Page 127 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

118 Harmonic oscillator

Table 4.1. Hermite polynomials

n H n (î)

01
12î
2
24î ÿ 2
38î ÿ 12î
3
2
4
416î ÿ 48î ‡ 12
5
532î ÿ 160î ‡ 120î
3
4
2
6
664î ÿ 480î ‡ 720î ÿ 120
5
3
7
7128î ÿ 1344î ‡ 3360î ÿ 1680î
8
4
6
2
8256î ÿ 3584î ‡ 13440î ÿ 13440î ‡ 1680
7
3
9
5
9512î ÿ 9216î ‡ 48384î ÿ 80640î ‡ 30240î
6
2
8
4
10 1024î 10 ÿ 23040î ‡ 161280î ÿ 403200î ‡ 302400î ÿ 30240
The functions ö n (î) in equation (4.40) are identical to those de®ned by
equation (D.15) and, therefore, form a complete set as shown in equation
(D.19). Substituting equation (4.16) into (D.19) and applying the relation
(C.5b), we see that the functions ø n (x) in equation (4.41) form a complete set,
so that
1
X
ø n (x)ø n (x9) ˆ ä(x ÿ x9) (4:42)
nˆ0
Physical interpretation
The ®rst four eigenfunctions ø n (x) for n ˆ 0, 1, 2, 3 are plotted in Figure 4.2
2
and the corresponding functions [ø n (x)] in Figure 4.3. These ®gures also
show the outline of the potential energy V(x) from equation (4.5) and the four
2
corresponding energy levels from equation (4.30). The function [ø n (x)] is the
probability density as a function of x for the particle in the nth quantum state.
2
The quantity [ø n (x)] dx at any point x gives the probability for ®nding the
particle between x and x ‡ dx.
We wish to compare the quantum probability distributions with those
obtained from the classical treatment of the harmonic oscillator at the same
energies. The classical probability density P(y) as a function of the reduced
distance y (ÿ1 < y < 1) is given by equation (4.10) and is shown in Figure
4.1. When equations (4.8), (4.14), and (4.30) are combined, we see that the
maximum displacement in terms of î for a classical oscillator with energy
p  p  p 
1
(n ‡ )"ù is 2n ‡ 1.For î , ÿ 2n ‡ 1 and î . 2n ‡ 1, the classical
2