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. S

    How Does the Klein-Gordon Hamiltonian Define the Evolution of Field Variables?

    Suppose that we take the Klein-Gordon Hamiltonian to be of the form H = \int d^3x \, \mathcal{H}(x) = \frac{1}{2}\int d^3x\, (\pi^2(x) + (\nabla\phi(x))^2- m^2\phi^2(x)) If we want to compute, say, the evolution equation for \phi(x) we use the Poisson bracket: \dot{\phi}(x) = \{\phi(x),H\} =...
  2. M

    Hamiltonian of charged particle

    A charged particle of mass m is attracted by a central force with magnitude F = \frac{k}{r^2} . Find the Hamiltonian of the particle. I'm just wondering if I did this correctly because it seemed too easy. First I used the fact that -dU/dr = F = k/r^2, so the potential (with infinite...
  3. T

    Hamiltonian and Green functions.

    Let be the 1-D Hamiltonian: \hat H = -\hat D ^{2} + V(\hat x) (1) and its associated 'Green function' so: -D^{2} G(x,s)+V(x)G(x,s)=\delta (x-s) (2) then my question is if there is a relationship between: Tr[exp(-a\hat H ) ] and Det[1-aG] (3) where a >0 and 'G' is the...
  4. P

    DERIVE a 6x6 Hamiltonian for bulk semiconductors

    URGENT x 10 DERIVE a 6x6 Hamiltonian for bulk semiconductors Okay here is a little challenge for you guys. Try and test your skill a little. First 10 people to properly derive a 6x6 Hamiltonian for bulk semiconductors will gain bragging rights in this forum.
  5. T

    DOes the trace determine the Hamiltonian

    if we define Z as: Z(s)=Tr[exp(-sH)] my 2 questions are.. a) is the trace unique and define the Hamiltonian completely? i mean if we have 2 Hamiltonians H and K then Tr[exp(-sH)]\ne Tr[exp(-sK) and if we use the 'Semiclassical approach' then Z(s)=Tr[exp(-sH)]\sim...
  6. quasar987

    Hamiltonian mechanics: canonical transformations

    Say I have a canonical transformation Q(q,p), P(q,p). In the {q,p} canonical coordinates, the Hamiltonian is H(q,p,t)=p\dot{q}-L(q,\dot{q},t) And the function K(Q,P,t)=H(q(Q,P),p(Q,P),t) plays the role of hamiltonian for the canonical coordinates Q and P in the sense that...
  7. R

    Can a system have the total energy conserved but the hamiltonian not conserved?

    Homework Statement Can a system have the total energy conserved but the hamiltonian not conserved? Homework EquationsIf the partial of the lagrangian w.r.t time is zero, energy is conserved. The hamiltonian is found by the usual method- get the generalized momentum from the lagrangian...
  8. L

    Hamiltonian Mechanics: Central Potential V(r) w/ Spherical Coordinates

    Homework Statement Using spherical coordinates (r, \theta, \phi), obtain the Hamiltonian and the Hamilton equations of motion for a particle in a central potential V(r). Study how the Hamilton equations of motion simplify when one imposes the initial conditions p_{\phi}(0) = 0 and \phi (0)...
  9. L

    How Do You Derive a Lagrangian from a Time-Dependent Hamiltonian?

    Consider the time-dependent Hamiltonian H(q,p;t) = \frac{p^2}{2m \sin^2{(\omega t})} - \omega pq \cot{(\omega t)} - \frac{m}{2} \omega^2 \sin^2{(\omega t)} q^2 with constant m and \omega. Find a corresponding Lagrangian L = L(q,\dot{q};t) Ok, I know that the Hamiltonian is given by...
  10. R

    Finding the Hamiltonian for the Be Atom without Nuclear Terms

    I have a problem that uses the QM Hamiltonian for the berylium atom, but I am having trouble finding this Hamiltonian using the Born-Oppenheimer approximation (leaving out the nuclear-nucler and nucler-electron terms). Any know how to get this?
  11. K

    Zero Hamiltonian and its energies

    Let's suppose we have a Hamiltonian of the form: \bold H =0 then it's obivious that if you want to get its energies you would get E_{n}=0 for every n, this is a non-sense since you must have positive energies (and a ground state) ...then if we "cheat" :rolleyes: and make: \bold H...
  12. U

    How Do You Find the Hamiltonian for a Single Particle in Different Coordinates?

