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. M

    Does a Sudden Change in Hamiltonian Affect Wave Function Continuity?

    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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. L

    Differentiating a Hamiltonian - Is this a typo?

    In equation 5.8 in this document http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdf I am trying to derive this Hamiltonian. I find H= \pi \dot{\psi} - L = i \psi^\dagger \dot{\psi} - \bar{\psi} ( i \gamma^\mu \partial_\mu - m ) \psi = i \bar{\psi} \gamma^0 \partial_0 \psi - i \bar{\psi}...
  7. N

    HiLets say I have a Hamiltonian which is invariant in e.g. the

    Hi Lets say I have a Hamiltonian which is invariant in e.g. the spin indices. Does this imply that spin is a conserved quantity? If yes, is there an easy way of seeing this? Niles.
  8. L

    Proving the Hamiltonian Operator in QFT with Klein Gordon Lagrangian

    The Hamiltonian operator in quantum field theory (of Klein Gordon Lagrangian) is H=\frac{1}{2} \int \frac{d^3p}{(2 \pi)^3} \omega_{\vec{p}} a_{\vec{p}}^\dagger a_{\vec{p}} after normal ordering Now we construct energy eigenstates by acting on the vacuum |0 \rangle with a_{\vec{p}}^\dagger...
  9. K

    Differentiation a Hamiltonian Vector Field

    Homework Statement Consider a smooth manifold M, and smooth functions H_i: T^*M \to \mathbb R, i=0,1 on the cotangent bundle. Further, let z:[t_1,t_2] \subset \mathbb R \to M define a trajectory on T^*M by \frac{dz}{dt} = \vec H_0(z(t)) + u(t) \vec H_1(z(t)) where \vec H_i is the...
  10. T

    Definition of action of quantum hamiltonian, two state system

    Homework Statement Consider a two-state system. We denote the two orthonormal states by |1>and |2>. The Hamiltonian of the system is given by a 2 × 2 matrix: [omitted in this post, has 4 entries of course, not very interesting] Write the action of H on the states |1> and |2>. 2. The...
  11. A

    Is the harmonic oscillator Hamiltonian an unbounded operator?

    My answer would be "yes," and here's my argument: If we let H = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + \frac 12 m \omega^2 x^2, it is a Hermitian operator with familiar normalized eigenfunctions \phi_n(x) (these are products of Hermite polynomials and gaussians) and associated...
  12. J

    Solve Hamiltonian Operator Question: What is [\hat{H}, i\hbar]?

    Homework Statement I'm completing a question regarding position and momentum operators, however I'm stuck on one term. What does [\hat{H}, i\hbar] equal? Or what does it mean? Thanks.
  13. B

    Quantum question - hamiltonian

    Homework Statement see attached Homework Equations The Attempt at a Solution so i don't see why you need H to be time independent.. if you use TDSE to differentiate <Ek| psi> then you get d/dt of that = 0 regardless of whether H is time indep? surely? Also not sure how to...
  14. O

    QFT Commutator (momentum and Hamiltonian) Issue

    Hi, I haven't posted this in the homework section, as I don't really see it as homework as such. I'm trying to derive the Heisenberg equations of motion for the Klein Gordon field (exercise 2.2 of Mandl and Shaw). I'm trying to derive the commutator of the Hamiltonian and canonical momentum...
  15. A

    Is every Hamiltonian necessarily Hermitean?

    I know the eigenvalues of a Hermitean operator are necessarily real, and we want energies to be real...but isn't it possible for non-Hermitean operators to have real eigenvalues? If that's so, shouldn't it be possible for at least some Hamiltonians to be non-Hermitean? Also, is it possible for...
  16. G

    What is the Hamiltonian for a bead on a rotating rod with fixed z and R?

