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. Daaavde

    Proof of Hamiltonian equations

    So, I should prove that: - \frac{\partial H}{\partial q_i} = \dot{p_i} And it is shown that: - \frac{\partial H}{\partial q_i} = - p_j \frac{\partial \dot{q_j}}{\partial q_i} + \frac{\partial \dot{q_j}}{\partial q_i} \frac{\partial L}{\partial \dot{q_j}} + \frac{\partial L}{\partial q_i} =...
  2. BiGyElLoWhAt

    Hamiltonian in Classical mechanics?

    I've read a couple of places that a hamiltonian can be a tool used in classical mechanics and that it's eigenvalues are useful pieces of information. I've tried finding info on the subject matter, as I want to see something that actually requires linear algebra, or at least makes good use of it...
  3. C

    Variables in lagrangian vs hamiltonian dynamics

    In the lagrangian formalism, we treat the position ##q## and the velocity ##\dot q## as dependent variables and talk about configuration space, which is just the space of positions. In the hamiltonian formalism we talk about canonical positions and momenta, and we consider them independent. Is...
  4. A

    Average of any operator with Hamiltonian

    Homework Statement Prove that for any stationary state the average of the commutator of any operator with the Hamiltonian is zero: \langle\left[\hat{A},\hat{H}\right]\rangle = 0. Substitute for \hat{A} the (virial) operator:\hat{A} = \frac{1}{2}\sum\limits_i\left(\hat{p}_ix_i...
  5. M

    Quantum Hamiltonian Question

    So, I was reading about the exchange interaction, and trying to work out what it referred to, and came across something strange in the treatment of the hydrogen molecule (I think it was on wikipedia): The hamiltonian given for the system included a term e2/(4πε0 * Rab) for the repulsion between...
  6. B

    Time evolution, Hamiltonian

    Homework Statement A magnetic field pointing in ##\hat{x}##. The Hamiltonian for this is: ##H= \frac{eB}{mc}\begin{pmatrix} 0 & \frac{1}{2}\\ \frac{1}{2} & 0 \end{pmatrix}## where the columns and rows represent ##{|u_z\rangle, |d_z\rangle}##. (a) Write this out in Dirac...
  7. N

    Hamiltonian mechanics: ∂H/∂t = ?

    According to my book, and wiki http://en.wikipedia.org/wiki/Hamiltonian_mechanics#As_a_reformulation_of_Lagrangian_mechanics, ##\frac{\partial{H}}{\partial{t}} = - \frac{\partial{L}}{\partial{t}}##, where ##L## is the Lagrangian. But how can this be? This assumes the generalized...
  8. P

    Hamiltonian Operator: Difference vs. E?

    Is there any difference between Hamiltonian operator and E? Or do we describe H as an operation that is performed over (psi) to give us E as a function of (psi)??
  9. C

    What are the main differences between Hamiltonian and Lagrangian mechanics?

    Homework Statement So I just learned how to derive the equation of motion under the Lagrangian formulation which involves finding the euler-lagrange equation when setting the change in action to zero, chain rule, integration by parts etc.. Then I learned how to find the equations of motion...
  10. K

    What is the name of this Hamiltonian?

    Hello (and sorry for this stupid question), Could someone tell me the name of this Hamiltonian H = \left(\dfrac{p^2+q^2}{2}\right)^2 Thanks in advance
  11. Greg Bernhardt

    Hamiltonian: Definition, Equations & Explanation

    Definition/Summary The Hamiltonian, like the Lagrangian, is a function that summarizes equations of motion. It has the additional interpretation of giving the total energy of a system. Though originally stated for classical mechanics, it is also an important part of quantum mechanics...
  12. ShayanJ

    Eigenvectors of this Hamiltonian

    I've got a problem which is asking for the eigenvalues and eigenstates of the Hamiltonian H_0=-B_0(a_1 \sigma_z^{(1)}+a_2 \sigma_z^{(2)}) for a system consisting of two spin half particles in the magnetic field \vec{B}=B_0 \hat z . But I think the problem is wrong and no eigenstate and...
  13. B

    Hamiltonian Noether's theorem in classical mechanics

    How does one think about, and apply, in the classical mechanical Hamiltonian formalism? From the Lagrangian perspective, Noether's theorem (in 1-D) states that the quantity \sum_{i=1}^n \frac{\partial \mathcal{L}}{\partial ( \frac{d y_i}{dx})} \frac{\partial y_i^*}{\partial \varepsilon} -...
  14. L

    Is any Hamiltonian system integrable?

