Homework 1
(due Friday, August, 30)
This homework assignment gives you some practice working with complex numbers, probabilities, expectation values, and simple programming in Matlab. We will be using all of
these skills throughout the course. I will be giving you for the most part the Matlab code,
but it is a very useful to have experience using it. Part of the assignment is to get Matlab
or one of the free equivalents, Octave (http://www.gnu.org/software/octave/) or FreeMat
(http://freemat.sourceforge.net/) up a running on a computer that you use.
1. Let z1 = 2 + 3i and z2 = 4 − 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
2. Consider the Gaussian wave function ψ(x) = Ce−αx , where α is a positive real number.
(a) Find the constant C so that ψ is normalized, i.e.
Z
∞
−∞
|ψ(x)|2 dx = 1.
(1)
(b) What is the expectation value of x, hxi?
(c) What is the expectation value of x2 , hx2 i?
(d) What is the expectation value of p, hpi?
(e) What is the expectation value of p2 , hp2 i?
(f) Show that a Gaussian wave function is a minimum uncertainty wave function by
computing ∆x∆p.
(g) Check your answers above using the following Matlab code. If you remove the
semicolons at the end, then you can see the result of a statement printed out. Try
to figure out what each statement does.
alpha = 1;
x = -5:0.01:5;
psi = exp(-alpha*x.*x);
area = trapz(x,abs(psi).^2);
C = 1/sqrt(area);
psi = C*psi;
trapz(x,abs(psi).^2) % This is a check.
expvalx = trapz(x,x.*abs(psi).^2);
expvalx2 = trapz(x,x.*x.*abs(psi).^2);
Deltax = sqrt(expvalx2 - expvalx^2);
% One can show analytically that the expectation value
% of the momentum is zero for any real wave function.
% Thus here we only compute the expecation value of p^2.
dx = x(2) - x(1);
dpsidx = diff(psi)/dx;
d2psidx2 = diff(dpsidx)/dx;
% This part is a little tricky. d2psidx2 is two shorter
% in length than psi and x. Can you see why?
N = length(x);
x = x(2:(N -1));
psi = psi(2:(N-1));
expvalp2 = - trapz(x,conj(psi).*d2psidx2); % with hbar = 1
Deltap = sqrt(expvalp2);
uncertainty = Deltax * Deltap % Should be greater or equal to 1/2.
3. Repeat question 2 for the wave function
ψ(x) = C2
!
1
i
2
2
√ e−(x−1) + √ e−(x+1) .
2
2
(2)
You may do this either analytically or numerically (or both). This will not be a
minimum uncertainty wave function.