Page 243 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

234 Approximation methods

ë 2 , ... , which can be varied. The quantity E is then a function of these
parameters: E (ë 1 , ë 2 , ...). For each set of parameter values, the corresponding
value of E (ë 1 , ë 2 , ...) is always greater than or equal to the true ground-state
energy E 0 . The value of E (ë 1 , ë 2 , ...) closest to E 0 is obtained, therefore, by
minimizing E with respect to each of these parameters. Selecting a suf®ciently
large number of parameters in a well-chosen analytical form for the trial
function ö yields an approximation very close to E 0 .

Ground-state eigenfunction
If the quantity E is identical to the ground-state energy E 0 , which is usually
non-degenerate, then the trial function ö is identical to the ground-state
eigenfunction ø 0 . This identity follows from equation (9.5), which becomes
X 2
ja n j (E n ÿ E 0 ) ˆ 0
n(6ˆ0)
where the term for n ˆ 0 vanishes because E n ÿ E 0 vanishes. This relationship
is valid only if each coef®cient a n equals zero for n 6ˆ 0. From equation (9.3),
the normalized trial function ö is then equal to ø 0 . Should the ground-state
energy be degenerate, then the function ö is identical to one of the ground-state
eigenfunctions.
When the quantity E is not identical to E 0 , we assume that the trial function
ö which minimizes E is an approximation to the ground-state eigenfunction
ø 0 . However, in general, E is a closer approximation to E 0 than ö is to ø 0 .

Example: particle in a box
As a simple application of the variation method to determine the ground-state
È
energy, we consider a particle in a one-dimensional box. The Schrodinger
equation for this system and its exact solution are presented in Section 2.5. The
ground-state eigenfunction is shown in Figure 2.2 and is observed to have no
nodes and to vanish at x ˆ 0 and x ˆ a. As a trial function ö we select
ö ˆ x(a ÿ x), 0 < x < a
ˆ 0, x , 0, x . a
which has these same properties. Since we have
a a 5

2 2
höjöiˆ x (a ÿ x) dx ˆ
0 30
the normalized trial function is
p 
30
ö ˆ x(a ÿ x), 0 < x < a
a 5=2