    This is related to classical Hamiltonian mechanics. There is something wrong in the following argument but I cannot pinpoint where exactly the pitfall is: Consider an arbitrary (smooth) Hamiltonian (let us assume conservative) and 2n phase space coordinates (q,p). The Hamiltonian flow gives...
  15. A

    Exceptation of energy by Hamiltonian

    Hello, The question says that we can write the Hamiltonian of the harmonic oscilator like this: H=0.5*[P^2/m + (4*h^2*x^2)/(m*σ^4)] where h is h-bar I need to calculate the expectation value of energy of the oscilator with the next function: ψ(x)=A*exp{-[(x-bi)^2]/σ^2}. I tried to the...
  16. J

    Hamiltonian for classical harmonic oscillator

    I am working through Leonard Susskinds 'the theoretical minimum' and one of the exercises is to show that H=ω/2(p^2+q^2). The given equations are H=1/2mq(dot)^2 + k/2q^2, mq(dot)=p and ω^2=k/m. q is a generalisation of the space variable x, and (dot) is the time derivative if this helps...
  17. genxium

    Relation between Hamiltonian of light ray and that of mechanics

    I'm learning ray optics and feeling so confused by the definition of "Hamiltonian of light". What I learned was that the "Hamiltonian of light" defined by H = n-|\vec{p}| = 0 indicates the momentum conservation, where n is refractive index and \vec{p} here is the canonical momentum. The...
  18. A

    Reading Hamiltonian (.HSX) files in SIESTA

    Hi all, I am working on DFT calculations using SIESTA. I have obtained the Hamiltonian file (.HSX) from the calculations. But, I cannot read the file it. How do I read the .HSX file .
  19. R

    Two Fourier transforms and the calculation of Effective Hamiltonian.

    Hi, The following contains two questions that I encountered in the books of Claude Cohen-Tannoudji, "Atom-Photon Interactions" and "Atoms and Photons: Introduction to Quantum Electrodynamics". The first one is about how to calculate two Fourier transforms, and the second one is a example of...
  20. S

    Roksar-Kivelson Hamiltonian for a Quantum Dimer Gas

    I completely do not understand How they work . Here is a link to my unanswered stack exchange question. Could anyone help me? Would you like any clarifications? http://physics.stackexchange.com/questions/112034/can-someone-explain-to-me-the-rocksar-kivelson-hamiltonian
  21. C

    Raising and lowering operators of the Hamiltonian

    Homework Statement a) The operators ##a## and ##a^{\dagger}## satisfy the commutation relation ##[a,a^{\dagger}] = 1##. Find the normalization of the state ##|\psi \rangle = C (a^{\dagger} )^2 |0\rangle##, where the vacuum state ##|0\rangle## is such that ##a|0\rangle = 0## b)A one...
  22. U

    Transition Radiation rates of Hamiltonian

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  23. Q

    Non Polynomial Hamiltonian Constraint

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  24. W

    Classical mechanics - particle in a well; Lagrangian and Hamiltonian

    Homework Statement I'm working (self-study) through Goldstein et al, Classical Mechanics, 3rd Edition, and I'm currently stuck on Problem 8.11: A particle is confined to a one-dimensional box. The ends of the box (let these be at \pm l(t)) move slowly towards the middle. By slowly we mean...
  25. E

    A Time Dependent Hamiltonian problem

    Homework Statement We want to get the time evolution of a wavefunction and the expectation value of the Hamiltonian, and from there we can show that it's the same as the time-independent result. So to be clear: given a wavefunction, get the time evolution of that function and the expectation...
  26. JonnyMaddox

    Infinitesimal transformations and the Hamiltonian as generator

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  27. W

    Hamiltonian of a pendulum constrained to move on a parabola

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  28. W

    Hamiltonian of two rotating and oscillating masses

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  29. B

    Lagrangian and Hamiltonian equations of motion

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  30. 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?
  31. 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...
  32. 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...
  33. D

