What is Hamiltonian: Definition and 894 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. A

    Eigenvectors of the Hamiltonian

    Hey guys (this is not a HW problem, just general discussion about the solution that is not required for the assignment), So I am doing this problem where I had to find the eigenvalues and eigenvectors of the Hamiltonian: H = A*S_{x}^{2} + B*S_{y}^{2} + C*S_{z}^{2}. Easy enough, just...
  2. V

    How to get lagragean when hamiltonian is given

    hai everyone, the question is " the hamiltonian of a particle is H = [(p*p)/2m + pq] where q is the generalised coordinate and p is the corresponding canonical momentum. the lagragean is ....? i know that H = p(dq/dt) - L. but the answer should not contain p. how can i solve it? answer is...
  3. M

    Non-diagonality of Dirac's Hamiltonian

    Hi all, My question is why Dirac's Hamiltonian isn't diagonal? As much as I understand, the momentum of the particle and it's spin belong to the complete set of commuting variables, which define the state of the particle, and their eigenstates must also be energy eigenstates. But because...
  4. W

    Constructing a hamiltonian for a harmonic oscillator

    Hello: I am trying to understand how to build a hamiltonian for a general system and figure it is best to start with a simple system (e.g. a harmonic oscillator) first before moving on to a more abstract understanding. My end goal is to understand them enough so that I can move to symplectic...
  5. M

    Hamiltonian for a particle moving on a plane tangent to the surface of the easth

    Homework Statement A point of mass m is placed on a frictionless plane that is tangent to the Earth’s surface. Determine Hamilton’s equations taking: (a) the distance x (b) the angle q as the generalized coordinate. Homework Equations The Attempt at a Solution Take the...
  6. T

    Hamiltonian system - qual question

    This was on a previous qualifying exam. Let H(x,y) be C^2. Assume that \lim_{x^2+y^2}{|H(x,y)|} = \infty and that the system \dot{x} = H_y, \quad \dot{y} = -H_x has only finitely many critical points. Prove that it has at least one Lyapunov stable critical point. Now, what I know is...
  7. L

    Derivation of geodesic equation from hamiltonian (lagrangian) equations

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  8. fluidistic

    Solving the Kepler problem with the Hamiltonian

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  9. fluidistic

    Finding the Hamiltonian if I'm given the Lagrangian

    Homework Statement Determine the Hamiltonian corresponding to the an-harmonic oscillator having the Lagrangian L(x,\dot x )=\frac{\dot x ^2}{2}-\frac{\omega ^2 x^2}{2}-\alpha x^3 + \beta x \dot x ^2. Homework Equations H(q,p,t)=\sum p_i \dot q _i -L. p _i=\frac{\partial L}{\partial \dot...
  10. 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...
  11. A

    Commutator of the Hamiltonian with Position and Hamiltonian with Momentum

    To prove: Commutator of the Hamiltonian with Position: i have been trying to solve, but i am getting a factor of 2 in the denominator carried from p2/2m Commutator of the Hamiltonian with Momentum: i am not able to proceed at all... Kindly help.. :(
  12. J

    Free-field Dirac Hamiltonian

    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 =...
  13. B

    Maths of Hamiltonian / Lagrangian mechanics

    Hello everyone I have difficulties in understanding some stuff in Lagrangian and Hamiltonian mechanics. This concerns the equations : \dot p = - \frac{\partial H}{\partial q} \frac{d}{dt} \frac{\partial L}{\partial \dot q} = \frac{\partial L}{\partial q} First I have to say that I'm a math...
  14. 1

    Eigevalues and eigevectors of an Hamiltonian

    Homework Statement Hi, I must find eigenvalues and eigenvector of this Hamiltonian, which describes a system of two 1/2-spin particles. H = A(S_{1z} - S_{2z}) + B(S_{1} · S_{2}) where S_{1} and S_{2} are the two spins, S_{1z} and S_{2z} are their z-components, and A and B are constants...
  15. 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...
  16. M

    Construction of Hamiltonian from Casimir operators

    In Greiner & Muller's 'Quantum Mechanics: Symmetries' (section 3.5) they explain that where a system possesses a symmetry, the corresponding Hamiltonian must be 'built up' from the Casimir operators of the corresponding symmetry group. Does anyone know of a reference where this is gone into...
  17. K

    The Jaynes-Cummings Hamiltonian

    This Hamiltonian popped up when I was reading an article, as a reference(wikipedia): http://en.wikipedia.org/wiki/Jaynes%E2%80%93Cummings_model#cite_note-1 I don't understand why the Hamiltonian \hat H_{atom} and \hat H_{int} look the way they are. Usually we we just take a classical Hamiltonian...
  18. D

    Fourier transform of scattering hamiltonian

    Hey, I am looking at the coupling hamiltonian for electrons in an EM field. In particular I'm interested in the inelastic scattering (this isn't the dominant part for inelastic scattering but it's confusing me). The part of the hamiltonian in the time/space domain that I'm interested in is...
  19. R

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  20. Simfish

    Legendre Transformation of the Hamiltonian

    It's given as this H\left(q_i,p_j,t\right) = \sum_m \dot{q}_m p_m - L(q_i,\dot q_j(q_h, p_k),t) \,. But if it's a Legendre transformation, then couldn't you also do this? H\left(q_i,p_j,t\right) = \sum_m \dot{p}_m q_m - L(p_i,\dot p_j(p_h, q_k),t) \,.
  21. B

