Numerical Methods in Quantum Mechanics - Dipartimento di Fisica

part and symmetric coordinate part), with the former lower in energy than the
latter for the same single electron levels. The ground state of He turns out to
be a singlet, so the coordinate wave function must be symmetric.
D.1 Perturbative Treatment for Helium atom
The Helium atom is characterized by a Hamiltonian operator
H = −¯h2 ∇ 2 1
2m e
− Zq2 e
− ¯h2 ∇ 2 2
− Zq2 e
r 1 2m e r 2
+ q2 e
r 12
(D.6)
where r 12 = |r 2 − r 1 | is the distance between the two electrons. The last term
corresponds to the Coulomb repulsion between the two electrons and makes the
problem non separable.
As a first approximation, let us consider the interaction between electrons:
V = q2 e
r 12
as a perturbation to the problem described by
(D.7)
H 0 = −¯h2 ∇ 2 1
2m e
− Zq2 e
− ¯h2 ∇ 2 2
− Zq2 e
r 1 2m e r 2
(D.8)
which is easy to solve since it is separable into two independent problems of
a single electron under a central Coulomb field, i.e. a Hydrogen-like problem
with nucleus charge Z = 2. The ground state for this system is given by the
wave function described in Eq.(2.29) (1s orbital):
φ 0 (r i ) = Z3/2
√ π
e −Zr i
(D.9)
in a.u.. We note that we can assign to both electrons the same wave function, as
long as their spin is opposite. The total unperturbed wave function (coordinate
part) is simply the product
ψ 0 (r 1 , r 2 ) = Z3
π e−Z(r 1+r 2 )
(D.10)
which is a symmetric function (antisymmetry being provided by the spin part).
The energy of the corresponding ground state is the sum of the energies of the
two Hydrogen-like atoms:
E 0 = −2Z 2 Ry = −8Ry
(D.11)
since Z = 2. The electron repulsion will necessarily raise this energy, i.e. make
it less negative. In first-order perturbation theory,
E − E 0 = 〈ψ 0 |V |ψ 0 〉 (D.12)
∫
= Z6 2
π 2 e −2Z(r 1+r 2 ) d 3 r 1 d 3 r 2
(D.13)
r 12
= 5 ZRy. (D.14)
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