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. 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)...
  2. L

    What Symmetry Group Does the Quantum Harmonic Oscillator Exhibit?

    ##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...
  3. 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...
  4. 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...
  5. 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...
  6. A

    How Do You Diagonalize a Hamiltonian Matrix?

    Homework Statement The exercise: https://www.physicsforums.com/attachment.php?attachmentid=64229&d=1385257430 Homework Equations Are in my attempt at a solution. I am sure it would be easier to use the transformation equation for the operator and plug it into the diagonalized Hamiltonian...
  7. D

    Hamiltonian Qn Homework: Find T+U is Not Equal

    Homework Statement The simple form H=T+U is true only if your generalized coordinates are "natural"(relation between generalized and underlying Cartesian coordinates is independent of time). If the generalized coordinates are not natural, you must use the definition H=Ʃpq'-L. To illustrate...
  8. E

    Proving The Hamiltonian Is Invariant Under Coordinate Transformation

    Homework Statement (a) Consider a system with one degree of freedom and Hamiltonian H = H (q,p) and a new pair of coordinates Q and P defined so that q = \sqrt{2P} \sin Q and p = \sqrt{2P} \cos Q. Prove that if \frac{\partial H}{\partial q} = - \dot{p} and \frac{\partial H}{\partial p} =...
  9. J

    Hamiltonian for elastic collision

    What would the Hamiltonian for a system of two classical point particles, with no interaction except for an elastic collision between them at a point look like? My gut says it's the usual T + V, with T = p12/2m1 + p22/2m2 and V = Kδ(r1-r2) With K approaching infinity -- each particle...
  10. Jalo

    Find the eigenvalues of the Hamiltonian - Harmonic Oscillator

    Homework Statement Find the eigenvalues of the following Hamiltonian. Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |RHomework Equations â|\phi_{n}>=\sqrt{n}|\phi_{n-1}> â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}> The Attempt at a Solution By applying the Hamiltonian to a random state n I...
  11. H

    Hamiltonian function vs. operator

    I've dealt with both the Hamiltonian function for Hamiltonian mechanics, and the Hamiltonian operator for quantum mechanics. I have a kind of qualitative understanding of how they're similar, especially when the Hamiltonian function is just the total energy of the system, but I was wondering if...
  12. S

    Two State System Described by a Time-Dependent Hamiltonian

    Homework Statement A two state system is described the time dependent Hamiltonian \hat{H}=|\psi\rangle E\langle\psi|+|\phi\rangle E \langle\phi|+|\psi\rangle V(t)\langle\phi|+|\phi\rangle V(t)\langle\psi| Where \langle \psi|\psi \rangle = 1=\langle \phi|\phi\rangle, \langle \phi|\psi...
  13. T

    Hamiltonian matrix off diagonal elements?

    I'm trying to understand how Hamiltonian matrices are built for optical applications. In the excerpts below, from the book "Optically polarized atoms: understanding light-atom interaction", what I don't understand is: Why are the \mu B parts not diagonal? If the Hamiltonian is \vec{\mu} \cdot...
  14. H

    SHO ladder operators & some hamiltonian commutator relations

    Homework Statement For the SHO, find these commutators to their simplest form: [a_{-}, a_{-}a_{+}] [a_{+},a_{-}a_{+}] [x,H] [p,H] Homework Equations The Attempt at a Solution I though this would be an easy problem but I am stuck on the first two parts. Here's what I did at first...
  15. F

    Hamiltonian for a free particle (Special relativity)

    I've seen that the lagrangian for a relativistic free particle is -mc \sqrt{\eta_{\mu\nu} \dot{x^{\mu}}{\dot{x\nu}} but when I construct the hamiltonian as p_{\mu} \dot{x^{\mu}} - L I seem to get zero. I am not really sure what I'm doing wrong. I find that if in the first term of...
  16. B

    Lagrangian, Hamiltonian coordinates

    Dear All, To give a background about myself in Classical Mechanics, I know to solve problems using Newton's laws, momentum principle, etc. I din't have a exposure to Lagrangian and Hamiltonian until recently. So I tried to read about it and I found that I was pretty weak in coordinate...
  17. B

    Primary constraints for Hamiltonian field theories

    I am currently trying to carry out the construction of the generalised Hamiltonian, constraints and constraint algebra, etc for a particular field theory following the procedure in Dirac's "Lectures on quantum mechanics". My question is the following: I have momentum variables that depend on the...
  18. K

    Hamiltonian Formulation of GR for Elliptic Spacetime?