    Showing that S^2 and Sz Commute with the System Hamiltonian

    Homework Statement A system with two spins of magnitude 1/2 have spin operators S1 and S2 and total spin S = S1 + S2 B is a B-field in the z direction (0,0,B) The Hamiltonian for the system is given by H = m S1 . S2 + c B.S where m,c are constants. By writing the Hamiltonian in...
  22. S

    Solving Heisenberg Hamiltonian for 1/2-Spin Systems

    I have various 1/2-spin systems and I should find energetic spectra (eigenvalues of the Hamiltonian matrix). Hamiltonian I use is in the form: H=JSiSj+t(c+ic-j+c+jc-i) The first part is interaction between two spins, so sum over every spin-pair, the second part is interaction between spin...
  23. Y

    LaTeX How to type Hamiltonian symbol in latex

    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...
  24. A

    Difference between Hamiltonian operator and Total energy operator?

    What is the difference between the Hamiltonian operator and the Total energy operator? If both is used when working with total energy, why are there two different operators?
  25. Z

    Physical interpretation of the Hamiltonian

    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?
  26. Z

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

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  27. N

    QM: Fluctuating Hamiltonian

    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 <>...
  28. P

    Hamiltonian of Van der Pol Oscillator

    Hi, wondering if anyone can help with this; we have the Hamiltonian of a linear harmonic oscillator H=(p2+w2q2)/2 now we apply the frictional force F=ap(bq-1) where a and b are constants. how do we alter the Hamiltonian to take that into account? if it helps, the total time derivative of H...
  29. 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...
  30. 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.
  31. J

    Canonical transformation in Hamiltonian

    Hamiltonian H=\frac{1}{2m}(P+\frac{e}{c}A)^{2} - e\phi and H^{'}=\frac{1}{2m}(P+\frac{e}{c}A^{'})^{2} - e\phi^{'} With gauge: A^{'}=A+\nabla\chi and \phi^{'}=\phi-\frac{1}{c}\dot{\chi} Why H^{'}-\frac{e}{c}\dot{\chi}=e^{-\frac{ie\chi}{\hbar c}}He^{\frac{ie\chi}{\hbar c}} ? Thanks.
  32. 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...
  33. 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...
  34. 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...
  35. 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...
  36. G

    Tight binding hamiltonian matrix

    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...
  37. 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.
  38. U

    Eigenvalues for a Hamiltonian

    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...
  39. 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...
  40. 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} \\...
  41. M

    Constructing Eigenstate for a given hamiltonian

    Homework Statement Hi; I am trying to construct eigenstate for the given hamiltonian. I have the energy eigenvalues and corresponding eigenvectors. But How can I construct eigenstates? Homework Equations The Attempt at a Solution I tried to use the H . Psi= E . Psi...
  42. 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...
  43. 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...
  44. Peeter

    Physical system for this two level Hamiltonian?

    An old QM exam question asked for consideration of a two-level quantum system, with a Hamiltonian of H = \frac{\hbar \Delta}{2} ( {\lvert {b} \rangle}{\langle {b} \rvert}- {\lvert {a} \rangle}{\langle {a} \rvert})+ i \frac{\hbar \Omega}{2} ( {\lvert {a} \rangle}{\langle {b} \rvert}- {\lvert...
  45. B

    Restrictions on Numbers in Hermitian Hamiltonian Equation

    Homework Statement In terms of the usual ladder operators A, A* (where A* is A dagger), a Hamiltonian can be written H = a A*A + b(A + A*) What restrictions on the values of the numbers a and b follow from the requirement that H has to be Hermitian? Show that for a suitably chosen...
  46. M

    Sudden change in Hamiltonian

    Homework Statement The Hamiltonian of a quantum system suddenly changes by a finite amount. Show that the wave function must change continuously if the time-dependent Schrodinger equation is to be valid throughout the change. Homework Equations Time independent Schrodinger equation and...
  47. C

    Why does the Hamiltonian generate time evolution?

    The first part of QFT seems to be almost entirely mathematical formalism, without really requiring a whole lot of physical insight to proceed. For instance, we can start with the free-field scalar Lagrangian, minimize it using the Euler-Lagrange equation to arrive at the Klein-Gordon equation...
  48. V

    Euler's path same as Hamiltonian cycle

    Homework Statement Find (classify) all graphs with n vertices who's Euler's path is the same as their Hamiltonian cycle. The Attempt at a Solution I'd say that any regular graphs with a degree greater than n/2 (Dirac's Theorem) with and n divisible by 2 has both and Euler's path and a...
  49. snoopies622

    Hamiltonian mechanics and the electric field

    The dimensions of action divided by the dimensions of electric field strength are distance x time x charge. Does this mean that distance x time x charge - whatever one might call that - is the "conjugate momentum" of an electric field? If so - is there any physical significance to this...
  50. E

    Can a quantum formalism exist without a Hamiltonian Formalism

    I was thinking about this. In every problem I have worked, we suppose a hamiltonian exists which can describe the system. There are obviously Hamiltonians which are not possible classically, such as in the 1-D ising model of paramagnetism, where the Hamiltonian contains terms of s_i. s_j where...
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