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

    Proving Hamiltonian ≠ Energy for Rotating Ball

    Consider a ball of mass m rotating around an axis Oz (vertical). This ball is on a circle whose center is the same O. Given: Angular velocity of ring is d∅/dt = ω. Mind explaining it so we can prove that Hamiltonian here is different from Energy?!
  2. O

    How Do I Find the Energy of a Non-Hermitian Hamiltonian with Bosonic Operators?

    Hi all, There is a Hamiltonian in terms of "a" and "a^{dagger}"bosonic operators H=ω*(a^{dagger}a+1/2)+alpha*a^2+β*a^{dagger}^2 and ω, alpha and β are real constants and its energy is E=(n+1/2)*epsilon where epsilon is ω^2-4*alpha*β. Now, I tried to find this energy but I couldn't. Would you...
  3. J

    Brain freeze on Dirac EQ v. Dirac Hamiltonian

    Alright. So the Dirac Eq is (i \gamma^{\mu} \partial_{\mu} - m) \psi = 0 or putting the time part on one side with everything else on the other and multiplying by \gamma^0 , i \partial_t \psi = (i \gamma^0 \vec{\gamma} \cdot \nabla + \gamma^0 m) \psi I would think that this is the...
  4. N

    Finding hamiltonian for spring/pulley problem

    Homework Statement a light, inextensible string passes over a small pulley and carries a mass of 2m on one end. on the other end is a mass m, and beneath it, supported by a spring w/ spring constant k, is a second mass m. using the distance x, of the first mass beneath the pulley, and the...
  5. R

    Hamiltonian and lagrangian mechanics

    i'm just ready to start QM and I looked at the text and I turned to Shro eq to see if I could understand it and they mentioned Hamiltonian operator. It looked like the book assumed knowledge of H and L mechanics. Do I need to know this stuff? I wasn't told by others that I needed this. I was...
  6. D

    Why are the Lagrangian and the Hamiltonian defined as they are?

    I have two somewhat related questions. First, why would we care about the Lagrangian L = T - V (or K - U)? I understand with the Hamiltonian H = T +V, the total energy is conserved. But with the Lagrangian, what does it actually mean? Mathematically, it only inverts the potential energy...
  7. B

    Unraveling the Mystery of Hamiltonian Mechanics

    Hi, A fundamental aspect in the Hamiltonian framework of mechanics is that the q's and p's are independent. I feel like I understand the steps in the Legendre transform from Lagrangian to Hamiltonian mechanics, but I don't see how you can go from a system where only the q's are independent...
  8. X

    Expanding the electromagnetic hamiltonian

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  9. B

    Two State Quantum System with a given Hamiltonian

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  10. A

    Why is there a sin^2(theta) factor in the diatomic molecule Hamiltonian?

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  11. H

    Expected Value of Hamiltonian in a Forced Quantum Harmonic Oscillator.

    Homework Statement Given an initial (t=-∞) Fock state , \left|n\right\rangle, and a function f(t), where f(±∞)=0, show that for a Harmonic Oscillator perturbed by f(t)\hat{x} the difference \left\langle H(+∞) \right\rangle - \left\langle H(-∞) \right\rangle is always positive.Homework Equations...
  12. M

    Breaking Hamiltonian of a particle

    Dear all, I have a fundamental question about breaking the Hamiltonian. Here is the description: Suppose a particle, \lambda^{0}, is produced in a high energy nuclear collision with proton beam. It is produced by strong interaction, and it has fixed energy (can be obtained from its...
  13. V

    Parity switching wave functions for a parity invariant hamiltonian?

