Helium

3

3.1 The ground state of helium 45

3.2 Excited states of helium 46

3.3 Evaluation of the integrals

in helium 53

Further reading 56

Exercises 58

Helium has only two electrons but this simplicity is deceptive. To treat
systems with two particles requires new concepts that also apply to
multi-particle systems in many branches of physics, and it is very worth-
while to study them carefully using the example of helium. There is
truth in the saying that atomic physicists count ‘one, two, many’ and a
detailed understanding of the two-electron system is sufficient for much
of the atomic structure in this book.^1

(^1) This book considers only those multi-
electron systems with one, or two, va-
lence electrons ‘outside’ a spherically-
symmetric core of charge.

3.1 The ground state of helium

Two electrons in the Coulomb potential of a chargeZe,e.g.thenucleus
of an atom, obey a Schr ̈odinger equation of the form

{

−
^2

2 m

∇^21 +

−
^2

2 m

∇^22 +

e^2

4 π 0

(

−

Z

r 1

−

Z

r 2

+

1

r 12

)}

ψ=Eψ. (3.1)

Herer 12 =|r 1 −r 2 |is the distance between electron 1 and electron 2 and
the electrostatic repulsion of electrons is proportional to 1/r 12. Neglect-
ing this mutual repulsion for the time being, we can write the equation
as

(H 1 +H 2 )ψ=E(0)ψ, (3.2)

where

H 1 ≡

−
^2

2 m

∇^21 −

Ze^2

4 π 0 r 1

(3.3)

andH 2 is a similar expression for electron 2. Writing the atomic
wavefunction as a product of the wavefunctions for each electron,ψ=
ψ(1)ψ(2), allows us to separate eqn 3.2 into two single-electron
Schr ̈odinger equations:

H 1 ψ(1) =E 1 ψ(1) (3.4)

and a similar equation forψ(2) with energyE 2. The solutions of these
one-electron equations are hydrogenic wavefunctions with energies given
by the Rydberg formula. Helium hasZ= 2 and in its ground state both
electrons have energyE 1 =E 2 =− 4 hcR∞=− 54 .4 eV. Thus the total
energy of the atom (neglecting repulsion) is

E(0)=E 1 +E 2 =−109 eV. (3.5)