Assignment 5

6.1.2(2) A grounded toroidal conducting shell has a shape given by the equation z2 2 + (r -R)2 = a2 where (r , , z) are cylindrical coordinates, and R = 1 meter , a=0.75 meter. Plot this shell using a Parametric-Plot3D command. The shell is filled with a uniform charge density /o = 10 V. Find and plot the contours of constant electrostatic potential in the (r , z)plane. At what r and z is the potential maximum, and what is its value, to three significant figures? (Hint: Solve Poisson's equation in cylindrical coordinates.)

6.1.6(6) Solve the time-dependent (dimensionless) Schrdinger equation for the wave function (x, t) of a particle in a potential V(x), subject to boundary conditions (a, t) = 0 and initial condition (x, 0) = o(x):

(x,t)t

=V(x) (x, t) - 12

2(x,t)x2

.

Take V(x) = x4 4, use trigonometric functions for the modes in the Galerkin method, and work out the matrix elements of V(x), Vnm, analytically for general n and m. (Note: Check to make sure the integrals for Vnn are evaluated properly.) Take o(x) = exp-2 (x - 1)2, and a = 4. Keep as many basis functions as necessary to obtain a converged answer. Plot the solution in time increments of 0.1 for 0 < t < 4.

6.2.5(5)(a) Use the CTCS method to finite-difference the wave equation in D dimensions (on a uniform Cartesian grid with the same stepsize x in each dimension). Perform a von Neumann stability analysis to show that this method is stable provided that ct /x 1 D .

(b) For the 1-D wave equation, 2y

t2= c2 (x)

2y(x,t)x2

on -4 < x < 4, where c(x) = 1 for x < 0 and c(x) = 1 /2

for x 0, use the CTCS method to solve the following problem for 0 < t < 20. Take x = 0.25, t = 0.2:

y(x, 0) = -2 (x-2)2, y(x,t)t t=0

= -4 (x - 2) -2 (x-2)2, y(-4, t) = y(4, t) = 0.

Make an animation of the result showing every second time step. Note: The CTCS scheme will require you to specify yk

1 directly from the two initial conditions. To determine yk1use the equation for uniform

acceleration,

yk1 = yk0 + v0 k t + ak t

2 2,

where vo k is the initial velocity of the k th point and ak is the initial acceleration, determined by

ak = c22y(x,0)

x2 x=xk .

6.2.8(8)(a) For the 1-D Schrdinger equation and for a particle of mass m in a uniform potential V(x) =Vo, show that the Crank-Nicolson method is stable for any time step size.

(b) Use the Crank-Nicolson method to solve the Schrdinger equation for a particle of mass m, described by a wave function (x, t), moving in a potential V(x) = 8 x on -4 < x < 4, over the time range 0 < t < 3. Animate ( )2 over this time interval. Take = 0 on the boundaries, and assume an initial condition (x, 0) = 5 x -2 (x+2)2. Also, take m = = 1.

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