    This has bothered me for some time. In the ADM formulation, we foliate spacetime into 3+1 dimensions by creating 3 dimensional hypersurfaces via ##T = constant## along the worldline of some observer whose proper time is ##T##. This allows us to write dynamical equations for the evolution of some...
  19. Einj

    Partition function for position-independent hamiltonian

    Hi everyone. Suppose I have an Hamiltonian which doesn't depend on the position (think for example to the free-particle one H=p^2/2m). I know that the classical partition function for the canonical ensemble is given by: $$ Z(\beta)=\int{dpdq e^{-\beta H(p,q)}}. $$ What does it happen to...
  20. G

    Given a Hamiltonian how do you pick the most convenient Hilbert space?

    For example, I have a 3D particle that experiences a harmonic oscillator potential only in the X,Y plane for all Z ie. a free particle in the Z direction. This seems like cylindrical coordinates but I'm not sure how to express the Hilbert space if I want to be able to describe states and...
  21. A

    Book for Hamiltonian and Lagrangian mechanics

    I am learning Hamiltonian and Lagrangian mechanics and looking for a book that starts with Newtonian mechanics and then onto Lagrangian & Hamiltonian mechanics. It should have some historical context explaining the need to change the approaches for solving equation of motions. Also it should...
  22. A

    Interaction Hamiltonian of Scalar QED

    Homework Statement Problem 7.15 from Aitchison and Hey, Volume I, 3rd Edition. Verify the forum (7.139) of the interaction Hamiltonian \mathcal{H_{S}^{'}}, in charged spin-0 electrodynamics. Equation 7.139 is \mathcal{H_{S}^{'}}= - \mathcal{L}_{int} - q^2 (A^0)^2 \phi^{\dagger} \phi...
  23. L

    Diagonalization of a hamiltonian for a quantum wire

    I try diagonalize the Hamiltonian for a 1D wire with proximity-induced superconductivity. In the case without a superconductor is all fine. However, with a superconductor I don't get the correct result for the energy spectrum of the Hamiltonian (arxiv:1302.5433) H=\eta(k)τz+Bσ_x+αkσ_yτ_z+Δτ_x...
  24. J

    MHB K5 Graph: Hamiltonian Circuits & Analysis

    Consider the complete graph with 5 vertices, denoted by K5. E.) Does K5 contain Hamiltonian circuits? If yes, draw them. I know that a Hamiltonian circuit is a graph cycle through a graph that visits each node exactly once. However, the trivial graph on a single node is considered to possesses...
  25. Roodles01

    Hamiltonian matrix and eigenvalues

    OK. An example I have has me stumped temporarily. I'm tired. General spin matrix can be written as Sn(hat) = hbar/2 [cosθ e-i∅sinθ] ...... [[ei∅sinθ cosθ] giving 2 eigenvectors (note these are column matrices) I up arrow > = [cos (θ/2)] .....[ei∅sin(θ/2)] Idown arrow> =...
  26. N

    Symplectic Structure of Thermodynamics and the Hamiltonian

    (inspired partially by this blog post: http://johncarlosbaez.wordpress.com/2012/01/19/classical-mechanics-versus-thermodynamics-part-1/ To my understanding, the thermodynamic configuration space has a nice symplectic structure. For example, using the language of classical mechanics, starting...
  27. B

    Can quantum mechanics predict the likelihood of chemical reactions in mixtures?

