The hydrogen atom

2

2.1 The Schr ̈odinger equation 22
2.2 Transitions 29
2.3 Fine structure 34

Further reading 42

Exercises 42

The simple hydrogen atom has had a great influence on the development

of quantum theory, particularly in the first half of the twentieth century

when the foundations of quantum mechanics were laid. As measurement

techniques improved, finer and finer details were resolved in the spec-

trum of hydrogen until eventually splittings of the lines were observed

that cannot be explained even by the fully relativistic formulation of

quantum mechanics, but require the more advanced theory of quantum

electrodynamics. In the first chapter we looked at the Bohr–Sommerfeld

theory of hydrogen that treated the electron orbits classically and im-

posed quantisation rules upon them. This theory accounted for many of

the features of hydrogen but it fails to provide a realistic description of

systems with more than one electron, e.g. the helium atom. Although

the simple picture of electrons orbiting the nucleus, like planets round

the sun, can explain some phenomena, it has been superseded by the

Schr ̈odinger equation and wavefunctions. This chapter outlines the ap-

plication of this approach to solve Schr ̈odinger’s equation for the hydro-

gen atom; this leads to the same energy levels as the Bohr model but

the wavefunctions give much more information, e.g. they allow the rates

of the transitions between levels to be calculated (see Chapter 7). This

chapter also shows how the perturbations caused by relativistic effects

lead to fine structure.

2.1 The Schr ̈odinger equation

The solution of the Schr ̈odinger equation for a Coulomb potential is

in every quantum mechanics textbook and only a brief outline is given

(^1) The emphasis is on the properties of here. (^1) The Schr ̈odinger equation for an electron of mass me in a
the wavefunctions rather than how to
solve differential equations.
spherically-symmetric potential is
{
−
^2
2 me
∇^2 +V(r)

}

ψ=Eψ. (2.1)

This is the quantum mechanical counterpart of the classical equation

for the conservation of total energy expressed as the sum of kinetic and

potential energies.^2 In spherical polar coordinates we have

(^2) Theoperatorforlinearmomentumis
p=−i
∇and for angular momentum
it is
l=r×p. This notation differs in
two ways from that commonly used in
quantum texts. Firstly,
is taken out-
side the angular momentum operators,
and secondly, the operators are written
without ‘hats’. This is convenient for
atomic physics, e.g. in the vector model
for the addition of angular momenta.

∇^2 =

1

r^2

∂

∂r

(

r^2

∂

∂r

)

−

1

r^2

l^2 , (2.2)