    Homework Statement A bead of mass, m is threaded on a frictionless, straight rod, which lies in the horizontal plane and is forced to spin with constant angular velocity, \omega, about a fixed vertical axis through the midpoint of the rod. Find the Hamiltonian for the bead and show that it...
  17. N

    Hamiltonian matrix eigenvalue calculation

    Homework Statement A spin system with only 2 possible states H = (^{E1}_{0} ^{0}_{E2}) with eigenstates \vec{\varphi_{1}} = (^{1}_{0}) and \vec{\varphi_{2}} = (^{0}_{1}) and eigenvalues E1 and E2. Verify this & how do these eigenstates evolve in time? Homework Equations...
  18. Y

    Trying to find the eigenvectors of a hamiltonian operator

    Homework Statement I am given the Hamiltonia operator of a system in two-dimensional Hilbert space: H = i\Delta(|w1><w2| + |w2><w1|) and am asked to find the corresponding eigenstates. I wrote this operator as a matrix, where H11 = 0, H22 = 0, and H12= i\Delta and H21= -i\Delta...
  19. M

    How Do Commutators Influence Eigenvalues in Quantum Mechanics?

    Homework Statement Homework Equations The Attempt at a Solution I think I have to use the fact that [a+ , a] = 1 but I don't know where to apply this.
  20. M

    Finding the Ground State of a Hamiltonian Operator

    When given a Hamiltonian operator (in this case a 3x3 matrix), how do you go about find the ground state, when this operator is all that is given? By the SE when have H\Psi=E\Psi. I can easily solve for Eigenvalues/vectors, but which correspond to the ground state, or am I missing something?
  21. Rasalhague

    Hamiltonian Flow: Meaning & Definition

    In The Road to Reality, § 20.2, Roger Penrose talks about a "vector field on the phase space T^*(C)", where C is a configuration space. He calls this vector field "the Hamiltonian flow", draws it as little arrows in Fig. 20.5 (that's his typical way of drawing tangent vectors, in contrast to the...
  22. P

    What Is the Difference Between Hamiltonian and Hermitian Operators?

    If anyone has time could they please answer this question. I was looking and concept of the the http://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)" , I was wonder is their a difference between the two terms? If so how are Hermitian and the Hamiltonian different? Can anyone give...
  23. I

    Hamiltonian Mechanics: Constants of Motion & Calculation

    1. A particle of mass m is in the environment of a force field with components: Fz=-Kz, Fy=Fx=0 for some constant K. Write down the Hamiltonian of the particle in Cartesian coordinates .What are the constant of motion? 2. H=kinetic energy +potential energy [b]3. Is the Hamiltonian...
  24. B

    Comp. GS of hubbard hamiltonian

    Hi, I'm trying to repeat the numerical calculation of D Jaksch's article PRL 81,3108. It is about using variational method for the ground state of bose hubbard hamiltonian: H=-J\sum{a+i+1ai}+U\sum{nini},where i denotes the lattice index the trial function is based on Gutzwiller ansatz...
  25. 4

    Using software to solve eigenvalues of Hamiltonian

    This is my first time posting in this forum, or any, so I'm sorry if something is out of place. I'm doing undergrad research with a professor on quantum supercomputing and I need to use some software to find the eigenvalues of the energy using the Hamiltonian. He suggested I used maplesoft...
  26. G

    Question concerning Hamiltonian and eigenstates

    Homework Statement Two spin-1/2 particles are placed in a system described by Hamiltonian H=S(x1)S(x2), (S(x) being the spin operator in the x-direction). States are written like |\uparrow\downarrow>, (and can be represented by 2 x 2 matrix) so that there are 4 possible states...
  27. W

    What is the hamiltonian in Klein-Gordon equation?

    since the time derivative is second order, the KG equation can not be put in the form i \dot{\psi}= H \psi so there is no H in KG equation? and no Heisenberg picture for KG equation?
  28. N

    Is Newtonian Mechanics more general than Hamiltonian Mechanics?

    Apparently things like the Lorentz' force can't be handled as a hamiltonian system. I heard other people describe the hamiltonian mechanics as an equivalent characterization of classical mechanics, but this is wrong, then?
  29. N

    Do forces always work in the hamiltonian mechanics?

