What is Hamiltonian: Definition and 893 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. 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...
  2. P

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

    The Hamiltonian for a scalar field contains the term $$\int d^3x m^2 \phi(x) \phi(x)$$, does it changs to the following form? $$\int d^3x' {m'}^2 \phi'(x') \phi'(x')=\int d^3x' \gamma^2{m}^2 \phi(x) \phi(x)$$? As it is well known for a scalar field: $$\phi'(x')=\phi(x)$$ .
  3. 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...
  4. Rayan

    The time-dependence of the expectation values of spin operators

    So first I derived the expressions for the dynamics of the spin operators and got: $$ \frac{d\hat{S}_y}{dt} = w\hat{S}_x^H $$ $$ \frac{d\hat{S}_x}{dt} = w\hat{S}_y^H $$ $$ \frac{d\hat{S}_z}{dt} = 0 $$ Now I want to calculate the time-dependence of the expectation values of the spin operators...
  5. T

    I Heisenberg Equations of Motion for Electron in EM-field

    Consider the Heisenberg picture Hamiltonian $$H(t) = \int_{\textbf{r}}\psi^{\dagger}(\textbf{r},t)\frac{(-i\hbar\nabla+e\textbf{A})^{2}}{2m}\psi(\textbf{r},t)$$ where ##\psi(\textbf{r},t)## is a fermion field operator. To find the equations of motion that ##\psi,\psi^{\dagger}## obey. I would...
  6. 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...
  7. P

    I Operators related to time

    We studied about the time translation operator and that its generator is the hamiltonian the question is could there be a time rotation operator in analogy with rotations in space and what would be it's relation to relativity?
  8. 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 -...
  9. M

    I Hamiltonian of the bead rotating on a horizontal stick

    Hi, In David Morin's "Introduction to classical mechanics", Problem 6.8, when deriving Hamiltonian of the bead rotating on a horizontal stick with constant angular speed, the Lagrangian derivative over angular speed isn't included. Why is that? Specifically, the Lagrangian takes form...
  10. 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...
  11. T

    I Interpreting Unitary Time Evolution

    Hi all, This should be a simple question but it has been bothering me for a bit: Consider 2 Hamiltonian terms ##H_{1},H_{2}## that satisfy ##[H_{1},H_{2}] = 0##. Suppose we are working in the Heisenberg picture and we time evolve some operator ##A## according to ##A(t) =...
  12. T

    I Quantum Circuit Confusion On Time Evolution

    Hi all, When working in the Heisenberg picture, we can represent implementing time evolution on an operator via a Hamiltonian H through a quantum circuit type picture like the following: where time is on the vertical axis and increases going up and the block represents the unitary gate...
  13. Coelum

    Two-level Quantum System - initial state

    TL;DR Summary: Find the initial state of a two-level quantum system, given the probability of measurements for two observables and the expected value of an operator. Dear PFer's, I have been struggling with the following problem. It was assigned at an exam last year. Problem Statement For a...
  14. A

    Classical Introductory books to Hamiltonian chaos

    I'm looking for books (or any other reference) to start studying the emergence of Hamiltonian chaos and KAM theory. You know, something that doesn't require a Ph.D in math to understand but is comprehensive enough to give a good understanding of the topic. Added bonus if it has a discussion on...
  15. H

    I The Hamiltonian elements in Anderson dimer

    In a system with two orbitals ##c## and ##d## (each with two spin degrees of freedom), consider the Hamiltonian ##H=V(d^{\dagger}_{\uparrow} c_{\uparrow} + c^{\dagger}_{\uparrow}d_{\uparrow}+d^{\dagger}_{\downarrow} c_{\downarrow} + c^{\dagger}_{\downarrow}d_{\downarrow})##. Also suppose that...
  16. E

    I The Hamiltonian and Galilean transformations

    In a classical example, for a system consisting of a mass attached to a spring mounted on a massless carriage which moves with uniform velocity U, as in the image below, the Hamiltonian, using coordinate q, has two terms with U in it. But if we use coordinate Q, ##Q=q−Ut##, which moves with the...
  17. T

    I Fourier Transform of Photon Emission Hamiltonian

    Hey all, I just wanted to double check my logic behind getting the Fourier Transform of the following Hamiltonian: $$H(x) = \frac{ie\hbar}{mc}A(x)\cdot\nabla_{x}$$ where $$A(x) = \sqrt{\frac{2\pi\hbar c^2}{\omega L^3}}\left(a_{p}\epsilon_{p} e^{i(p\cdot x)} + a_{p}^{\dagger}\epsilon_{p}...
  18. H

    How to find the eigenvector for a perturbated Hamiltonian?

