Page 244 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

9.1 Variation method 235

The quantity E is, then

a 2 2
30 ÿ" d
^ 2 2
E ˆhöjHjöiˆ (ax ÿ x ) (ax ÿ x )dx
a 5 0 2m dx 2
30" 2 a 2 5" 2
ˆ (ax ÿ x )dx ˆ
ma 5 0 ma 2
The exact ground-state energy E 1 is shown in equation (2.39) to be
2 2
2
ð " =2ma . Thus, we have
10
E ˆ E 1 ˆ 1:013E 1 . E 1
ð 2
giving a 1.3% error.

Example: harmonic oscillator
We next consider an example with a variable parameter. For the harmonic
oscillator, discussed in Chapter 4, we select
ö ˆ e ÿcx 2
as the trial function, where c is a parameter to be varied so as to minimize
E (c). This function has no nodes and approaches zero in the limits x ! 1.
Since the integral höjöi is
1=2
1 2 ð
höjöiˆ e ÿ2cx dx ˆ
2c
ÿ1
where equation (A.5) is used, the normalized trial function is
1=4
2c 2
ö ˆ e ÿcx
ð
^
The Hamiltonian operator H for the harmonic oscillator is given in equation
(4.12). The quantity E (c) is then determined as follows
1=2 2 2 1=2 2
2c " 1 ÿcx 2 d ÿcx 2 2c mù 1 2 ÿ2cx 2
E (c) ˆÿ e e dx ‡ x e dx
ð 2m ÿ1 dx 2 ð 2 ÿ1
1=2 2 1=2
2c " c 1 2 ÿ2cx 2 c 2 1 2 ÿ2cx 2
ˆ (1 ÿ 2cx )e dx ‡ mù x e dx
ð m 2ð
ÿ1 ÿ1
" #
1=2 2 1=2 1=2 1=2 2 1=2
2c " c ð ð c mù ð
ˆ ÿ c ‡
ð m 2c 8c 3 2ð 2 8c 3
2
" c mù 2
ˆ ‡
2m 8c