Homework 1
(due Wednesday, August, 29)
Normally, the homework assignments will be on what was covered the previous week.
This assignment covers background material that we will be using throughout the course, specifically working with complex numbers, finding eigenvalues and eigenvectors analytically of small matrices, and finding eigenvalues and eigenvectors numerically using Matlab.
1. Let z
1
= 3 + 2 i and z
2
= 1 − 2 i . Compute the following:
(a) z
1
+ z
2
(b) z
1
− z
2
(c) z
1 z
2
(d) z ∗
1
(e) | z
1
| 2 = z ∗
1 z
1
(f) z
1
/z
2
Hint: Multiply the numerator and the denominator by z ∗
2
.
(g) Any complex number may be written as re iθ tan( θ ) = y/x . Determine r and θ for z
1 and z
2
=
.
x + iy , where r =
√ x 2 + y 2 and
(h) Use the representation of the previous step to take the square root of z
1
.
2. Find the eigenvalues and eigenvectors of the following two matrices analytically.
(a)
A =
2 1
1 − 2
!
(1)
(b)
B =
2 − i i − 2
!
(2)
3. There many interesting quantum mechanics problems involving systems larger than
2 × 2 or 3 × 3 matrices! To do some of these problems in this course we are going to use
Matlab or the equivalent free program Octave. These programs can also be used in a wide range of problems in science and engineering so these are useful skills in general.
(a) Find a way to get access to either Matlab or Octave. It is probably going to be best to use your personal computer; however, there are computer labs in the
Physics building and on campus that have Matlab.
(b) Use Matlab or Octave to numerically get the eigenvectors and eigenvalues in problem 2, print out your code, and compare the results.