    How do I write the non-approximated Schrodinger equation Hamiltonian for a mixture containing 25% by partial pressure of H2 gas and 75% by partial pressure of He gas, at 100 KPa pressure and 298 K?
  28. R

    Hamiltonian as applied to the grand canonical partition function

    Does anyone know of any REALLY good derivations of the grand canonical partition function(T,V,μ) from the hamiltonian. I am using the graduate level thermodynamics book by tester and there appears to be some algebric manipulation that occurs going from the ensemble to the partition function...
  29. S

    Hamiltonian for an unknown dissipative system

    Homework Statement Consider the following Hamiltonian H=\frac{p^2}{2m}e^{\frac{-q}{a}} a: constant m: mass of the particle q corresponds to the coordinate, and p its momentum. note: q' stands for the derivative of q. a) Prove that for p(t) > 0 this system seems to describe a particle...
  30. L

    Question about L.S component of hamiltonian

    Homework Statement I need to find the energies of the lowest level of the following hamiltonian H = p^2/2m + (mw^2r^2)/2 + L.S (spin 1/2 particle of mass m) The tridimensional harmonic oscilator with a coupling term Homework Equations The first two terms are solved, my...
  31. N

    Quantization of hamiltonian with complex form

    In most of textbooks, the canonical quantization procedure is used to quantize the hamiltonian with a simple form, the quadratic form. I just wonder how should we deal with more complex form hamiltonian, such like the ones including interaction terms?
  32. S

    Feynman clock's Hamiltonian matrix reduction

    Homework Statement I have this 2^n*2^n matrix that represent the evolution of a system of $n$ spin. I know that I can have only one excited spin in my configuration a time. (eg: 0110 nor 0101 ar not permitted, but 0100 it is) s_+ is defined with s_x+is_y and s_- is defined with s_x-is_y...
  33. H

    Derivation of ideal gas law by Hamiltonian mechanics

    Hi! I am trying to understand the statistical mechanics derivation of the ideal gas law shown at: http://en.wikipedia.org/wiki/Ideal_gas_law inder "Derivations". First of all, the statement "Then the time average momentum of the particle is: \langle \mathbf{q} \cdot \mathbf{F} \rangle=...
  34. 1

    When Will the Particle Reach Infinity with Given Initial Values?

    Homework Statement At what time does the particle reach infinity given that H(p,x)=(1/2)p^2 -(1/2)x^4. And initial values are x(0)=1 and p(0)=1 Homework EquationsThe hamiltonian equations i believe are given by the partial derivatives let d mean partial derivative so x'=dH/dp and p'=-dH/dx...
  35. maverick280857

    Is the Spin Orbit Hamiltonian really Hermitian?

    The regular spin orbit Hamiltonian is H_{SO} = \frac{q\hbar}{4 m^2 c^2}\sigma\cdot(\textbf{E}\times \textbf{p}) If I consider a 2D system where E = E(x,y) and p is treated as an operator, i.e. \hat{p} = \hat{i}p_x + \hat{j}p_y then, clearly E and p do not commute, so this doesn't look like...
  36. D

    How Are Eigenstates Determined for a Given Hamiltonian Matrix?

    Homework Statement Assume a Hilbert space with the basis vectors \left| 1 \right\rangle, \left| 2 \right\rangle and \left| 3 \right\rangle, and a Hamiltonian, which is described by the chosen basis as: H=\hbar J\left( \begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \\...
  37. L

    Hamiltonian of linear harmonic oscilator

    Could hamiltonian of linear harmonic oscilator be written in the form? ##\hat{H}=\sum^{\infty}_{n=0}(n+\frac{1}{2})\hbar\omega |n\rangle \langle n| ##
  38. E

    Help with electron evolution governed by Hamiltonian

    help with electron evolution governed by Hamiltonian ,,, Homework Statement an electron evolution governed by Hamiltonian H=(p^2) /2m +(1/(4Piε))* (e^2)/(r1-r2) give an energy approximation and what's the physical interpretation of the such a Hamiltonian Homework Equations The...
  39. U