    Hi, I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian. However, I don't see how to find the eigenvectors. To find the eigenvalues for the first excited state I build this matrix ##...
  19. Y

    A Learning DFT: Inhomogeneous Electron Gas (Hohenberg) Question

    I'm reading through Hohenberg's seminal paper titled: "Inhomogeneous Electron Gas" that help lay the foundation for what we know of as Density Functional Theory (DFT) by proving the existence of a universal functional that exactly matches the ground-state energy of a system with a given...
  20. graviton_10

    Expected value of variance of Hamiltonian in coherent states

    I am trying to find the expected value of the variance of energy in coherent states. But since the lowering and raising operators are non-hermitian and non-commutative, I am not sure if I am doing it right. I'm pretty sure my <H>2 calculation is right, but I'm not sure about <H2> calculation...
  21. patric44

    Parameters in Bohr-Mottelson Collective Hamiltonian

    Hi all I was reading a certain paper that involves solving the Bohr-Mottelson Hamiltonian for a 5dimential square well potential, the B-M Hamiltoian reads: my question is just how do I calculate the mass parameter "B" for a certain nuclei, and with a 5D infinite potential well how do I get the...
  22. 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...
  23. H

    I Momentum and Action: Understanding Lagrangian Mechanics

    Hi, In my book I have and expression that I don't really understand. Using the definition of action ##\delta S = \frac{\partial L}{\partial \dot{q}} \delta q |_{t_1}^{t_2} + \int_{t_1}^{t_2} (\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}}) \delta q dt## Where L...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. 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...
  29. 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...
  30. 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.
  31. 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##...
  32. 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...
  33. 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?
  34. O

    Hamiltonian open string

    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...
  35. 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...
  36. 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...
  37. 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...
  38. Mayhem

    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...
  39. 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...
  40. 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...
  41. 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...
  42. 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...
  43. 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.
  44. 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...
  45. 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...
  46. AzAlomar

    A Reference for empirical Tight-binding Hamiltonian of spds* vs sps*

    Is there a clear reference article/note for the 20X20 Hamiltonian matrix of the spds* Zinc-Blende system similar to the sps* reference in [1] Table (A) of Vogl P, Hjalmarson HP, Dow JD. A Semi-empirical tight-binding theory of the electronic structure of semiconductors†. J Phys Chem Solids...
  47. Salmone

    I How to obtain Hamiltonian in a magnetic field from EM field?

    To calculate the Hamiltonian of a charged particle immersed in an electromagnetic field, one calculates the Lagrangian with Euler's equation obtaining ##L=\frac{1}{2}mv^2-e\phi+e\vec{v}\cdot\vec{A}## where ##\phi## is the scalar potential and ##\vec{A}## the vector potential, and then we go to...
  48. L

    A SO(3) group, Heisenberg Hamiltonian

    We have commutation relation ##[J_j,J_k]=i \epsilon_{jkl}J_l## satisfied for ##2x2##, ##3x3##, ##4x4## matrices. Are in all dimensions these matrices generate ##SO(3)## group? I am confused because I think that maybe for ##4x4## matrices they will generate ##SO(4)## group. For instance for...
  49. K

    I Hamiltonian of a particle in a magnetic field

    I've just started Quantum mechanics by McIntyre and have understood the following about operators which the author wrote till chapter 2: Each observable has an operator Operators act on kets to produce another kets. Only eigenvalues of an operator are possible values of a measurement. Now...
  50. Hari Seldon

    A Deriving Navier-Stokes: Lagrangian & Hamiltonian Methods

    Is that possible to derive the Navier-Stokes equations with Lagrangian and Hamiltonian methods? If yes, how? and if it is not possible, why?
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