Hamiltonian Definition and 901 Threads

  1. K

    B Question about Hamiltonian and kinetic energy

    basically, as far as I know we can derive 1/2mv2 from ∫F⋅ds=1/2mv2=(p2)/2m for wave equation we use Hamiltonian H=P2/2m+V where P and V are both operators However, I wonder how we can say that P2/2m is the term for kinetic energy because ∫F⋅ds=∫(dp/dt)⋅ds=1/2mv2 is saying that knowing F and...
  2. S

    A Symmetry of hamiltonian under renormalization

    Hi everyone, Currently, I am self-learning Renormalization and its application to PDEs, nonequilibrium statistical mechanics and also condensed matter. One particularly problem I face is on the conservation of symmetry of hamiltonian during renormalization. Normally renormalization of...
  3. J

    I Hamiltonian for mass on a smooth fixed hemisphere

    I am trying to figure out how to get the Hamiltonian for a mass on a fixed smooth hemisphere. Using Thorton from example 7.10 page 252 My main question is about the Potential energy= mgrcosineθ is the generalized momenta Pdotθ supposed to be equal to zero because θ is cyclic? Or is Pdotθ=...
  4. Konte

    I Hamiltonian matrix - Eigenvectors

    Hello everybody, From a complete set of orthogonal basis vector ##|i\rangle## ##\in## Hilbert space (##i## = ##1## to ##n##), I construct and obtain a nondiagonal Hamiltonian matrix $$ \left( \begin{array}{cccccc} \langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\...
  5. redtree

    A Conjugate variables in the Fourier and Legendre transforms

    In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...
  6. P

    I Yes, thank you for explaining it to me. I have a much better understanding now.

    Hi, I am a physics student and i was asked to answer some questions about Hydrogen atom wavefunctions. I hope you can help me (sorry for my english, is not my motherlanguage, i will try to explain myself properly) 1. In order to find hamiltonian eigenfunctions of Hydrogen atom, we make then be...
  7. T

    I Permutation operator and Hamiltonian

    The permutation operator commutes with the Hamiltonian when considering identical particles, which implies: $$ [\hat{P}_{21}, \hat{H}] = 0 \tag{1}$$ Now given a general eigenvector ##{\lvert} {\psi} {\rangle}##, where $$ \hat{P}_{21} (\hat{H}{\lvert}{\psi}){\rangle} = (\hat{P}_{21} \hat{H})...
  8. L

    Perturbed Hamiltonian Matrix for Quantum Harmonic Oscillator

    Homework Statement How to calculate the matrix elements of the quantum harmonic oscillator Hamiltonian with perturbation to potential of -2cos(\pi x) The attempt at a solution H=H_o +H' so H=\frac{p^2}{2m}+\frac{1}{2} m \omega x^2-2cos(\pi x) I know how to find the matrix of the normal...
  9. binbagsss

    Real Scalar Field, Hamiltonian, Conjugate Momentum

    ## L(x) = L(\phi(x), \partial_{u} \phi (x) ) = -1/2 (m^{2} \phi ^{2}(x) + \partial_{u} \phi(x) \partial^{u} \phi (x))## , the Lagrange density for a real scalar field in 4-d, ##u=0,1,2,3 = t,x,y,z##, below ##i = 1,2,3 =x,y,z## In order to compute the Hamiltonian I first of all need to compute...
  10. G

    Hamiltonian formulation and the Kepler problem

    This isn't exactly homework, but something which you'd get as an assignment, so I'll still post it here in order to reach the right people.. I'm attempting to freshen up my knowledge on Hamiltonian systems, so I've tried to formulate the Kepler problem in Hamiltonian dynamics...
  11. P

    Many particle physics - Hamiltonian for Fermi system

    Homework Statement Working through problems in Mahan's 'Many Particle Physics' book, and at the end of the 1st chapter there's a question where we're asked to consider a fermion system with three energy states with eigenvalues E1, E2, E3, and matrix elements M12, M23, M13 which connect them and...
  12. P

    Classical Lagrangian and Hamiltonian Celestial Mechanics

    Hi folks, I am looking to learn the Lagrangian and Hamiltonian approach to celestial mechanics - I have previous experience in Newtonian numerical solutions for orbital motion but am looking to achieve similar things but through the use of Hamiltonian formulations. After having a poke around...
  13. S

    I Vanishing Hamiltonian for quantum path integral

    The path integral in quantum mechanics involves a factor ##e^{iS_{N}/\hbar}##, where ##S_{N}\equiv \sum\limits_{n=1}^{N+1}[p_{n}(x_{n}-x_{n-1})-\epsilon H(p_{n},x_{n},t_{n})].## In the limit ##N \rightarrow \infty##, ##S_{N}## becomes the usual action ##S## for a given path.When the...
  14. H

    How Is the Hamiltonian Derived from the Proca Lagrangian?