    I need to find the Hamiltonian for a single particle under the influence of potential U in different coordinates: I have found the Hamiltonian for Cartesian coordinates fairly easily and would just like a check if it is: L=\frac{1}{2} m \dot{q}^2 -U with p=m \dot{q} which means...
  13. L

    Hamiltonian Math Problem: Finding the Time-Derivative of <p^2/2m>

    One more question. Sorry! Here's the problem: An electron moves in a straight line under the influence of a conservative force so that the Hamiltonian is H = \frac{p\wedge^2}{2m} + V(x), where p\wedge means the momentum operator and I think V(x) is the potential energy. I need to find an...
  14. J

    How to Determine the Eigenenergy of This Hamiltonian?

    The Hamiltonian is given: H=Aâ†â + B(â + â†) where â is annihilation operator and ↠is creation operator, and A and B are constants. How can I get the eigenenergy of this Hamiltonian? The given hint is "Use new operator b = câ + d, b† =c↠+ d (c and d are constants, too) But...
  15. J

    How can I calculate the eigenvalues of a Hamiltonian with spin 1/2 objects?

    Find the eigenvalues of the hamiltonian H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A) where S_A, S_B, S_C, S_D are spin 1/2 objects _________________________ I rewrite it as H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2] then i define...
  16. W

    Why do symmetry and Hamiltonian operators commute?

    Why do operators representing some symmetry commute with the Hamiltonian?
  17. J

    Angular Momentum and Hamiltonian Commutation

    I am working on a problem for homework and am supposed to show that the angular momentum operator squared commutes with H and that angular momentum and H also commute. This must be done in spherical coordinates and everything I see says "it's straightforward" but I don't see it. At least not...
  18. B

    Solution of 1D Schrodinger using discretized hamiltonian

    Hi all, I'm trying to solve the 1D Schrodinger equation for an arbritary potential, to calculate Franck Condon factors for absorption and emmision spectra. I can do this using iterative techniques (e.g. the Numerov method), but I can't seem to get it to work by discretrizing the hamiltonian...
  19. kakarukeys

    Conserved charge generates symmetry transformation in Hamiltonian Mechanics

    Q is a conserved charge if \{Q, H\} = 0 Show that q+\epsilon\delta q satisfies the equation of motion. \delta q = \{q, Q\} I couldn't find the proof. Anybody knows? My workings: \delta q = \{q, Q\} \delta\dot{q} = \{\{q,Q\},H\} = - \{\{Q,H\},q\} - \{\{H,q\},Q\} \delta\dot{q} = \{\{q,Q\},H\} =...
  20. E

    Obtaining the Hamiltonian for Einstein's Lagrangian: A Wheeler-De Witt Approach

    If we have the Einstein Lagrangian... L= \sqrt (-g)R my question is how do you get the Hamiltonian?..the approach by Wheeler-De Witt is to consider the line element: ds^2 = N(t)dt^^2 + g_ij dx^i dz^ j (Einstein sum convention) and then substitute it into the Lagrangian above and perform...
  21. K

    Helium Hamiltonian: Derive Ground State & Quantum Numbers

    Hi, I have a question in a past exam paper which I can't quite understand how to prove. It reads: Give an expression for the Hamiltonian of the Helium atom. Neglecting the interaction between the electrons, derive the state function for the Helium ground state in terms of hydrogen-like...
  22. W

    Defination of eFz in Hamiltonian

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  23. P

    How do you a find a Hamiltonian?

