Fermi passed an examination

In the spring of 1927 Orso Mario Corbino, the director of the Institute of Physics
at Rome and an influential politician, launched a
famous appeal to the students of the Faculty of Engineering
to entice the most brilliant young minds into studying
Physics. Segrè and his friend Amaldi rose
to the challenge, joining Fermi and Franco Rasetti’s
group and telling them of Ettore’s exceptional gifts.
After some encouragement from Segrè and Amaldi,
Majorana eventually decided to meet Fermi in the
autumn of that year. The pair immediately started talking
about the statistical model of atoms that Fermi was
working on, later to be known as the Thomas–Fermi
model, which describes the energy of an atom in terms
of the density of its surrounding electrons. The model
involves a complicated non-linear differential equation.
The analytical solution of the equation was then
unknown, but Fermi had managed to obtain a numerical
table of approximate values for it. Majorana carefully
followed what Fermi said and, after asking a few
questions, left the Institute. The following morning he
returned to Fermi’s office and asked for a closer look
at the numerical table so that he could compare it with
an analogous table he had drawn up the previous
evening. Once he had established agreement between
the two tables, Majorana noted that Fermi’s table was
correct and left the Institute with no further comment.

What seems nothing more than an amusing anecdote,
as recalled by Rasetti, Segrè and Amaldi, has
since been carefully tested on scientific grounds. From
previously unpublished study notes written by Majorana
between 1927 and 1932, we now know that Majorana
had arrived at a “series solution” of the Thomas–
Fermi equation using a peculiar method that applies to
an entire class of mathematical problems. It is testament
to Majorana’s genius that although some of his
results anticipated by several years those of renowned
mathematicians and physicists, several others (including
his solution of the Thomas–Fermi equation) have
not been obtained by anyone else since.

As if satisfied that Fermi had passed his “examination”,
Majorana decided to leave engineering and join
what became known as the “via Panisperna boys” –
Fermi’s influential group that was named after the
street in Rome where the Physics department was
located.