    Hi guys, I'm reading Shankar and he's talking about the Variational method for approximating wave functions and energy levels. At one point he's using the example V(x) = λx^4, which is obviously an even function. He says "because H is parity invariant, the states will occur with alternating...
  14. L

    Time Dependant Hamiltonian Jacob question

    Homework Statement I have a question given to me by my prof that is a time dependent Hamiltonian H(q,p,t) = g(t)(p2/(2m) + kq2/2) where f(t) has 2 different forms i need to solve 1) eat 2) cos(gt) problem is goldstein only covers conserved hamiltonians in chapter 10 for the H-J...
  15. D

    Generating function, hamiltonian dynamics

    Homework Statement A canonical transformation is made from (p,q) to (P,Q) through a generating function F=a*cot(Q), where 'a' is a constant. Express p,q in terms of P,Q. Homework Equations The Attempt at a Solution A generating function is supposed to be a bridge between (p,q) and...
  16. K

    How to solve for Hamiltonian gradient?

    I am trying to understand how Hamiltonian gradient works. H(q,p)=U(q)+K(p) U(q): potential energy K(p): kinetic energy q: position vector p: momentum vector both p and q are functions of time H(q,p): total energy \frac{d{{q}_{i}}}{dt}=\frac{\partial H}{\partial {{p}_{i}}}...
  17. fluidistic

    Why Use Spherical Coordinates for Hydrogen Atom's Hamiltonian in 3D?

    If I consider the problem of for example the hydrogen atom. I.e. a central force problem with an effective potential V(r) that depends only of r, the distance between the positively charged nucleous and the negatively charged electron. In the Schrödinger's equation, one considers the...
  18. B

    What Does a Zero Hamiltonian Indicate About System Dynamics?

    Homework Statement Basically, I'm given a Hamiltonian H = H(p,q) and asked to find a new Hamiltonian K = K(Q,P,t) using the generating functions method H = 1/2 (p^2 + q^2) Generating function f(q,P,t) = qp sec ( t ) - 1/2 (q^2 + P^2) tan ( t ) So, I have no problem finding the new...
  19. fluidistic

    Hamiltonian, generating function, canonical transformation

    Homework Statement Consider a harmonic oscillator with generalized coordinates q and p with a frequency omega and mass m. Let the transformation (p,q) -> (Q,P) be such that F_2(q,P,t)=\frac{qP}{\cos \theta }-\frac{m\omega }{2}(q^2+P^2)\tan \theta. 1)Find K(Q,P) where \theta is a function of...
  20. L

    Lagrangian hamiltonian mech COC Goldstein 8.27

    Homework Statement a) the lagrangian for a system of one degree of freedom can be written as. L= (m/2) (dq/dt)2sin2(wt) +q(dq/dt)sin(2wt) +(qw)2 what is the hamiltonian? is it conserved? b) introduce a new coordinate defined by Q = qsin(wt) find the lagrangian and hamiltonian...
  21. conquest

    Finding Hamiltonian as Legendre transform on SO(3)

    Homework Statement We need to find the Hamiltonian that corresponds to a given Lagrangian by finding the Legendre transform. The system is a rigid body pinned down in some point. This means the motion is described essentialy by SO(3). So the Lagrangian is given in terms of these matrices and...
  22. J

    Motion of a mass m confined to the x-axis (Hamiltonian)

    Consider a mass m confined to the x-axis and subject to a force Fx=kx where k>0. Write down and sketch the potential energy U(x) and describe the possible motions of the mass. (Distinguish between the cases that E>0 and E<0. It is the part in parenthesis that confuses me. I can't...
  23. W

    Lagrangian vs Hamiltonian in QFT vs QM

    In QFT, Lagrangian is often mentioned. While in QM, it's the Hamiltonian, Is the direct answer because in QFT "position" of particle is focused on and Lagrangian is mostly about position and coordinate while in QM, potential is mostly focus on and Hamiltonian is mostly about potential and...
  24. A

    Hamiltonian Cylinder: Mass m, Radius R, Force F=-kx

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  25. C

    Hamiltonian of the Quantum Harmonic Oscillator-Eigenfunction & Eigenvalue

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  26. C

    Does the Hamiltonian is always equal to the energy of the system?

