Problems, set 2. PHYSICS 212A: Mathematical Physics
•Problems #1-4 are to be done using Mathematica, problems #5-8 (and 9*) without.
•In Mathematica, look up the following functions (and the ones cross-listed with them) :
(a) ContourPlot, Plot3D, ParametricPlot, ListPlot; check out commands that affect various op-
tions (color, line/point thickness, range of plotted function, etc.)
(b) Table, List and functions related to linear algebra
(c) Solve, NSolve, FindRoot
1. Plot the following function
f(x, y) = 3 exp Ã−(x+ 2)2+ (y+ 2)2
3!−4
(x+ 1)2+ (y+ 2)2+ 1
as a
(a) contour plot
(b) 3D plot
Do the same with the options changed from the default settings (with the aim of making
plots nicer):
(c) in (a) change PlotRange, ColorFunction, Contours, ContourLines
(d) in (b) change PlotRange, PlotPoints, ColorFunction, Mesh, ViewPoint, AspectRatio
2. (a) Using ParametricPlot, plot a circle with the center at C= (x0, y0) = (2,3.5) and radius
r=√3. Make sure the resulting picture does look like a circle.
(b) Using Solve find intersection points of that circle with the line y= 1.5x.
(c) Using FindRoot (and with the help of Plot) find intersection(s) of the same circle with
the function y= 4 sin2(x).
3. (a) Solve the following system of linear equations:
x−2y+z= 5
2x+ 2y−2z=−3
x−y+z= 1
(b) define the matrix:
M=
1−2 1
2 2 −2
1−1 1
and the vector b= (5,−3,1). Using linear algebra tools solve M·x=bfor x= (x, y, z).
Compare with (a).
(c) Use Do (or any other programming tools in Mathematica) to define an n×nmatrix
(with n > 10 of your choice), whose elements are all ai,j ≡0 except the first diagonal
below, and the first diagonal above the main diagonal, where the elements of the matrix are
ai,i±1= 1 (a special case of the 3-diagonal matrix). Obtain the list of eigenvalues of this
matrix. Order it from the lowest to the highest. Show, by a direct comparison of the lists,
that the eigenvalues are given by Ej=−2 cos ³πj
n+1 ´, where j= 1..n.
(Energies of the free electron on a chain of natoms.)
4. (a) Define a Mathematica function to give you f(n) = 1 + 1/2+1/3 + . . . + 1/n. Create a
list and use PlotList to plot f(n) from 1 to 100.
(b) Plot ln(n) (natural log) on a separate PlotList plot and then together with f(n) (use
Show). Explain why they are similar.
(c) By manipulating already defined lists, create a list of their difference, ln(n)−f(n), and
P lotList it together with −γ(where γis the Euler’s gamma constant, EulerGamma).