    Homework Statement Starting from the proca lagrangian $$L=-\frac14 F_{uv}F^{uv}+\frac12 m^2 A_uA^u$$ Homework Equations $$H=\sum p_i\dot{q_i}-L$$ The Attempt at a Solution $$L=-\frac14F_{uv}F^{uv}+\frac12m^2A_uA^u\rightarrow\partial_uF^{uv}+m^2A^v=0$$ $$p^i=\frac{\partial L}{\partial...
  15. 4

    I QFT for the Gifted Amateur Question (2)

    In chapter 11, Lancaster takes us through the 5 steps for canonical quantization of fields, and in example 11.3 he derives a mode expansion of the Hamiltonian which ends in this: $$E=\int d^3 p E_p (a _p^{\dagger} a_p + \frac{1}{2} \delta^{(3)}(0)) $$ Which I have no problem with, but then...
  16. Milsomonk

    Determine the potential V(x) from the Hamiltonian

    Homework Statement Assuming psi is an eigenstate of the Hamiltonian (TISE) and that E=0, determine the potential V(x) appearing in the Hamiltonian. Homework Equations Time Independant Schrodinger Equation - 1 Dimensional (x) I am given the wavefunction psi = N/(1+x^2) I have found the...
  17. TheCapacitor

    Classical Best analytical mechanics textbook recommandation

    Hello, I'm a second year physics student. We are going to use "hand and finch analytical mechanics", however the reviews I saw about this book are bad. I've already taken calculus for mathematicians, linear algebra, classical mechanics, special relativity, and electromagnetism. The topics it...
  18. A

    I Finite difference Hamiltonian

    Suppose I am given some 1D Hamiltonian: H = ħ2/2m d2/dx2 + V(x) (1) Which I want to solve on the interval [0,L]. I think most of you are familiar with the standard approach of discretizing the interval [0,L] in N pieces and using the finite difference formulas for V and the...
  19. Gopal Mailpalli

    Classical Good book for Lagrangian and Hamiltonian Mechanics

    This book should introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I know about Goldstein's Classical Mechanics, but don't know how do I approach the book.
  20. F

    Hamiltonian as the generator of time translations

    In literature I have read it is said that the Hamiltonian ##H## is the generator of time translations. Why is this the case? Where does this statement derive from? Does it follow from the observation that, for a given function ##F(q,p)##, $$\frac{dF}{dt}=\lbrace F,H\rbrace +\frac{\partial...
  21. M

    How to Find Ψ(x,t) for a Given Hamiltonian Matrix and Initial State?

    Homework Statement I have the matrix form of the Hamiltonian: H = ( 1 2-i 2+i 3) If in the t=0, system is in the state a = (1 0)T, what is Ψ(x,t)? Homework Equations Eigenvalue equation The Attempt at a Solution So, I have diagonalized given matrix and got...
  22. A

    I How to diagonalize Hamiltonian with Zeeman field

    Recently I have been asked to solve the problem of an electron in a Zeeman-field that couples the spin of the electron to the magnetic field. I am not sure how to correctly set up the problem. I think, however, that what I have done on the picture is correct. The usual p^2/2m + V term in the...
  23. H

    I Meaning of the Hamiltonian when it is not energy

    Suppose the initial radial position and radial velocity of the bead are ##r_0>0## and ##0## respectively. Then ##E## is negative. Is there any significance to the negative value of ##E##? Note that ##E## is defined by (5.52) and given by (5.144) below.
  24. I

    I How to change the Hamiltonian in a change of basis

    Dear all, The Hamiltonian for a particle in a magnetic field can be written as $$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$ where ##\boldsymbol\sigma## are the Pauli matrices. This Hamiltonian is written in the basis of the eigenstates of ##\sigma_z##, but how is it...
  25. I

    Magnetic field Hamiltonian in different basis

    Homework Statement A spin-1/2 electron in a magnetic field can be regarded as a qubit with Hamiltonian $$\hat{H} = \frac{1}{2}g\mu_B\textbf{B}\cdot\boldsymbol\sigma$$. This matrix can be written in the form of a qubit matrix $$ \begin{pmatrix} \frac{1}{2}\epsilon & t\\ t^* &...
  26. I

    Eigenfunction of a spin-orbit coupling Hamiltonian

    Dear all, The Hamiltonian for a spin-orbit coupling is given by: \mathcal{H}_1 = -\frac{\hbar^2\nabla^2}{2m}+\frac{\alpha}{2i}(\boldsymbol \sigma \cdot \nabla + \nabla \cdot \boldsymbol \sigma) Where \boldsymbol \sigma = (\sigma_x, \sigma_y, \sigma_z) are the Pauli-matrices. I have to...
  27. M

    What are Lagrangian and Hamiltonian mechanics?