    So when I first learned about Hamiltonians, my teacher presented them as something which you derived based Newton's second law, with the intermediate step of the Lagrangian. Doing it this way gets a Hamiltonian of the form H = T + V. Now I have a new teacher, who says that that these sorts of...
  24. S

    The Hamiltonian: Solving dH/dt Formally

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  25. P

    Hamiltonian formulation of general relativity

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  26. R

    .Solving [X,H] for the SHO Hamiltonian

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  27. X

    How Can Two Qubits Interacting with Photons Be Simplified Using Pauli Operators?

    please see the following Hamiltonian mode. two two-level atoms(or two qubits) interact with photon. á、à are the creation and annihilation operators of photon please rewriting H in the way of pauli operators , will looks very simple H = E1 ( |e1><e1| －|g1><g1| ) + E2 ( |e2><e2| －|g2><g2|...
  28. quasar987

    Hamiltonian of a spin 1/2 particle in a constant mag. field

    Why is there no kinetic energy term in said hamiltonian? Suppose I take a magnetic dipole in my hand, and I throw it in the field. Then surely its classical energy is E = p²/2m - \vec{\mu} \cdot \vec{B}. Then why is the p²/2m term absent in the hamiltonian?
  29. Q

    Spin Orbit Interaction Hamiltonian

    Spin Orbit Interaction Hamiltonian is defined as follows: H_{SO}=\frac{1}{2m_{e}c^2}\frac{1}{r} \left(\frac{\partial V}{\partial r}\right)L\cdot S How does one derive the above Spin Orbit Interaction Hamiltonian from relativistic treatment? Is there a good textbook that elaborates on...
  30. S

    What Is Heisenberg Hamiltonian?

    Hi. Can anyone tell me what exactly the "Heisenberg Hamiltonian" is? I found it in an article related to: the renormalization group. Thanks in advance. Somy:smile:
  31. F

    Angular momentum commutes with Hamiltonian

    How can I prove that the Hamiltonian commutes with the angular momentum operator? In spherical coordinates it is straightforward but I'd like to understand the physical meaning of it. Thanks.
  32. marcus

    Infinite number of Hamiltonian constraints

    http://www.arxiv.org/abs/gr-qc/0510011 "Recently the Master Constraint Programme (MCP) for Loop Quantum Gravity (LQG) was launched which replaces the infinite number of Hamiltonian constraints by a single Master constraint. The MCP is designed to overcome the complications associated with the...
  33. Y

    Lagrange and Hamiltonian question

    I can't understand what the question is asking~ hope somebody can help me~ A particle of mass m moves in a plane under the influence of Newtonian gravitation force, described by the potential V(r) = - GmM/r (symbol in conventional meaning) Now introduce a new variable u(theta) = 1/r(theta)...
  34. H

    Poisson brackets and EM Hamiltonian

    Consider the following general Hamiltonian for the electromagnetic field: H = \int dx^3 \frac{1}{2} E_i E_i + \frac{1}{4}F_{ij}F_{ij} + E_i \partial_i A_0 + \lambda E_0 where \lambda is a free parameter and E_0 is the canonical momentum associated to A_0, which defines a constraint (E_0 =...
  35. E

    Is the Potential in Quantum Hamiltonian II Always Real?

    Let be a Hamiltonian in the form H=T+V we don,t know if V is real or complex..all we know is that if E_n is an energy also E*_n=E_k will be another energy, my question is if this would imply V is real... my proof is taking normalized Eigenfunctions we would have that...
  36. E

    Hamiltonian with complex potential

    let,s suppose we have the Hamiltonian H=T+V but V is V=F(z) being z=a+ix then would the energies be real?..thanks.
  37. N

    Is the Hamiltonian Cycle an NP Problem?

    sup, just realized there is not graphtheory/combinatorial subforums. is the hamiltonian cycle still considered and NP problem?
  38. L

    What are Hamiltonian Mechanics?

