Page 70 - Electrical Properties of Materials

The hydrogen atom and the

4
periodic table

I see the atoms, free and fine,
That bubble like a sparkling wine;
I hear the songs electrons sing,
Jumping from ring to outer ring;
Lister The Physicist

4.1 The hydrogen atom
Up to now we have been concerned with rather artificial problems. We said:
let us assume that the potential energy of our electron varies as a function of
distance this way or that way without specifying the actual physical mechanism
responsible for it. It was not a waste of time. It gave an opportunity of becoming
acquainted with Schrödinger’s equation, and of developing the first traces of a
physical picture based, perhaps paradoxically, on the mathematical solution.
It would, however, be nice to try our newly acquired technique on a more
physical situation where the potential is caused by the presence of some other
physical ‘object’. The simplest ‘object’ would be a proton, which, as we know,
becomes a hydrogen atom if joined by an electron.
We are going to ask the following questions: (i) What is the probability that
the electron is found at a distance r from the proton? (ii) What are the allowed
energy levels?
The answers are again provided by Schrödinger’s equation. All we have to
do is to put in the potential energy due to the presence of a proton and solve
the equation.
The wave function is a function of time, and one might want to solve prob-
lems, where the conditions are given at t = 0, and one is interested in the
temporal variation of the system. These problems are complicated and of little
general interest. What we should like to know is how a hydrogen atom behaves
on the average, and for that purpose the solution given in eqn (3.7) combined
with (3.11) is adequate. We may then forget about the temporal variation,
because
2
|w(t)| = 1, (4.1)
and solve eqn (3.13), the time-independent Schrödinger equation.
The proton, we know, is much heavier than the electron; so let us regard it as
infinitely heavy (that is immobile) and place it at the origin of our coordinate
system.
The potential energy of the electron at a distance, r, from the proton is
known from electrostatics:
e 2
V(r)=– . (4.2)
4π 0 r