    Only thing I know about them is that they are alternate mechanical systems to bypass the Newtonian concept of a "force". How do they achieve this? Why haven't they replaced Newtonian mechanics, if they somehow "invalidate" it or make it less accurate, by the Occam's razor principle? Thanks in...
  28. V

    Bohr frequency of an expectation value?

    Homework Statement Consider a two-state system with a Hamiltonian defined as \begin{bmatrix} E_1 &0 \\ 0 & E_2 \end{bmatrix} Another observable, ##A##, is given (in the same basis) by \begin{bmatrix} 0 &a \\ a & 0 \end{bmatrix} where ##a\in\mathbb{R}^+##. The initial state of the system...
  29. Arnd Obert

    I Density of states with delta function

    Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...
  30. P

    I Why do we differentiate in physics and why twice?

    I have a basic understanding of the reason why we look for derivative or integration in Physics, based on the water flow example, where integration is the process of accumulating the varying water flow rate "2x" , while we reverse to the water flow rate by differentiating " x squared " the...
  31. AwesomeTrains

    Rayleigh–Ritz method - Yukawa coulomb potential

    Hello everyone Homework Statement I have been given the testfunction \phi(\alpha, r)=\sqrt{(\frac{\alpha^3}{\pi})}exp(-\alpha r) , and the potential V(r,\theta, \phi)=V(r)=-\frac{e^2}{r}exp(\frac{-r}{a}) Given that I have to write down the hamiltonian (in spherical coordinates I assume), and...
  32. B

    A Effect of time reversed hamiltonian acting on a state?

    Hi, I have been trying to get my head around the effect of a time reversed hamiltonian ##H^B(t)=H(-t)=T^{-1}H^F T ## on a state ket ##|\psi>##, where ##H^F=H## is the regular hamiltonian for the system (energy associated with forward time translation) and ##H^B=H(-t)## is the time reversed...
  33. U

    How many constants-of-motion for a given Hamiltonian?

    I am using Jose & Saletan's "Classical Dynamics", where they introduce a rather contrived Hamiltonian in the problem set: H(q_1,p_1,q_2,p_2) = q_1p_1-q_2p_2 - aq_1^2 + bq_2^2 where a and b are constants. This Hamiltonian has several constants-of-motion, including f = q1q2, as can be easily...
  34. H

    Finding a matrix representation of a Hamiltonian.

    Homework Statement The Hamiltonian H for a certain physical quantum mechanical system has three eigenvectors {|v1>, |v2>, |v3>} satisfying: H|vj> = (2-j)a|vj> Write down the matrix representing H in the representation {|v1>, |v2>, |v3>} . Homework EquationsThe Attempt at a Solution I though...
  35. Z

    Lagrangian Mechanics: Find Lagrangian & Hamiltonian of Pendulum

    Homework Statement We have a mas m attached to a vertical spring of length (l+x) where l is the natural length. Homework Equations Find the Lagrangian and the hamiltonian of the system if it moves like a pendulum The Attempt at a Solution we know that the lagrangian of a system is defined as...
  36. R

    Classifaction of equilibrium points for a Hamiltonian System

    Homework Statement For the system: \frac{dx}{dt}=x\cos{xy} \: \: \frac{dy}{dt}=-y\cos{xy} (a) is Hamiltonian with the function: H(x,y)=\sin{xy} (b) Sketch the level sets of H, and (c) sketch the phase portrait of the system. Include a description of all equilibrium points and any saddle...
  37. I

    Eigenvalues of a spin-orbit Hamiltonian

    Good day everyone, The question is as following: Consider an electron gas with Hamiltonian: \mathcal{H} = -\frac{\hbar^2 \nabla^2}{2m} + \alpha (\boldsymbol{\sigma} \cdot \nabla) where α parameterizes a model spin-orbit interaction. Compute the eigenvalues and eigenvectors of wave vector k...
  38. T