    Can anyone give me a basic definition of Hamiltonian Mechanics without all the fancy mathematics, and perhaps could supply a few examples as to this? I am trying to make sense of this, but everywhere I go, I run into very large mathematical equations and no defintions I can understand...
  39. Z

    Understanding the Relationship between the Hamiltonian and Magnetic Field

    I have a question regarding the Hamiltonian in a magnetic field. First Hamiltonian with potential V is given by Ho = (1/2m)*p^2 + V but if a vector potential A is also present then H1 = (1/2m)*(p+eA)^2 + V there is a way to write H1 interm of Ho H1 = exp(-ier.A) Ho exp(ier.A) where r...
  40. H

    P&S Chapter 2: Derivation of the KG Hamiltonian

    I am not familiar with calculations such as the following one, and I want to be sure I do the right steps before going on with Peskin & Schröder. I want to derive step by step the Hamiltonian of the real scalar field. I have no problem to arrive at the first part of (2.31) in page 21: H =...
  41. H

    Symplectic runge kutta for hamiltonian system

    Hi ! I'm trying to solve the restricted problem of three bodies, where a negligeable mass particule is moving in the gravitationnal field of two heavy objects which are in circular orbit around their common center of mass. this is a plane problem... I describe the mouvment in the mobile...
  42. S

    Hamiltonian of flyball governor

    Hamilton equations of flyball governor I'm trying to find 1. The Hamiltonian 2. The Hamilton equation of motion for the flyball governor shown in problem 2 here http://www.srl.caltech.edu/phys106/1999/Homework3.pdf This is what i have. Can someone tell me if I'm right...
  43. C

    Angular Momentum vs Hamiltonian in Dirac Field Theory (Canonical)

    I need some suggestions and/or corrections if I understand this correct? My questions are based on the book by Mandl and Shaw. Conserved currents are based on Noethers theorem and directly connected to spacetime and field transformations (rotations, translations, phase, ...). One can...
  44. R

    Quantum Mechanics and General Relativity: Understanding H Y = E Y and T Y = G Y

    Generally we write H Y = E Y in quantum mechanics. Would it make any sense to use T ( stress-energy tensor) from general relativity and G (curvature) to write : T Y = G Y
  45. I

    Relativistic Lagrangian -> Hamiltonian

    I came cross this problem when solving some older tests from Classical Mechanics, so I was hoping anybody can help me. I have expression for relativistic lagrangian of a particle in some potential that is function only of coordinates and not velocities: L = - mc^2* sqrt( 1 - (x'^2 + y'^2 +...
  46. T

    Can Identical Quantum Particles Have Different Energies?

    Let say I have prepared two identical particle, both describable by a wavefunction Psi, whereby, Psi = a*1 + b*2, where, 1 and 2 are two stationary wavefunctions. If I perform an experiment to find out the systems' energy, this is equivalent to operating a Hamiltonian on Psi. Operating...
  47. E

    Hamiltonian is commutable with momentum operator

    [H,P]=0 , where P is momentum operator. Hamiltonian is commutable with momentum operator. so H and p have wave function simultaniously, but in 1-dimensional potential well degeneracy not exist. what is the reason?
  48. suyver

    Standard Model: Lagrangian vs. Hamiltonian

    I was wondering: why is the SM always written with a Lagrangian? Couldn't you just as well write it with a Hamiltonian? The way I understand, the Lagrangian gives me the kinetic energy minus the potential energy (basically a measure for the "free energy", though not in the thermodynamical...
  49. E

    How Do You Quantize a Classical Hamiltonian of the Form H=f(x)P^n?

    given a classic hamiltonian of the form H=f(x)P**n what would be it quantum version of it?..how do you quantizy this?... (n is an integer) Should you take all the possible permutation of it?..i have this problem...thanks.
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