Biomedical Engineering Reference
In-Depth Information
This is called a
generalized matrix eigenvalue problem
and there are exactly
n
possible
solutions
c
1
, ε
1
;
c
2
, ε
2
; ...
c
n
, ε
n
. Each of the
n
solutions is an upper bound to
n
electronic
states (including the ground state), and so the linear variation method has the added bonus
of giving approximations to each of
n
states simultaneously. Not only that, but if we add
further Ψ 's and repeat the calculation, each of the energies will at worst stay the same or
possibly get closer to the true energy. I have illustrated
MacDonald's theorem
for the case
n
=
6 in Figure 14.2.
Ψ
3
Ψ
1
n
=
6n
=
7
true
Figure 14.2
MacDonald's theorem
Addition of Ψ
7
has had no effect on the
n
=
6 approximation for Ψ
3
but has (for example)
lowered the energy of the Ψ
1
approximation.
I have stressed the special property of atoms; they are spherical and their electronic
Hamiltonian commutes with the square of the angular momentum operator, together with
the
z
component. The Schrödinger equation makes no mention of electron spin and so the
Hamiltonian also commutes with the square of the spin angular momentum operator and
the
z
component. In Table 14.4, I have summarized the relevant quantum numbers for the
six states of helium considered.
Table 14.4
Angular momentum quantum numbers for helium
states
L
M
L
S
M
S
Ψ
1
0
0
0
0
Ψ
2
0
0
0
0
Ψ
3
0
0
1
1
Ψ
4
0
0
1
0
Ψ
5
0
0
1
−
1
0
0
0
0
Ψ
6