    I Schrödinger equation and interaction Hamiltonian

    Given 1A.1 and 1A.2, I have been trying to apply the Schrödinger equation to reproduce 1A.3 and 1A.4 but have been struggling a bit. I was under the assumption that by applying ##\hat{W} \rvert {\psi} \rangle= i\hbar \frac {d}{dt} \rvert{\psi} \rangle## and then taking ##\langle{k'} \lvert...
  39. T

    A Interaction Hamiltonian coupling question

    System is composed of two qubits and the bath is one bath qubit. The interaction Hamiltonian is: $$\sigma_1^x\otimes B_1 + \sigma_2^x\otimes B_2$$ where $$B_i$$ is a 2 by 2 matrix. I try to interpret and understand this, is it the same as: $$(\sigma_1^x\otimes B_1)\otimes I_2 +...
  40. L

    For given Hamiltonian, is spin conserved?

    A system consisting of two spins is described by the Hamiltonian (b>0) H = aσ1 ⋅ σ2 + b(σ1z - σ2z) where a and b are constants. (a) Is the total spin S = ½ (σ1 + σ2) conserved? Which components of S, if any, are conserved? (b) Find the eigenvalues of H and the corresponding...
  41. 1

    How do I set up this Legendre Transform for Hamiltonian

    Homework Statement Im trying to understand the Legendre transform from Lagrange to Hamiltonian but I don't get it. This pdf was good but when compared to wolfram alphas example they're slightly different even when accounting for variables. I think one of them is wrong. I trust wolfram over the...
  42. A

    I Is the Hamiltonian always the total energy?

    I'm working on some classical mechanics and just got a question stated: Is the Hamiltonian for this system conserved? Is it the total energy? In my problem it was indeed the total energy and it was conserved but it got me thinking, isn't the Hamiltonian always the total energy of a system...
  43. S

    A Hamiltonian of the quantised Klein-Gordon theory

    The Klein-Gordon field ##\phi(\vec{x})## and its conjugate momentum ##\pi(\vec{x})## is given, in the Schrodinger picture, by ##\phi(\vec{x})=\int \frac{d^{3}p}{(2\pi)^{3}}...
  44. S

    Equations of motion and Hamiltonian density of a massive vector field

    Homework Statement The Lagrangian density for a massive vector field ##C_{\mu}## is given by ##\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^{2}C_{\mu}C^{\mu}## where ##F_{\mu\nu}=\partial_{\mu}C_{\nu}-\partial_{\nu}C_{\mu}##. Derive the equations of motion and show that when ##m...
  45. S

    Courses Interest in Areas of Classical Mechanics

    What are Hamiltonian/Lagrangian Mechanics and how are they different from Newtonian? What are the benefits to studying them and at what year do they generally teach you this at a university? What are the maths required for learning them?
  46. M

    Paramagnetic system: computing number of microstates

    Homework Statement We are given a paramagnetic system of N distinguishable particles with 1/2 spin where we use N variables s_k each binary with possible values of ±1 where the total energy of the system is known as: \epsilon(s) = -\mu H \sum_{k=1}^{N} s_k where \mu is the magnetic moment...
  47. M

    A Hamiltonian and constant potential

    H=p^2/2m+c What's c? It's of course a shift in energy, but can be thought also as a smoother and smoother real-space local potential that becomes a constant all over the space. On the other hand, why couldn't one think about it as a constant potential in reciprocal space? It's a shift in energy...
  48. S

    I Why is the KE operator negative in QM?

    In the Hamilonian for an H2+, the kinetic energy of the electron (KE of nucleus ignored due to born-oppenheimer approximation) has a negative sign in front of it. I understand the signs for the potential energy operators but not for the KE apart from the strictly mathematical point of view. Can...
  49. S

    I Symmetry of Hamiltonian and eigenstates

    Suppose we have an electron in a hydrogen atom that satisfies the time-independent Schrodinger equation: $$-\frac{\hbar ^{2}}{2m}\nabla ^{2}\psi - \frac{e^{2}}{4\pi \epsilon_{0}r}\psi = E\psi$$ How can it be that the Hamiltonian is spherically-symmetric when the energy eigenstate isn't? I was...
  50. S

    QM:Finding the probabilities of a Hamiltonian measurement

    Homework Statement This problem is from Zetelli 3.21 http://imgur.com/wYTNVwz http://imgur.com/wYTNVwz Homework Equations Just the standard probability via product between the eigenfunction and the wavefunction The Attempt at a Solution I've found the eigenvectors for the Hamiltonian...
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