Page 60 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.5 Particle in a one-dimensional box 51

The remaining constant of integration A is determined by the normalization
condition (2.32),
a ð
1 nðx a a
2 2 2 2 2 2
jø n (x)j dx ˆjAj sin dx ˆjAj sin nè dè ˆjAj ˆ 1
ÿ1 0 a ð 0 2
where equation (A.15) was used. Therefore, we have
2
2
jAj ˆ
a
or
r 
2
A ˆ e iá
a
Setting the phase á equal to zero since it has no physical signi®cance, we
obtain for the normalized wave functions
r 
2 nðx
ø n (x) ˆ sin , 0 < x < a (2:40)
a a
ˆ 0, x , 0, x . a
È
The time-dependent Schrodinger equation (2.30) for the particle in a box has
an in®nite set of solutions ø n (x) given by equation (2.40). The ®rst four wave
functions ø n (x) for n ˆ 1, 2, 3, and 4 and their corresponding probability
2
densities jø n (x)j are shown in Figure 2.2. The wave function ø 1 (x) corre-
sponding to the lowest energy level E 1 is called the ground state. The other
wave functions are called excited states.
If we integrate the product of two different wave functions ø l (x) and ø n (x),
we ®nd that
a 2 a lðx nðx 2 ð
ø l (x)ø n (x)dx ˆ sin sin dx ˆ sin lè sin nè dè ˆ 0
0 a 0 a a ð 0
(2:41)
where equation (A.15) has been introduced. This result may be combined with
the normalization relation to give

a
ø l (x)ø n (x)dx ˆ ä ln (2:42)
0
where ä ln is the Kronecker delta,

ä ln ˆ 1, l ˆ n
(2:43)
ˆ 0, l 6ˆ n

Functions that obey equation (2.41) are called orthogonal functions. If the
orthogonal functions are also normalized, as in equation (2.42), then they are