the Hamiltonian without affecting the wave function

    How many ways can we change the Hamiltonian without affecting the wave functions (eigenvectors) of it. Like multiply all the elements in the matrix by a constant. I'm facing a very difficult Hamiltonian,:cry: I want to simplify it, so the wave function will be much easier to derive. Thanks in...
  37. C

    Harmonic oscillator Hamiltonian.

    I know I've seen this point about roots discussed somewhere but I can't for the life of me remember where. I'm hoping someone can point me the right direction. Here's the situation:- The standard derivation of the quantum HO starts with the classic Hamiltonian in the form H = p2 + q2...
  38. D

    Generating Functions in Hamiltonian Mechanics

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  39. A

    Can I get Bandgap of 3D material with 1D Hamiltonian

    Hi All, Greetings! I have a 3d material and I use result from first principal for getting the potential (U(x,y,z)). I then find average U(x) from U(x,y,z). Now if I write one dimensional Hamiltonian in X direction and use this value of U(x), can I get bandgap of the original 3d material (I...
  40. A

    Tight Binding Hamiltonian and Potential (U)

    Hi All, Greetings ! Here is what I wish to know. Specifying a tight binding hamiltonian requires values of potential (U). Consider a 3d solid. If I have potential profile in x direction (U1, U2, U3...so on) can I directly plug in these U values into the tight binding hamiltonian or do I...
  41. M

    Hamiltonian For The Simple Harmonic Oscillator

    I am reading an article on the "energy surface" of a Hamiltonian. For a simple harmonic oscillator, I am assuming this "energy surface" has one (1) degree of freedom. For this case, the article states that the "dimensionality of phase space" = 2N = 2 and "dimensionality of the energy surface" =...
  42. L

    Heisenberg Hamiltonian for 2 Electron System: Get Relation (1)

    For two electron system ##\vec{S}_1##, ##\vec{S}_2## \mathcal{H}=J\vec{S}_1\cdot \vec{S}_2=J(\frac{1}{2}(S_{tot})^2-\frac{3}{4}) (1) How you get relation (1)?
  43. carllacan

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    Just a little doubt. When we are performing a canonical transformation on a Hamiltonian and we have the equations of the new coordinates in terms of the old ones we have to find the Kamiltonian/new Hamiltonian using K = H +\frac{\partial G}{\partial t}. My question is: do we have to derive the...
  44. A

    Understanding Hamiltonian with Even/Odd Bonds

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  45. H

    Diagonalization of a Hamiltonian for two fermions

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  46. H

    How to select the good basis for the special Hamiltonian?

    How to select the good basis for the special Hamiltonian?? For the Hamiltonian H=\frac{P^2}{2\mu} -\frac{Ze^2}{r}+ \frac{\alpha}{r^3} L.S (which we can use L.S=\frac{1}{2} (J^2-L^2-S^2)in the third term) how to realize that the third term,\frac{\alpha}{r^3} L.S, commutes with sum of the...
  47. A

    What is the meaning of the pauli matrices in the Hamiltonian summation?

    For an exercise I am given the attached Hamiltonian, but I don't understand it completely. We sum over spin -½ and ½ and the paulimatrices seem to be dependent on this since they are labeled by σσ'. What does this mean? I mean the pauli matrices are just operators for the spin in the...
  48. U

    Showing energy is expectation of the Hamiltonian

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  49. A

    Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

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  50. A

    Commutate my hamiltonian H with a fermionic anihillation operator

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