Biomedical Engineering Reference
In-Depth Information
as no surprise when I tell you that exact solution of the many-electron atomic Schrödinger
problem seems to be impossible, because of the electron-electron repulsions, and that we
are apparently in a weaker position than the astronomers. On the positive side, I have
stressed the Born interpretation of quantum mechanics. Here we do not focus attention
on the trajectories of the individual particles, rather we ask about the probability that a
region of space is occupied by any one of the particles. The quantum mechanical problem
therefore seems to be more hopeful than the astronomical one.
14.3 Orbital Model
Let me now investigate the Schrödinger equation that would result from an atom consisting
of electrons that did not repel each other. We can think of this as some 'zero-order' approx-
imation to a true many-electron atom, just as the astronomers might have investigated their
simple model of planetary motion.
We therefore write
H
Ψ (
r
1
,
r
2
, ...
r
n
)
=
εΨ (
r
1
,
r
2
, ...
r
n
)
n
h
(1)
(
r
i
)
Ψ (
r
1
,
r
2
, ...
r
n
)
=
εΨ (
r
1
,
r
2
, ...
r
n
)
i
=
1
and this appears to be a candidate for separation of variables. I write
Ψ (
r
1
,
r
1
, ...
r
1
)
=
ψ
1
(
r
1
) ψ
2
(
r
2
) ...ψ
n
(
r
n
)
(14.6)
Substitution and separation gives
n
identical one-electron atom Schrödinger equations, and
so the total wavefunction is a product of the familiar 1s, 2s, 2p atomic orbitals discussed
in Chapter 13 (with nuclear charge
Ze
). The energy is given by the sum of the orbital
energies.
We then have to take account of electron spin and the Pauli principle, as discussed in
Chapter 13. I can remind you of the principles by writing down some of the lowest energy
solutions for helium, in particular those that formally involve the 1s and 2s orbitals. I will
label the atomic orbitals 1s and 2s for obvious reasons, and I will adopt the habit of writing
s
for the spin variable (sorry about the double use of the same symbol) as in Chapter 13.
The allowed wavefunctions are given in Table 14.1.
Ψ
1
describes the spectroscopic ground state, giving energy 2ε
1
s
. The remaining wave-
functions describe excited states. Ψ
2
is the first excited singlet state whilst Ψ
3
, Ψ
4
and Ψ
5
are the three components of the first triplet state. We refer to Ψ
2
through Ψ
5
as
singly excited
wavefunctions
because they have been formally produced from the ground-state wavefunc-
tion by exciting a single electron. Due to my neglect of electron repulsion, the energies of
Ψ
2
through Ψ
5
are the same. Ψ
6
is a
doubly excited wavefunction
and so on.
Our zero-order model predicts that the singlet and triplet excited states derived from a
1s
1
2s
1
orbital configuration will have energy
1
1
2
−
Z
2
m
e
e
4
8
h
2
ε
0
1
2
2
ε
2s
−
ε
1s
=
