QUESTION 1 Consader the variational problem with Lagrangan function Lis. )+ky-)-+ Obtaun the Ealer-Tagrange equatio...
QUESTION 1 Consider the variational problem with Lagrangian function (1,3,2,3,») - ++y* - 328y. (1.1) Obtain the Euler-Lagrange equations and solve them. (10) (1.2) For this system (a) Construct the Hamiltonian function from L (b) Obtain the Hamilton's equations. (e) Write down the Hamilton-Jacobi equation. (6) (4) (4) (24)
1. This problem is all about the variational method, as applied to the particle in a box. Remember that we discussed in class how to compute the variational energy for this problem, using (2) = x(L - 2) as the trial wavefunction. (a) Let's take instead the trial wavefunction (x) = x(L2 - 22). Sketch carefully this function from 0 to L, and show that it satisfies the particle in a box boundary conditions. (b) Compute the variational energy for...
1. Explain the variational principle and illustrate it with some example (different from the one in the following point) 2. A trial function for ls electron in hydrogen atom has a form of (r) = e-ara. Derive the nor- malization constant. Explain the difference between this trial function and the true l-electron hydrogen-like orbital. 3. The expression for energy as a function of a for the H-atom using above trial function is given by: E(a) = 3h2a 2me e2a1/2 21/26073/2...
1. Variational method In this problem, you will approximate the ground state wave function of a quantum system using the variational theory. Use the trial wave function below 2 cos/T) , 1x1 trial a/2 to approximate the ground state of a harmonic oscillator given by 2.2 2 using a as an adjustable parameter. (a) Calculate the expectation value for the kinetic energy, (?) trial 4 points (b) Calculate the expectation value for the potential energy, Virial. Sketch ??tria, (V)trial, and...
Question 3 (a) Grven that f(-2)= 46, f(-1) 4, J(1) 1, f(3)= 156, and f(4)= 484, formula to estimate f(0) Use four-decimal arithmetac with rounding use the Lagrange interpolation (8) (b) Why should the Lagrange formula be used in practice only with caution" (2) (e) Wnte down the system of equations that need to be solved in order Function for the following data to construct the natural cubic spline 30 -5 6790 -3 6674 3 1 32-22178 (8) Note You...
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
QUESTION 4 (4.1) Consider the variational problem with Lagrangan function 2r Sin t and endpont conditions z(0) 0, z(*/2) = 1 (a) Find the smooth extremal of the gven vanational problem (7) (6) (b) Compare the value of the fundamental integral dt along an extremal between those two endpoints wnth the value along the curve z 1-cost through the two points Hint: You may use integration by parts and also remember that cost-(1+cos 2t), ant (1-cos 2t) and sin 2t...
Given the following two point boundary value problem: ty" + 2y + (3 - t)y = 4, y(2) = -1, y(8) = 1. Divide the given interval (3.7] into three equal sub-intervals, and apply the finite difference method (i,e: use the formulas for approximating y' and y" derive from Taylor series erpansion) to SETUP ( do not solve) a system of linear equations (write it in "A.r = b" form that will allow you to approximate the function value of...
question (c), (d), (e), (f) please. Thanks.
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
1. (10 points, part I) Consider the following initial boundary value problem lU (la) (1b) (1c) 0L, t> 0 3 cos ( a(x, 0) (a) Classify the partial differential equation (1a) (b) What do the equations (la)-(1c) model? (Hint: Give an interpretation for the PDE, boundary conditions and intial condition.) c) Use the method of separation of variables to separate the above problem into two sub- problems (one that depends on space and the other only on time) (d) What...