Page 271 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

262 Approximation methods

Calculate the ®rst-order perturbation correction to the ground-state energy level
using the particle in a box with V(x) ˆ 0for 0 < x < a, V(x) ˆ1 for x , 0,
x . a as the unperturbed system. Then calculate the ®rst-order perturbation
correction to the ground-state wave function, terminating the expansion after the
term k ˆ 5. (See Appendix A for trigonometric identities and integrals.)
9.9 Using ®rst-order perturbation theory, determine the ground-state energy of a
hydrogen atom in which the nucleus is not regarded as a point charge. Instead,
regard the nucleus as a sphere of radius b throughout which the charge ‡e is
evenly distributed. The potential of interaction between the nucleus and the
electron is

ÿe9 2 r 2
V(r) ˆ 3 ÿ , 0 < r < b
2b b 2
ÿe9 2
ˆ , r . b
r
The unperturbed system is, of course, the hydrogen atom with a point nucleus.
(Inside the nuclear sphere, the exponential
2
2
e ÿ2r=a 0 ˆ 1 ÿ (2r=a 0 ) ‡ (2r =a ) ÿ
0
may be approximated by unity because r is very small in that region.)
9.10 Using ®rst-order perturbation theory, show that the spin±orbit interaction energy
for a hydrogen atom is given by
1
1 2 (0) for j ˆ l ‡ , l 6ˆ 0
1
2 á jE j 1 2
n
n(l ‡ )(l ‡ 1)
2
1
1 2 (0) 1
ÿ á jE j for j ˆ l ÿ , l 6ˆ 0
2 n nl(l ‡ ) 2
1
2
^
The Hamiltonian operator is given in equation (7.33), where H 0 represents the
^
unperturbed system and H so is the perturbation. Use equations (6.74) and (6.78)
to evaluate the expectation value of î(r).