    Do forces always "work" in the hamiltonian mechanics?... Hello, I'll phrase my question two times: once for the people in a hurry, and a second time in a broader way: 1) short If I write down the equation of a force, can it be that Hamiltonian mechanics doesn't 'apply' to it? I was...
  30. X

    Critical points of a Hamiltonian system

    Hello. I'm studying for an ODE/PDE qualifier and I'm wondering how to do this problem. I feel like it should be pretty easy, but anyway.. Show that a Hamiltonian system in \mathbb{R}^{2n} has no asymptotically stable critical points. Any suggestions? Thanks..
  31. K

    Uncertain Hamiltonian and Values Therein

    Hey all, I've got a Hamiltonian of the form H = \omega (\sigma_z^1 - \sigma_z^2) + J \sigma_z^1 \sigma_z^2 where \omega is a frequency ( I think), J is the indirect dipole-dipole coupling, and \sigma_z^i is the Pauli Z operator on the ith particle. Does anybody know what this...
  32. P

    Hamiltonian in cylindrical coordinates

    Hi, I'm trying to find the Hamiltonian for a system using cylindrical coordinates. I start of with the Lagrangian L=\frac{1}{2}m(\dot{r}^2+r^2\dot{\theta}^2+\dot{z}^2)-U(r,\theta,z) From that, using H=\sum p\dot{q}-L...
  33. C

    Hamiltonian Algebras: What Do We Mean By "Generated"?

    I keep hearing jargon like "the algebra generated by the hamiltonian", and I'd like to get to the bottom of it. Given a set of hamiltonians, does the "algebra that they generate" refer to the unitaries that they generate and their subsequent combinations? Or does it refer to the hamiltonians...
  34. Z

    A proposed Hamiltonian operator for Riemann Hypothesis

    HERE http://vixra.org/pdf/1007.0005v1.pdf is my proposed proof of an operator whose Eigenvalues would be the Imaginary part of the zeros for the Riemann Hypothesis the ideas are the following* for semiclassical WKB evaluation of energies the number of levels N(E) is related to the integral of...
  35. V

    Is Hamiltonian operator a Hermitian operator?

    Hi, there. It should be yes, but I'm very confused now. Consider a simple one-dimensional system with only one particle with mass of m. Let the potential field be 0, that's V(r) = 0. So the Hamiltonian operator of this system is: H = -hbar^2/(2m) * d^2/dx^2 \hat{H} =...
  36. antibrane

    Diagonalizing Spin Hamiltonian

    How would one find the eigenstates/values for the following Hamiltonian? H=A S_z + B S_x where A,B are just constants. Any help is appreciated. Thanks.
  37. K

    How Do You Determine Spin States from a Hamiltonian?

    Sorry if this question is very general/vague, but I would prefer a general answer rather than a specific solution... I'll put more detail in if necessary though. So, say we have a Hamiltonian for a system (of fermions, spin 1/2); then we find its eigenvalues and hence eigenstates. These are...
  38. P

    Interacting Spins , Heisenberg Hamiltonian , Tensor product

    Hello, I'm studying the Heisenberg Model. Given the Hamiltonian H = - 2 \frac{J}{\hbar^2} \vec{S}_1 \vec{S}_2 with \begin{equation} \vec{S} = \frac{\hbar}{2} \; \left( \begin{array}{ccc} \sigma_x \\ \sigma_y \\ \sigma_z \end{array} \right) \end{equation} \sigma_{x,y,z} \quad...
  39. B

    How is Hamiltonian energy of orbital differ from orbital in certain

    how is Hamiltonian energy of orbital differ from orbital in certain atom with another atom although the two orbital have the same no. of electrons... ex...Hamiltonian energy differ in lithium and hydrogen although both have one electron in last orbital please help me quickly
  40. K

    Hamiltonian for a Dissipative System/ Liouville's Theorem

    Homework Statement Given is the Hamiltonian for a particle in free fall: H(z,p) = P^2/(2m) + mgz At time t=0 there is an region given by the constrains: p1 less than or equal to p less than or equal to p2 E1 less than or equal to E less than or equal to E2 What is the area of...
  41. pellman

    Can a Hamiltonian be formed from this Lagrangian?

