Hamiltonian Definition and 898 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

    A Question about perturbation theory

    Hello! I have a situation where I have time dependent Hamiltonian, ##H_0(t)## which I can solve for exactly and thus get ##\psi_0## as its eigenfunction (given the initial conditions). Now, on top of this, I add a time independent Hamiltonian, ##H_1## much smaller than ##H_0##. How can I get the...
  2. M

    How should I show that the transformation makes the system Hamiltonian?

    Proof: Consider the transformation ## x=\frac{1}{\sqrt{1+e^{-2q}}} ## and ## y=\frac{1}{\sqrt{1+e^{-2p}}} ## with the Hamiltonian function ## H(q, p)=ap-b\cdot ln(e^{p}+\sqrt{1+e^{2p}})+cq-d\cdot ln(e^{q}+\sqrt{1+e^{2q}}) ##. Let ## \dot{x}=\frac{dx}{dt}=(a-by)x(1-x^2)=(a-by)(x-x^3) ## and ##...
  3. M

    I Matrix elements of Hamiltonian include Dirac delta term

    Currently reading a textbook on non-equilibrium green's functions and I'm stuck in chapter 1 where it recaps just general quantum mechanics because of dirac deltas included in the matrix elements of a generalized hamiltonian. The textbook gives this: I just don't understand how to think about...
  4. R

    Solve this Hamiltonian System in Several Ways

    Good evening, unfortunately I can't get to the solution of my task I wrote for the system: ## \frac{dz}{dt} = \nabla_p H ; \\ \frac{dp}{dt} = - \nabla_z H ## Then the solution would be (as ## \nabla_p H = 0) ##: ## \frac{dz}{dt} = 0 \Rightarrow z = const. ## and ## p = zt + p_0 ##. But...
  5. flyusx

    What To Project Upon When Doing QM Measurements?

    I have read that if one measures the Hamiltonian and receives a value of h2, then the quantum state will be in ##|h2\rangle##. Finding the probability of a1 is done by projecting ##|a1\rangle## upon ##|h2\rangle## divided by ##\langle h2|h2\rangle##. In other words: $$\frac{|\langle...
  6. J

    Eigenvalues of Hamiltonian operator

    Hello, I try to solve this problem, and I think a) wasn't too hard, I have the following solution: ##H = \lambda (\frac{\vec{S^2-(\vec{S_1}^2+\vec{S_2}^2)}{2})##. I struggle with 2. I find it very abstract. When I have H as a matrix I know how to calculate eigenvalues, but I don't know how...
  7. P

    A How the mass term of the Hamiltonian for a scalar fields transform?

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  8. T

    Recommendation for a book on Hamiltonian Mechanics

    Homework Statement: Practical examples of Hamiltonian Mechanics sought Relevant Equations: Hamilton Jacobi Equations, MTW Hi, I'm currently a bit stuck on Box 24.2 in MTW. I really need to get a better understanding of Hamiltonian Mechanics to be able to work my way through this and I...
  9. Rayan

    The time-dependence of the expectation values of spin operators

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  10. T

    I Heisenberg Equations of Motion for Electron in EM-field

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  11. P

    I Effective Hamiltonian to Rotational Term Values

    Hello, I am fairly new to the world of molecular spectroscopy, so I apologize for any ignorance on my part. For the last few months, I've been working on a diatomic spectral simulation tool and have reached a point where I want to incorporate more advanced theory to model complex interactions in...
  12. P

    I Is a Time Rotation Operator Analogous to Spatial Rotations Possible?

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  13. Rayan

    Possible energy values given Hamiltonian

    So first I rewrote H as a matrix: $$ H = \begin{pmatrix} a & b \\ b & c \end{pmatrix} $$ And tried to find the eigenvalues/energies of H, so I solved $$ det (H - \lambda I ) = \begin{vmatrix} a-\lambda & b \\ b & c-\lambda \end{vmatrix} = (a-\lambda)(c-\lambda) - b^2 = ac - a\lambda -...
  14. M

    I Hamiltonian of the bead rotating on a horizontal stick

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  15. M

    A Confused about going from relativistic to non-relativistic Hamilonian

    Hello! My question is related to going from Eq. 32 to Eq. 33 in this paper (however I have seen this in other papers, too). In summary, starting with: $$H \propto \bar{e}\gamma_\mu\gamma_5 e \bar{q}\gamma^\mu q$$ where we have the gamma matrices, e is the electron field and q is the...
  16. T

    I Interpreting Unitary Time Evolution

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  17. T

    I Quantum Circuit Confusion On Time Evolution

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  18. Coelum

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  19. A

    Classical Introductory books to Hamiltonian chaos

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  20. H

    I The Hamiltonian elements in Anderson dimer

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  21. E

    I The Hamiltonian and Galilean transformations

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  22. T

    I Fourier Transform of Photon Emission Hamiltonian

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  23. H

    How to find the eigenvector for a perturbated Hamiltonian?