    Does the Hamiltonian is always equal to the energy of the system?? I have this doubt since a few weeks ago. For the Newtonian case we have that H=K+U, kinetical energy plus potential energy, but given that the definition of the Hamiltonian is H=\dot{q}P-L, my question is, Does exist a system or...
  27. V

    Hamiltonian Commutator: Finding [H,P_x] for Polarization Operator

    Homework Statement How do I obtain [H,P_x]? P_x is the polarization operator. Homework Equations H=-\frac{\hbar^2}{2m}\frac{\partial^2 }{\partial x^2}+V(x) P_x=2Re[c_+^*c_-] The Attempt at a Solution I know how to commute H and x. But somehow can't think of a way to...
  28. N

    Does the time dependent hamiltonian have stationary states?

    It doesn't seem like a time dependent hamiltonian would have stationary states, am I wrong? I've run into conflicting information.
  29. A

    Construction of Hamiltonian for interacting optical fields

    Hi. I am going to start my MSc in a couple of months majoring in nonlinear quantum optics. I have a good basic in quantum mechanics, but have never looked at quantum optics before. My topic will be to investigate quantum properties of nonlinear optical coupler but i have problem with the...
  30. J

    Spin-dependent Hamiltonian of two particles

    Homework Statement Two spin-half particles with spins S1 and S2 interact with a spin-dependent Hamiltonian H=λS1*S2 (the multiplication is a dot product and is a positive constant). Find the eigenstates and eigenvalues of H in terms of |m1,m2>, where (hbar)m1 and (hbar)m2 are the z-components...
  31. L

    Hamiltonian mechanics electromagnetic field

    Homework Statement Let (V(x,t) , A(x,t)) be a 4-vector potential that constructs the electromagnetic field (in gaussian Units) by E(x,t) = -∇V(x,t) - (1/c)δtA(x,t) , B = ∇xA , (x,t) elements of R3xRt Consider the lagrangian L=.5mv2 - eV(x,t) + (ev/c)(dot)A(x,t) a) compute and interpret the...
  32. M

    Find Hamiltonian & EOM for 2 DOF System w/ Lagrangian

    Homework Statement Find the Hamiltonian and Hamilton's equations of motion for a system with two degrees of freedom with the following Lagrangian L = 1/2m1\dot{}xdot12 + 1/2m2\dot{}xdot22 + B12\dot{}xdot1x2 + B21\dot{}xdot1x1 - U(x1, x2) Explain why equations of motion do not depend...
  33. N

    Proof of orthonormality of the hamiltonian when not real

    Hi, I know how to prove the orthonormality of the hamiltonian when it is real but am struggling to work out how to prove it when the hamiltonian is not real. When proving for a real hamiltonian the lefthand side equals zero as H(mn)=H(nm)complexconjugate. but if the hamiltonian is not...
  34. D

    Hamiltonian and Commuting operators

    Hi, A general question.. In analytical mechanics, we take a given hamiltonian and re-write it in term of generalzed coordinates. In a way- we recode the hamiltonian to concern only the "essence" of the problem. However, it seems to me, that in QM we do the opposite- we look for operators that...
  35. A

    What is the difference between Langrangian, Hamiltonian and Newtonian Mechanics?

    What is the main difference between Langrangian, Hamiltonian, and Netwonian Mechanics in physics, and what are the most important uses of them? I'm currently a high school senior, with knowledge in calculus based physics, what would the prerequisites be in order for me to begin Langrangian...
  36. N

    Learning Dirac Notation: Writing Hamiltonian for 3 States

    I am new to quantum physics. My question is how to write the Hamiltonian in dirac notation for 3 different states say a , b , c having same energy. I started with Eigenvaluee problem H|Psi> = E|psi> H = ? for state a? SO it means that indvdually H= E (|a><a|) for state a and for all three...
  37. Y

    Hamiltonian with position spin coupling

    I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation} The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where L is angular momentum operator and S is spin operator. The eigenvalue for \begin{equation}L^2 ...
  38. Y

    How to Solve a Hamiltonian with a Complex Term?

    Homework Statement I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation} Homework Equations The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where L is angular momentum operator and S is spin operator. The...
  39. V

    Is the ISW Hamiltonian Diagonal in the Energy Basis?

