Page 263 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

254 Approximation methods
g n
X (1)
^
ÿ c áã H
kâ,nã
ãˆ1
a ná,k⠈ (0) (9:70)
(0)
(E ÿ E )
k n
The eigenfunctions ø ná for the perturbed system to ®rst order are obtained by
combining equations (9.61), (9.69), and (9.70)
g n
X (1)
^
c áã H
g k kâ,nã
X X ãˆ1 (1)
ø ná ˆ ö (0) ÿ ë (0) (0) ø kâ (9:71)
ná
n
k(6ˆn) âˆ1 (E k ÿ E )
Example: hydrogen atom in an electric ®eld
As an illustration of the application of degenerate perturbation theory, we
consider the in¯uence, known as the Stark effect, of an externally applied
electric ®eld E on the energy levels of a hydrogen atom. The unperturbed
^ (0)
Hamiltonian operator H for the hydrogen atom is given by equation (6.14),
and its eigenfunctions and eigenvalues are given by equations (6.56) and
(6.57), respectively. In this example, we label the eigenfunctions and eigenva-
^ (0)
lues of H with an index starting at 1 rather than at 0 to correspond to the
^
principal quantum number n. The perturbation H9 is the potential energy for
the interaction between the atomic electron with charge ÿe and an electric ®eld
E directed along the positive z-axis

^
^
H9 ˆ H (1) ˆ eE z ˆ eE r cos è (9:72)
If spin effects are neglected, the ground-state unperturbed energy level E (0)
1
is non-degenerate and its ®rst-order perturbation correction E (1) is given by
1
equation (9.24) as
(1)
E 1 ˆ eE h1sjzj1siˆ 0

This integral vanishes because the unperturbed ground state of the hydrogen
atom, the 1s state, has even parity and z has odd parity.
(0)
The next lowest unperturbed energy level E , however, is four-fold degen-
2
erate and, consequently, degenerate perturbation theory must be used to
determine its perturbation corrections. For simplicity of notation, in the
^
(0)
(0)
quantities ø , ö , and H (1) we drop the index n, which has the value
ná
ná
ná,nâ
n ˆ 2 throughout. As the initial set of eigenfunctions for the unperturbed
system, we select the 2s, 2p 0 ,2p 1 , and 2p ÿ1 atomic orbitals as given by
equations (6.59) and (6.60), so that