Homework 1
(due Wednesday, September, 7)
The goals of this homework assignment are to review how to manipulate complex numbers, to get practice taking expectation values, and to run a simple Matlab code. All three
of these skills we will use throughout the course.
1. Let z1 = 1 + 2i and z2 = 2 − i. Compute the following:
(a) z1 + z2
(b) z1 − z2
(c) z1 z2
(d) z1∗
(e) |z1 |2 = z1∗ z1
(f) z1 /z2 Hint: Multiply the numerator and the denominator by z2∗ .
(g) Any complex number may be written as reiθ = x + iy, where r =
tan(θ) = y/x. Determine r and θ for z1 and z2 .
√
x2 + y 2 and
(h) Use the representation of the previous step to take the square root of z1 .
2. Consider the wave function ψ(x) = C sin(πnx/a) on the interval 0 ≤ x ≤ a, where n
is a positive integer.
(a) What is C so ψ is normalized to unity?
(b) Compute the expectation values of x and x2 as well as σx .
(c) Compute the expectation values of p and p2 as well as σp .
(d) Check that the uncertainty principle is satisfied.
3. Consider the wave function ψ(x) = x(a − x) on the interval 0 ≤ x ≤ a.
(a) What is C so ψ is normalized to unity?
(b) Compute the expectation values of x and x2 as well as σx .
(c) Compute the expectation values of p and p2 as well as σp .
(d) Check that the uncertainty principle is satisfied.
4. So that we will not be restricted to solving problems with polynomials and trigonometric functions as above, we are going to start using Matlab or one of the free compatible
programs: Octave (www.octave.org) or SciLab (www.scilab.org) .
(a) Find a place where you have access to one of these programs. I am interested
where that is so please make note here which program you are using and where
you are using it.
(b) Create a file myplot.m with the text below and run it.
x = 0:0.01:1;
psi = sin(pi*x);
area = trapz(x,psi.*psi);
psi = psi/sqrt(area);
plot(x,psi)
xlabel(’x’)
ylabel(’C sin(pi x)’)
Print out the result.
(c) Modify the code to use the wavefunction of question 3. Also plot the result.