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  24. Y

    A Learning DFT: Inhomogeneous Electron Gas (Hohenberg) Question

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  25. graviton_10

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  26. patric44

    Parameters in Bohr-Mottelson Collective Hamiltonian

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  27. H

    Unitary vector commuting with Hamiltonian and effect on system

    Hi, I'm not sure to understand what ##| \phi_n \rangle = \sum_i \alpha_i |\psi_n^i## means exactly or how we get it. From the statement, I understand that ##[U,H] = 0## and ##H|\psi_n \rangle = E_n|\psi_n \rangle## Also, a linear combination of all states is also an solution. If U commutes...
  28. H

    I Momentum and Action: Understanding Lagrangian Mechanics

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  29. S1000

    Hamiltonian of a Point particle on a frictionless plane

    I am stuck on Question e and then how to proceed to f. I cannot seem to show this using the steps in the prior questions. My answers are: a) b) c) c) continued - and d) at the bottom of the page d)I am not sure where I have gone wrong, as I am not sure how to apply the relevant...
  30. James1238765

    A Hamiltonian V and T of a lattice?

    A toy model of a QFT lattice (in 1 dimension) is given in [here] (at 5:55): We assume that ##\Psi## is a vector set of four complex numbers having some values at every point on the grid, for instance: $$\Psi_{100} = \begin{bmatrix} 1+2i \\ 3+4i \\ 5+6i \\ 7+8i \end{bmatrix}$$ and...
  31. yucheng

    A Jaynes-Cummings Hamiltonian: Where did the time dependence go?

    Consider the interaction of a two level atom and an electric field (semiclassically, we treat the field as 'external' i.e. not influenced by the atom; the full quantum treats the change in the field as well) Electric field in semiclassical Hamiltonian: plane wave ##H_{int,~semiclassical}=-\mu...
  32. Omega0

    B Lagrangrian and Hamiltonian mechanics: A historical picture

    Hi, I believe that I have an acceptable level of understanding where SRT, GRT, QM and QFT come from. This is not true for me regarding the "good old stuff". Newton, okay, this is relatively (:wink:) clear to me but do you know something about the historical motivation for Lagrangian and...
  33. yucheng

    Canonical transformations of a quantized Hamiltonian?

    Source: Scully and Zubairy, Quantum Optics, Section 1.1.2 Quantization Questions: 1. Why are the destruction and creation operators considered a canonical transformations? 2. If these are canonical transformations, does it suggest that we are also canonically transforming the Hamiltonian...
  34. codebpr

    A The kinetic term of the Hamiltonian is not positive definite

    I am trying to reproduce the results from this paper. On page 10 of the paper, they have an equation: $$ \frac{S}{T}=\int dt\sum _{n=0,1} (\dot{c_n}{}^2-c_n^2 \omega _n^2)+11.3 c_0^3+21.5 c_0 c_1^2+10.7 c_0 \dot{c_0}{}^2+3.32 c_0 \dot{c_1}{}^2+6.64 \dot{c_0} c_1 \dot{c_1} \tag{B12} $$ where they...
  35. A

    Orbital angular momentum Hamiltonian

    I think that the quantum numbers are l=1 and ml=0, so I write the spherical harmonic Y=Squareroot(3/4pi)*cos(theta). I would like to know how to compute the wave function at t=0, then I know it evolves with the time-evolution operator U(t), to answer the first request.
  36. hilbert2

    A Changing Hamiltonian with some eigenvalues constant

    Suppose some quantum system has a Hamiltonian with explicit time dependence ##\hat{H} := \hat{H}(t)## that comes from a changing potential energy ##V(\mathbf{x},t)##. If the potential energy is changing slowly, i.e. ##\frac{\partial V}{\partial t}## is small for all ##\mathbf{x}## and ##t##...
  37. PhysicsRock

    I How does one compute the Fourier-Transform of the Dirac-Hamiltonian?

    Greentings, I've dealt with Quantum Theory a lot, but there's one thing I don't quite understand. When deriving the Fermion-Propagator, say ##S_F##, all the authors I've read from made use of the Fourier-Transform. Basically, it always goes like $$ \begin{align} H_D S_F(x-y) &= (i \hbar...
  38. lindberg

    B What is a non-local Hamiltonian?

