Page 72 - PRINCIPLES OF QUANTUM MECHANICS as Applied to Chemistry and Chemical Physics

2.8 Particle in a three-dimensional box 63

Table 2.1. Energy levels for a particle in a three-
dimensional box with a ˆ b ˆ c

Energy Degeneracy Values of n x , n y , n z
2
2
3(h /8ma ) 1 1,1,1
2
2
6(h /8ma ) 3 2,1,1 1,2,1 1,1,2
2
2
9(h /8ma ) 3 2,2,1 2,1,2 1,2,2
2
2
11(h /8ma ) 3 3,1,1 1,3,1 1,1,3
2
2
12(h /8ma ) 1 2,2,2
2
2
14(h /8ma ) 6 3,2,1 3,1,2 2,3,1 2,1,3 1,3,2 1,2,3
Table 2.2. Energy levels for a particle in a three-
dimensional box with b ˆ a=2,c ˆ a=3
Energy Degeneracy Values of n x , n y , n z
2
2
14(h /8ma ) 1 1,1,1
2
2
17(h /8ma ) 1 2,1,1
2
2
22(h /8ma ) 1 3,1,1
2
2
26(h /8ma ) 1 1,2,1
2
2
29(h /8ma ) 2 2,2,1 4,1,1
2
2
34(h /8ma ) 1 3,2,1
2
2
38(h /8ma ) 1 5,1,1
2
2
41(h /8ma ) 2 1,1,2 4,2,1
the integers n x , n y , n z equal to 2 and the remaining ones equal to unity. Thus,
2
2
there are three ways of obtaining the value 6h =8ma , namely, E 2,1,1 , E 1,2,1 ,
and E 1,1,2 . Each of these three possibilities corresponds to a different wave
function, respectively, ø 2,1,1 (r), ø 1,2,1 (r), and ø 1,1,2 (r). An energy level that
corresponds to more than one wave function is said to be degenerate. The
second energy level in this case is threefold or triply degenerate. The zero-
point energy level is non-degenerate. The energies and degeneracies for the
®rst six energy levels are listed in Table 2.1.
The degeneracies of the energy levels in this example are the result of
symmetry in the lengths of the sides of the box. If, instead of the box being
cubic, the lengths of b and c in terms of a were b ˆ a=2, c ˆ a=3, then the
values of the energy levels and their degeneracies are different, as shown in
Table 2.2 for the lowest eight levels.
Degeneracy is discussed in more detail in Chapter 3.