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. Salmone

    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...
  2. AzAlomar

    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...
  3. Salmone

    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...
  4. L

    A SO(3) group, Heisenberg Hamiltonian

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  5. K

    I Hamiltonian of a particle in a magnetic field

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  6. Hari Seldon

    A Deriving Navier-Stokes: Lagrangian & Hamiltonian Methods

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  7. M

    Understanding Fixed Points in Hamiltonian Systems

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  8. S

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  9. D

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  10. L

    Relativity Special relativity in Lagrangian and Hamiltonian language

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  11. AfonsoDeAlbuquerque

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  12. Saptarshi Sarkar

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  13. hello_world30

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  14. ubergewehr273

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    Now from the relevant equations, $$U(t) = \exp(-i \omega \sigma_1 t)$$ which is easy to compute provided the Hamiltonian is diagonalized. Writing ##\sigma_1## in its eigenbasis, we get $$\sigma_1 = \begin{pmatrix} 1 & 0\\ 0 & -1\\ \end{pmatrix} $$ and hence the unitary ##U(t)## becomes...
  15. A

    I References for Hamiltonian field theory and Dirac Brackets

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  16. D

    A Cat state acting on given Hamiltonian

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  17. LCSphysicist

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  18. Mohammad-gl

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  19. T

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  20. M

    I Proving Imaginary Hamiltonian is P-Odd, T-Even & Imaginary

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  21. Q

    A How to systematically find the symmetry operator given a Hamiltonian?

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  22. sophiatev

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  23. zhangnous

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  24. B

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  25. B

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  26. Llukis

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  27. tanaygupta2000

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  28. P

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  29. lelouch_v1

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  30. Llukis

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  31. D

    Simple harmonic oscillator Hamiltonian

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  32. CptXray

    I Coherent state evolution - nonlinear Hamiltonian

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  33. Llukis

    I The time evolution of a Hamiltonian

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  34. L

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    If Hamiltonian commutes with a parity operator ##Px=-x## are then all eigenstates even or odd? Is it true always or only in one-dimensional case?
  35. K

    A linear combination of states that diagonalize the Hamiltonian

    He told me I "need to show that the Hamiltonian matrix elements you get by using those states have nonzero elements only on the diagonal." I understand what and how a diagonal matrix works, but what I don't understand is what those states are. Are they states I put in my "quantum mechanical...
  36. H

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    Hello! I need some help with this problem. I've solved most of it, but I need some help with the Hamiltonian. I will run through the problem as I've solved it, but it's the Hamiltonian at the end that gives me trouble. To find the Lagrangian, start by finding the x- and y-positions of the...
  37. J

    I Energy in the Hamiltonian formalism from phase space evolution

    The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions...
  38. Mayan Fung

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    What I have tried is a completing square in the Hamiltonian so that $$\hat{H} = \frac{\hat{p}^2}{2} + \frac{(\hat{q}+\alpha(t))^2}{2} - \frac{(\alpha(t))^2}{2}$$ I treat ##t## is just a parameter and then I can construct the eigenfunctions and the energy eigenvalues by just referring to a...
  39. E

    What is the derivation for the probability of energy in quantum mechanics?

    If we can identify ##|c_n|^2## as the probability of having an energy ##E_n##, then that equation is just the bog standard one for expectation. But the book has not proved this yet, so I assumed it wants a derivation from the start. I tried $$ \begin{align*} \Psi(x,t) = \sum_n c_n...
  40. F

    I Can a Hamiltonian with non-spherical potential commute with l^2?

    I know that in the case of central potential V(r) the hamiltonian of the system always commutes with l^2 operator. But what happends in this case?
  41. LuccaP4

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  42. Tanmoy

    Ground State Energy: Explanation & Calculation

  43. LCSphysicist

    What is the minimum mathematic requirement for learning Lagrangian and Hamiltonian mechanics?

    Homework Statement:: ... Relevant Equations:: . What is the minimum mathematic requirement to the Lagrangian and hamiltonian mechanics? Maybe calc 3 and linear algebra?
  44. D

    Phase space of a harmonic oscillator and a pendulum

    Hello everybody, new here. Sorry in advance if I didn't follow a specific guideline to ask this. Anyways, I've got as a homework assignment two cannonical transformations (q,p)-->(Q,P). I have to obtain the hamiltonian of a harmonic oscillator, and then the new coordinates and the hamiltonian...
  45. E

    Quantum motion of a charged particle in a magnetic field

    Once I know the Hamiltonian, I know to take the determinant ##\left| \vec H-\lambda \vec I \right| = 0 ## and solve for ##\lambda## which are the eigenvalues/eigenenergies. My problem is, I'm unsure how to formulate the Hamiltonian. Is my potential ##U(r)## my scalar field ##\phi##? I've seen...
  46. SamRoss

    Not sure where this final Hamiltonian came from

    Here's the problem and the solution provided online by the author (the problem numbers are different but it's the same question). I think I'm okay up until the last step where he declares the Hamiltonian is (1 1 1 -1). Where did he get those components?
  47. M

    Converting a Lagrangian to a Hamiltonian

    Given the following $$L(\theta,\dot{\theta},\phi,\dot{\phi}) = \frac12ml^2((\dot{\theta})^2 + (sin(\theta)^2)\dot{\phi}^2) + k\theta^4$$ This is my attempt: I am not understanding if the conserved quantities (like angular momentum about the z-axis) impacts my formulation of the Hamiltonian or...
  48. P

    How Does Charge and Elasticity Affect Hamiltonian Dynamics?

    Let a mass m charged with q, attached to a spring with constant factor k = mω ^2 in an electric field E(t) = E0(t/τ) x since t=0. (Equilibrium position is x0 and the deformation obeys ξ = x - x0) What would the hamiltonian and motion equations be in t ≥ 0, in terms of m and ω?? Despise magnetic...
  49. A

    I When does the exchange operator commute with the Hamiltonian

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  50. O

    What was the intuition behind the development of Hamiltonian mechanics?

    Is Hamiltonian mechanics a mathematical generalization of Newtonian mechanics or is it explaining some fundamental relationship that has a meaning that extends into our nature ? I guess my question is what would led William Rowan Hamilton to come up with his type of mechanics or anything...
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