    If I understand it correctly, the Hamiltonian represents the total energy of the system. Can it be non-local? If yes, doesn't this contradict relativistic locality?
  39. O

    How can negative integers be used in deriving the Hamiltonian for open strings?

    On ***page 38*** of Becker Becker Schwarz, we're given ***equation 2.69*** which is the Hamiltonian for a string given as $$H=\frac{T}{2}\int_{0}^{\pi}(\dot{X}^{2}+X^{'2})$$ Considering the open string we have...
  40. T

    I Numerical Calculation of Hamiltonian Overlaps

    Hi all, I am currently reading through this paper: https://iopscience.iop.org/article/10.1088/1367-2630/10/4/045030 and would like to reproduce their results for N=5. My roadblock is with (9), which models the classical motion of the system. Now symbolically finding the eigenstates of the matrix...
  41. Simobartz

    I Hamiltonian formalism and partition function

    In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##. For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##. Can i consider ##\theta## or ##cos...
  42. Salmone

    I Strange Hamiltonian of two particles on the surface of a sphere

    I have a problem with this Hamiltonian: two identical particles of mass ##m## and spin half are constrained to move on the surface of a sphere of radius ##R##. Their Hamiltonian is ##H=\frac{1}{2}mR^2(L_1^2+L_2^2+\frac{1}{2}L_1L_2+\frac{1}{2}S_1S_2)##. By introducing the two operators...
  43. Mayhem

    Where Can I Find Information on the Total Molecular Hamiltonian?

    Hello. As an assignment, I have to explain the total molecular Hamiltonian. Problem is, I can't find it anywhere in my book (Atkins, Physical Chemistry: Quanta, Matter, and Change, 2nd Edition), even when I access the index for "Hamiltonian -> polyatomic molecules". They do give the electronic...
  44. T

    A Secular Approximation of Dipole-Dipole Hamiltonian

    Hey folks, I'm looking for a derivation of the secular approximation of the dipole-dipole Hamiltonian at high magnetic fields. Does anybody know a reference with a comprehensive derivation or can even provide it here? Given we have the dipolar alphabet, I'd like to understand (in the best...
  45. alexmurillo242

    Mathematica Piecewise Time-Dependent Hamiltonian in Mathematica Strategy

    Hi all, I'm doing some light simulations for an experiment I'm going to be running soon. I've ran through the math symbolically on paper but I'm not exactly eager for handling this large of matrices by hand so I'm trying to work through it and see if I can generate a simulated signal to compare...
  46. Salmone

    I Separability of a Hamiltonian with spin

    I'd like to know if this Hamiltonian ##\hat{H}=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2+\frac{A}{\hbar^2}(J^2-L^2-S^2)## is separable into two parts ##H_1=\frac{p^2}{2m}+\frac{1}{2}m\omega^2r^2## and ##H_2=\frac{A}{\hbar^2}(J^2-L^2-S^2)## and ##[H_1,H_2]=0##. Here A is a constant. I did so...
  47. Salmone

    I Separable Hamiltonian for central potential

    In a central potential problem we have for the Hamiltonian the expression: ##H=\frac{p^2}{2m}+V(r)## and we use to solve problems like this noting that the Hamiltonian is separable, by separable I mean that we can express the Hamiltonian as the sum of multiple parts each one commuting with the...
  48. J

    I How to get the energy eigenvalue of the Hamiltonian: H0+λp/m ?

    Someone says we can choose the new eigenstate: exp(-iλx/hbar)*ψ,and let the momentum operator p acts upon this new state. At the same time, so does p^2. Something miraculous will happen afterwards. My question is: how to image this point? Thank you very much.
  49. Samama Fahim

    I Deriving the Commutator of Exchange Operator and Hamiltonian

    In the boxed equation, how would you get the right hand side from the left hand side? We know that ##H(1,2) = H(2,1)##, but we first have to apply ##H(1,2)## to ##\psi(1,2)##, and then we would apply ##\hat{P}_{12}##; the result would not be ##H(2,1) \psi(2,1)##. ##\hat{P}_{12}## is the exchange...
  50. Salmone

    I Hamiltonian of a particle moving on the surface of a sphere

    In a quantum mechanical exercise, I found the following Hamiltonian: Consider a particle of spin 1 constrained to move on the surface of a sphere of radius R with Hamiltonian ##H=\frac{\omega}{\hbar}L^2##. I knew that the Hamiltonian of a particle bound to move on the surface of a sphere was...
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