Page 248 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.3 Non-degenerate perturbation theory 239

Equation (9.12) has the form
N
X
a ki x i ˆ 0, k ˆ 1, 2, ... , N (9:13)
iˆ1
for which a trivial solution is x i ˆ 0 for all i. A non-trivial solution exists if,
and only if, the determinant of the coef®cients a ki vanishes

a 11 a 12 a 1N

a 21 a 22 a 2N
ˆ 0

a N1 a N2 a NN
This determinant or its equivalent algebraic expansion is known as the secular
equation. In equation (9.12) the parameters c i correspond to the unknown
quantities x i in equation (9.13) and the terms (H ki ÿ E S ki ) correspond to the
coef®cients a ki . Thus, a non-trivial solution for the N parameters c i exists only
if the determinant with elements (H ki ÿ E S ki ) vanishes

H 11 ÿ E S 11 H 12 ÿ E S 12 H 1N ÿ E S 1N

H 21 ÿ E S 21 H 22 ÿ E S 22 H 2N ÿ E S 2N
ˆ 0 (9:14)

H N1 ÿ E S N1 H N2 ÿ E S N2 H NN ÿ E S NN
The secular equation (9.14) is satis®ed only for certain values of E . Since
this equation is of degree N in E , there are N real roots
E 0 < E 1 < E 2 < < E Nÿ1
According to the variation theorem, the lowest root E 0 is an upper bound to the
1
ground-state energy E 0 : E 0 < E 0 . The other roots may be shown to be upper
bounds for the excited-state energy levels
E 1 < E 1 , E 2 < E 2 , ... , E Nÿ1 < E Nÿ1

9.3 Non-degenerate perturbation theory

Perturbation theory provides a procedure for ®nding approximate solutions to
È
the Schrodinger equation for a system which differs only slightly from a system
^
for which the solutions are known. The Hamiltonian operator H for the system
of interest is given by
N
" 2 X 1
^ 2
H ˆÿ = ‡ V(r 1 , r 2 , ... , r N )
i
2 m i
iˆ1
1 J. K. L. MacDonald (1933) Phys. Rev. 43, 830.