    Generating Functions in Hamiltonian Mechanics

    Hi I have been looking at canonical transformation using generating functions. I am using the Goldstein book and it gives the following example - F1=qQ ⇔ Q=p and P=-q F2=qP ⇔ Q=q and P=p F3=pQ ⇔ Q=-q and P=-p F4=pP ⇔ Q=p and P=-q I'm confused ! Obviously functions 1 and 4 give the...
  34. 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...
  35. 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...
  36. 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" =...
  37. 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)?
  38. carllacan

    Order of steps on Hamiltonian canonical transformations

    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...
  39. A

    Understanding Hamiltonian with Even/Odd Bonds

    I have a question if you have an Hamiltonian given by H = \sum_{i,i+1} \sigma_i \cdot \sigma_{i+1} where i can even or odd bonds so in a 1D lattice so if you have 4 sites(1 2 3 4 1) then (12) and (34) are even bonds and (23) and (41) are odd bonds. and I was checking if [H_{x...
  40. H

    Diagonalization of a Hamiltonian for two fermions

    Homework Statement Hi, I want to diagonalize the Hamiltonian: Homework Equations H=\phi a^{\dagger}b + \phi^{*} b^{\dagger}a a and b are fermionic annihilation operators and \phi is some complex number. The Attempt at a Solution Should I use bogoliubov tranformations? I...
  41. 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...
  42. 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...
  43. U

    Showing energy is expectation of the Hamiltonian

    Homework Statement The vector \psi =\psi_{n} is a normalized eigenvector for the energy level E=E_{n}=(n+\frac{1}{2})\hbar\omega of the harmonic oscillator with Hamiltonian H=\frac{P^{2}}{2m}+\frac{1}{2}m\omega^{2}X^{2}. Show that...
  44. A

    Bogoliubov transformation / Interpretation of diagonalized Hamiltonian

    Hey, I consider a diagonalized Hamiltonian: H=\sum\limits_{k} \underbrace{ (\epsilon_{k} u_{k}^2 -\epsilon_{k} v_{k}^2 -2\Delta u_{k} v_{k} )}_{E_{k}}(d_{k \uparrow}^{\dagger}d_{k \uparrow} + d_{k \downarrow}^{\dagger}d_{k \downarrow}) +const with fermionic creation and annihilation...
  45. A

    Commutate my hamiltonian H with a fermionic anihillation operator

    Homework Statement I have problem where I need to commutate my hamiltonian H with a fermionic anihillation operator. Had H been written in terms of fermionic operators I would know how to do this, but the problem is that it describes phonon oscillations, i.e. is written in terms of bosonic...
  46. G

    Unbounded Hamiltonian leading to finite ground state

    If a Hamiltonian is unbounded from below, say the hydrogen atom where the Hamiltonian is -∞ at r=0, is there a way to tell if the ground state is bounded (e.g. hydrogen is -13.6 eV and not -∞ eV)? It seems if the potential is 1/r^2 or less, then the energy will be finite as: \int d^3 r (1/r^2)...
  47. L

    Group symmetry of Hamiltonian

    ##H=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+\frac{1}{2}m\omega^2x^2## Parity ##Px=-x## end ##e## neutral are group of symmetry of Hamiltonian. ## PH=H## ##eH=H## so I said it is group of symmetry because don't change Hamiltonian? And ##e## and ##P## form a group under multiplication. Is there...
  48. A

    Is the Hamiltonian in the Exercise Truly Time-Dependent?

    Consider the attached exercise. I am having some trouble understanding exactly what time dependent hamiltonian it refers to. Because from the equation it refers to it seems that the hamiltonian is by definition time independent. Am I to assume that the H diagonal is a time independent...
  49. D

    Matrix of Hamiltonian, system's state - quantum

    Hello I'm reaaaaaally stuck here, if someone can explain me it'd be great because i just dn't know how to do all that Homework Statement Consider a system with the moment of inertia l=1 A base of the space of the states is constituted by the three eigenvectors of Lz |+1> , |0>, |-1> of...
  50. R

    Separating a hamiltonian into C.O.M and relative hamiltonians

    Homework Statement Show that the two-body hamiltonianH_{\text{sys}}=\frac{\mathbf{p}_1^2}{2m_1}+\frac{\mathbf{p}_2^2}{2m_2}+V( \mathbf{r}_1,\mathbf{r}_2)can be separated into centre of mass and relative...
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