BioPHYSICAL chemistry

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