    Homework Statement Find the matrix elements of the Hamiltonian in the energy basis for the ISW. Is it diagonal? Do you expect it to be diagonal? Homework Equations H=\frac{p^2}{2m}+V \frac{d}{dt}\langle Q \rangle = \frac{i}{\hbar} \langle[\hat H, \hat Q] \rangle + \langle...
  40. MathematicalPhysicist

    Lagrange & Hamiltonian mech => Newtonia mech.

    My question is simple is every classical mechanics problem which is solvable by Lagrangian & Hamiltonian methods also solvable by Newtonian methods of forces and torques? And why does it seem that LH make solutions to be a lot more easier than Newtonian methods, and is it always this way?
  41. K

    Calculate Expectation Value of Hamiltonian using Dirac Notation?

    Homework Statement I have the state: |\psi>=cos(\theta)|0>+sin(\theta)|1> where \theta is an arbitrary real number and |\psi> is normalized. And |0> and |1> refer to the ground state and first excited state of the harmonic oscillator. Calculate the expectation value of the Hamiltonian...
  42. P

    What Does h.c. Signify in Quantum Mechanics Formulas?

    A textbook gives the following interaction Hamiltonian describing the interaction of an atom (having transition dipole moment \mu) with a photon whose polarization can be \epsilon_{1} or \epsilon_{2}): H = g \Sigma^{2}_{s=1}\mu\bullet\epsilons\sigma^{-}a^{+}_{s} + h.c. where \sigma^{-} is...
  43. T

    Solving a Linear Time-Dependent Hamiltonian Problem

    Homework Statement Suppose the potential in a problem of one degree of freedom is linearly dependent on time such that the Hamiltonian has the form: H= p^2/2m - mAtq where m is the mass of the object and A is contant Using Hamilton's canonical equations that are give below. Find the...
  44. D

    Hamiltonian describing energy transfer to bloch Electron from EM field

    Hi all, I've a hamiltonian that describes the coupling of electrons in a crystal (bloch electrons) to an EM field described by a vector potential A \begin{equation} \mathscr{H} = \frac{e}{mc}\left[\mathbf{p}(-\mathbf{k}) \cdot \mathbf{A}(\mathbf{k}, \omega)\right] \end{equation}...
  45. B

    Commutation of Hamiltonian and time evolution operator

    Can anyone explain how the time evolution operator commutes with the Hamiltonian of a system ( given that the the Hamiltonian does not depend explicitly on t ) ?
  46. A

    Prove the Hamiltonian Operator is Hermitian

    Homework Statement Show that the Hamiltonian operator (\hat{H})=-((\hbar/2m) d2/dx2 + V(x)) is hermitian. Assume V(x) is real Homework Equations A Hermitian operator \hat{O}, satisfies the equation <\hat{O}>=<\hat{O}>* or ∫\Psi*(x,t)\hat{O}\Psi(x,t)dx =...
  47. F

    Energy spectrum from a hamiltonian

    Hi there, just wondered if anyone could help me... If I am given a hamiltonian describing a particle in one dimension H=p^2/2m +1/2 (γ(x-a)^1/2) + K(x-b) how do I go about finding the eigenstates and eigenvalues of this hamiltonian? Many thanks
  48. J

    Weyl ordering of the hamiltonian

    Hi , I can't understand the general formula for weyl ordering of the hamiltonian . It is written in Srednicki field theory book in page 68 . Can someone explain how to derive this formula ?
  49. T

    Eigenvalues and ground state eigenfunction of a weird Hamiltonian

    Hello again everyone! I would like to ask a question regarding this Hamiltonian that I encountered. The form is H = Aa^+a + B(a^+ + a). Then there is this question asking for the eigenvalues and ground state wavefunction in the coordinate basis. The only given conditions are, the commutator...
  50. MathematicalPhysicist

    A question regarding a Hamiltonian.

    In page (3) of the next link:http://www.phy.bris.ac.uk/people/berry_mv/the_papers/Berry401.pdf he writes that one can transform the hamiltonian H=xp to p^2-x^2 with a simple rotation transformation. Which one is that?
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