    L=\frac{1}{2}m(\dot{q}_1-\dot{q}_2)^2-V(q_1,q_2) Because if we put p_1=\frac{\partial L}{\partial \dot{q}_1} p_2=\frac{\partial L}{\partial \dot{q}_2} we get p_1=-p_2=m(\dot{q}_1-\dot{q}_2) We can't invert to get \dot{q_1} in terms of the two momenta. We can still write down a...
  42. M

    Periodic orbit of Hamiltonian dynamical system in R^2 is stable

    Homework Statement We are given a Hamiltonian dynamical system with a smooth Hamiltonian H:\mathbb{R}^2\to\mathbb{R} on \mathbb{R}^2, with canonical symplectic structure. Suppose this Hamiltonian has a periodic orbit H^{-1}(h_0). Prove that there exists an \epsilon>0 such that for all h\in...
  43. E

    How Do Symmetries Determine Expectation Values in Quantum Mechanics?

    [QM] Hamiltonian and symmetries Homework Statement Let there be the hamiltonian: H=\frac{P^2}{2m}+\frac{1}{2}m\omega^2(x^2+y^2+z^2)+kxyz+\frac{k^2}{\hbar \omega}x^2y^2z^2 Find the expectation value of the three components of \vec r in the ground state using ONLY the symmetry properties of...
  44. pellman

    Hamiltonian as Legendre transformation?

    The definition of a Legendre transformation given on the Wikipedia page http://en.wikipedia.org/wiki/Legendre_transformation is: given a function f(x), the Legendre transform f*(p) is f^*(p)=\max_x\left(xp-f(x)\right) Two questions: what does \max_x mean here? And why is it not...
  45. Z

    Can a Hamiltonian be unbounded ?

    the idea is can a Hamiltonian in 1-D of the form H=p^2 + V(x) for a certain function V(x) be unbounded and have NEGATIVE energies , for example a Hamiltonian whose spectra may be E_{n} = ...,-3,-2,-1,1,2,3,... and so on, so we have an UNBOUNDED Hamiltonian with positive and negative energies...
  46. J

    Hamiltonian of spin 1/2 in tangential magnetic field

    Hi, I have this article in which I saw that for a spin 1/2 particle confined to move along a ring positiond in a magnetic field with a z and \varphi The Hamiltonian is given by: (in second attacment) What I do not understand is how do you get the last term in the Hamiltonian. Any help...
  47. M

    Conjugate momentum in the hamiltonian

    Hey, I just have a quick question that I haven't quite been able to find a definitive answer to, regarding conjugate momenta in the Hamiltonian. Ok, so it regards the following term for the hamiltonian in a magnetic field: H=\frac{1}{2m}(p-qA)^2 I'd like to ask whether p is the conjugate...
  48. Z

    Is the hamiltonian in the coordinate representation always diagonal?

    In the coordinate representation of a quantum mechanical system, is it always true that the Hamiltonian of the system is diagonal? If so, can someone explain to me why this is true?
  49. P

    Hamiltonian & Generalised cooordinates

    Homework Statement Let T = \frac{1}{2}T_{ij}\dot{q_{i}}\dot{q_{j}} and V = \frac{1}{2}V_{ij}q_{i}q_{j}. Verify that the equation of motion T_{ij}\ddot{q_{j}} + V_{ij}q_{j} = 0 imply that the energy T + V is conserved. Can the constancy of T+V be used to deduce the equations of motion...
  50. R

    Finding 2x2 Hamiltonian Matrix for Second-Quantized Hamiltonian

    Homework Statement I need to find the 2x2 Hamiltonian matrix for the Hamiltonian, which is written in second-quantized form as below for a system consisting of the electrons and photons. H = h/ωb†b + E1a†1a1 + E2a†2a2 + Ca†1a2b† + Ca†2a1b, a's are creation and annihilation operator for...
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