PHZ 6358: Standard Model I
Fall Term 2013
Homework Set 2
Due Friday, September 13, 2013
Reading: Anything.
Problem 1. 1-dimensional Monte Carlo Integration. Calculate π by means of Monte
Carlo integration. To this end, consider a definite integral whose numerical value is proportional
to π, then calculate the integral by Monte Carlo methods. For example, we know that the area
of the unit circle is equal to π, hence the area of half a circle is π/2:
π
=
2
Z
1
−1
Z 1
√
√
dx 1 − x2 = 2
dx 1 − x2 ,
0
(1)
and a convenient integral representation of π is
π=4
Z
0
1
√
dx 1 − x2 .
(2)
Now write a simple program which integrates the right-hand side. Hint: In fortran the function
rand() generates a random number between 0 and 1. Notice that the largest value of the
integrand is y = 1, so it is convenient to take “the box” encompassing the integrand to be
simply the unit square
0 ≤ x ≤ 1,
0≤y≤1
in which case we do not need to rescale the values returned by rand(). Investigate the numerical
as
precision of the Monte Carlo integration as a function of the number of tries N, i.e. plot ∆π
π
a function of N.
Problem 2. 6-dimensional Monte Carlo Integration. The volume of a ball of radius
R in 6 dimensions goes like
V6 ∼ π 3 R 6 .
Use Monte Carlo integration to guess the value of the correct numerical prefactor (do not look
it up!).
OK, now you can look it up and see if your guess was correct.
1