11.7 Exercises 377
and
dNY(t)
dt
=−ωYNY(t)+ωXNX(t).
We assume that att= 0 we haveNY( 0 ) = 0. In the beginning we will have an increase ofNY
nuclei, however, they will decay thereafter. In this project we let the nucleus^210 Bi represent
X. It decays throughβ-decay to^210 Po, which is theYnucleus in our case. The latter decays
through emision of anα-particle to^206 Pb, which is a stable nucleus.^210 Bi has a mean lifetime
of 7.2 days while^210 Po has a mean lifetime of 200 days.
- Find closed form solutions for the above equations assuming continuous variables and setting
the number of^210 Po nuclei equal zero att= 0.
- Make a program which solves the above equations. What is a reasonable choice of timestep
∆t? You could use the program on radioactive decay from the web-page of the course as
an example and make your own for the decay of two nuclei. Compare the results from your
program with the exact answer as function ofNX( 0 ) = 10 , 100 and 1000. Make plots of your
results.
- When^210 Po decays it produces anαparticle. At what time does the production ofαparticles
reach its maximum? Compare your results with the closed formsolutions forNX( 0 ) = 10 , 100
and 1000.
11.7.The task here is to integrate in a brute force manner a six-dimensional integral which
is used to determine the ground state correlation energy between two electrons in a helium
atom. Furthermore, you will need to parallelize your code for the Monte-Carlo integration.
We assume that the wave function of each electron can be modelled like the single-particle
wave function of an electron in the hydrogen atom. The single-particle wave function for an
electroniin the 1 sstate is given in terms of a dimensionless variable (the wavefunction is
not properly normalized)
ri=xiex+yiey+ziez,
as
ψ 1 s(ri) =e−αri,
whereαis a parameter and
ri=
√
x^2 i+y^2 i+z^2 i.
We will fixα= 2 , which should correspond to the charge of the helium atomZ= 2.
The ansatz for the wave function for two electrons is then given by the product of two 1 s
wave functions as
Ψ(r 1 ,r 2 ) =e−α(r^1 +r^2 ).
Note that it is not possible to find a closed form solution to Schrödinger’s equation for two
interacting electrons in the helium atom.
The integral we need to solve is the quantum mechanical expectation value of the correla-
tion energy between two electrons, namely
〈
1
|r 1 −r 2 |
〉=
∫∞
−∞
dr 1 dr 2 e−^2 α(r^1 +r^2 )
1
|r 1 −r 2 |
. (11.23)
Note that our wave function is not normalized. There is a normalization factor missing, but
for this project we don’t need to worry about that.
- Set up a program which performs a Monte Carlo integration of the above integral, but with-
out using importance sampling. That is, use only the uniformdistribution. An example of a
program which implements this can be written as