    Spectrum of the Hamiltonian in QFT

    I know in ordinary QM, the spectrum of the Hamiltonian \{ E_{n}\} gives you just about everything you need for the system in question (roughly speaking). So what happens to this spectrum in QFT where |\psi\rangle is now a multiparticle wavefunction in some Fock space? I've been trying to...
  40. G

    Difference hamiltonian and energy

    Hi I am looking for the MOST GENERAL statement that determines, when the Hamilton function and the energy are equal to each other in classical mechanics.
  41. V

    Help with loops in Mathematica (Hamiltonian)

    Homework Statement I have an assignment for my thesis to make Hamiltonian for Schrodinger equation. I won't go into physics part of it, because that is well understood. I need to somehow generate a specific matrix for Hamiltonian (H). Please see the attached file of what I need to get...
  42. B

    Hamiltonian Path - Induction Proof

    Homework Statement Let G be a graph. 1. Let n be a natural number. Use induction to show for all n >= 2 Kn has a Hamiltonian path. 2. Explain how you could use the proof from #1 to show that for all n (natural number) n > 2 Kn has a Hamiltonian cycle. Homework Equations The...
  43. S

    How can the virial theorem be applied to a quantum particle in one dimension?

    Homework Statement A quantum particle, i.e. a particle obeying Schrodinger equation and moving in one dimension experiences a potential ˆV (x). In a stationary state of this system show that ⟨x∂/∂x(ˆV(x)⟩ = ⟨ˆp2/2m⟩ Hint: Consider the time dependence of ⟨ˆxˆp⟩. Homework Equations...
  44. S

    Hamiltonian, hisenberg's eqn of motion etc

    Homework Statement A particle of mass m and charge q is subject to a uniform electrostatic eld ~E . (a) Write down the Hamiltonian of the particle in this system (Hint: consider the potential energy of an electric dipole); (b) Find the Heisenberg equation of motion for the...
  45. C

    Hamiltonian eigenstate problem

    Hi PF members, I am stuck with a problem about larmor precession. I cannot find the eigenstates of the hamiltonian given as H = \frac{\hbar}{2}\begin{pmatrix} \omega_{0} & \omega_{1}\delta(t-t') \\ \omega_{1}\delta(t-t') & \omega_{0} \end{pmatrix} Can anyone help me? Since it has time...
  46. M

    Two two-level atoms and form of the Hamiltonian

    Hello, If we look at a system of two two-level atoms interacting with light, most papers start with a Hamiltonian H_{int}=(\sigma_{1}^{+}+\sigma_{2}^{+})a_{\textbf{k},\lambda} + h.c. That is, we absorb a photon and lost one excitation in the atoms or vice versa. Why do we never...
  47. C

    Perturbation Theory for a Hamiltonian

    Hi guys, this is my first time posting, I'm studying physics at uni, in my third year and things are getting a bit tough, so basically my question is in relation to solving problem 1, (i included a picture...) I missed the class and don't really know what I'm doing. Any help would be appreciated.
  48. A

    Changing the Hamiltonian to a new frame of reference

    Suppose I'm considering particles of mass \mu_i, 1 \leq i \leq 3, located at positions r_i. Suppose I ignore the potential between \mu_1 and \mu_2. Then the Hamiltonian I'd write down would be H = -\frac{1}{2\mu_1}\Delta_1 -\frac{1}{2\mu_2}\Delta_2 - \frac{1}{2\mu_3}\Delta_3 + V_1(r_3 -...
  49. P

    Deriving Dirac Hamiltonian with (+,---) Metric Signature

    Hi can anyone explain how to derive an expression for the Dirac Hamiltonian, I thought the procedure was to use \mathcal{H}= i\psi^{\dagger}\Pi -\mathcal{L}, but in these papers the have derived two different forms of the Dirac equation H=\int d^{3}x...
  50. B

    Finding eigenstates and eigenvalues of hamiltonian

    Hey there, the question I'm working on is written below:- Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by: H = |a'>∂<a''| + |a''>∂<a'| where ∂ is just a real number. Write down the eigenstates...
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