Note that the variational calculation still uses first order perturbation theory. It just adds a variable
parameter to the wavefunction which we use to minimize the energy.This only works for the ground
state and for other special states.
There is only one allowed (1s)^2 state and it is the ground state. Forexcited states, the spatial
states are (usually) different so they can be either symmetric or antisymmetric (under interchange
of the two electrons). It turns out that the antisymmetric statehas the electrons further apart so
the repulsion is smaller and the energy is lower. If the spatial state isantisymmetric, then the spin
state is symmetric, s=1. So the triplet states are generally significantly lower in energy than the
corresponding spin singlet states. Thisappears to be a strong spin dependent interaction
but is actually just the effect of the repulsion between the electronshaving a big effect
depending on the symmetry of the spatial state and hence on the symmetry of the spin state.
Thefirst exited statehas the hydrogenic state content of (1s)(2s) and has s=1. We calculated the
energy of this state.
We’ll learn later that electromagnetictransitions which change spin are strongly suppressed
causing the spin triplet (orthohelium) and the spin singlet states (parahelium) to have nearly separate
decay chains.
1.33 Atomic Physics
The Hamiltonian for an atom with Z electrons and protons (See section 26) has many terms
representing the repulsion between each pair of electrons.
∑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 physics ofclosed shellsand angular momentum enable us to make sense of even the most
complex atoms. When we have enough electrons to fill a shell, say the1s or 2p, The resulting
electron distribution is spherically symmetric because
∑ℓm=−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 shelland the quantum numbers are
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