26 Atomic Physics
This material is covered inGasiorowicz Chapter 19,and inCohen-Tannoudji et al. Com-
plementAXIV.
26.1 Atomic Shell Model
The Hamiltonian for an atom withZelectrons and protons is
∑Z
i=1(
p^2 i
2 m−
Ze^2
ri)
+
∑
i>je^2
|~ri−~rj|
ψ=Eψ.We have seen that the coulomb repulsion between electrons is a verylarge correction in Helium and
that the three body problem in quantum mechanics is only solved by approximation. The states
we have from hydrogen are modified significantly. What hope do we have to understand even more
complicated atoms?
The physics ofclosed shellsand angular momentum enable us to make sense of even the most
complex atoms. Because of the Pauli principle, we can put only one electron into each state. When
we have enough electrons to fill a shell, say the 1s or 2p, The resulting electron distribution is
spherically symmetric because
∑ℓ
m=−ℓ|Yℓm(θ,φ)|^2 =2 ℓ+ 1
4 π.
With all the states filled and the relative phases determined by the antisymmetry required by Pauli,
the quantum numbers of the closed shell are determined. There is only one possible state
representing a closed shell.
As in Helium, the two electrons in the same spatial state,φnℓm, must by symmetric in space and
hence antisymmetric in spin. This implies each pair of electrons has a total spin of 0. Adding these
together gives a total spin state withs= 0, which is antisymmetric under interchange. The spatial
state must be totally symmetric under interchange and, since all the states in the shell have the
samenandℓ, it is the differentmstates which are symmetrized. This can be shown to give us a
totalℓ= 0 state.
So theclosed shell contributes a spherically symmetric charge and spin distributionwith
the quantum numbers
s= 0
ℓ= 0
j= 0The closed shell screens the nuclear charge. Because of thescreening, the potential no longer has
a pure^1 rbehavior. Electrons which are far away from the nucleus see less ofthe nuclear charge and
shift up in energy. This is a large effect and single electron states withlargerℓhave larger energy.
From lowest to highest energy, the atomic shells have the order
1 s, 2 s, 2 p, 3 s, 3 p, 4 s, 3 d, 4 p, 5 s, 4 d, 5 p, 6 s, 4 f, 5 d, 6 p.