Page 71 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

62 Schro Èdinger wave mechanics

Thus, the three-dimensional problem has been reduced to three one-dimen-
sional problems.
The differential equations (2.78) are identical in form to equation (2.34) and
the boundary conditions are the same as before. Consequently, the solutions
inside the box are given by equation (2.40) as
r 
2 n x ðx
X(x) ˆ sin , n x ˆ 1, 2, 3, ...
a a
r 
2 n y ðy
Y(y) ˆ sin , n y ˆ 1, 2, 3, ... (2:80)
b b
r 
2 n z ðz
Z(z) ˆ sin , n z ˆ 1, 2, 3, ...
c c
and the constants E x , E y , E z are given by equation (2.39)
2 2
n h
E x ˆ x , n x ˆ 1, 2, 3, ...
8ma 2
2 2
n h
y
E y ˆ , n y ˆ 1, 2, 3, ... (2:81)
8mb 2
2 2
n h
E z ˆ z , n z ˆ 1, 2, 3, ...
8mc 2
The quantum numbers n x , n y , n z take on positive integer values independently
of each other. Combining equations (2.76) and (2.80) gives the wave functions
inside the three-dimensional box
r 
8 n x ðx n y ðy n z ðz
(r) ˆ sin sin sin (2:82)
v a b c
ø n x ,n y ,n z
where v ˆ abc is the volume of the box. The energy levels for the particle are
obtained by substitution of equations (2.81) into (2.79)
2
h 2 n 2 n y n 2
ˆ x ‡ ‡ z (2:83)
E n x ,n y ,n z 2 2 2
8m a b c

Degeneracy of energy levels
If the box is cubic, we have a ˆ b ˆ c and the energy levels become
h 2 2 2 2
ˆ (n ‡ n ‡ n ) (2:84)
E n x ,n y ,n z 2 x y z
8ma
2
2
The lowest or zero-point energy is E 1,1,1 ˆ 3h =8ma , which is three times the
zero-point energy for a particle in a one-dimensional box of the same length.
The second or next-highest value for the energy is obtained by setting one of