into Schrödinger’s equation, yielding
(11.11)
As shown above, this second-order differential equation is called the
Hermite equation and has solutions of the form:
ψ 9 (y) =N 9 H 9 e−y
(^2) /2
(11.12)
where
(11.13)
N 9 is the normalization constant and is different for each term. It is equal to:
(11.14)
where 9! is the factorial term:
9! = 9 ( 9 −1)( 9 −2)( 9 −3)... (1) (11.15)
The Hermite functions,H 9 (y), are polynomials. The first four are listed in
Table 11.1. The ground-state wavefunction and its probability distribu-
tion are shown in Figure 11.3. Both terms have a maximal value at the
origin. Since the potential is symmetrical about x=0, the solutions are
also symmetrical. Note that because of the exponential dropoff both func-
tions quickly approach a zero value but remain positive and non-zero for
all values of x.
N 9
/ (^9)!
=
1
απ^1229
y
x
mk
, , , , ,..
/
==
⎛
⎝
⎜⎜
⎞
⎠
⎟⎟ =
α
α
Z^2
14
9 012.
−+=
Z^22
2
2
22 mx
x
k
xx Ex
d
d
ψψψ() () ()
226 PART 2 QUANTUM MECHANICS AND SPECTROSCOPY
Table 11.1
The first four solutions for the Hermite equation.
0 H 0 (y)
01
12 y
24 y^2 − 2
38 y^3 − 12 y