Hence we may tabulate the three quantum numbers n,l, and mtogether with their
permissible values as follows:Principal quantum number n1, 2, 3,Orbital quantum number l0, 1, 2, , (n1) (6.17)
Magnetic quantum number ml0, 1, 2, , lIt is worth noting again the natural way in which quantum numbers appear in quantum-
mechanical theories of particles trapped in a particular region of space.
To exhibit the dependence of R, , and upon the quantum numbers n,l,m, we
may write for the electron wave functions of the hydrogen atomRnl (^) lml (^) ml (6.18)
The wave functions R, , and together with are given in Table 6.1 for n1, 2,
and 3.
206 Chapter Six
Table 6.1Normalized Wave Functions of the Hydrogen Atom for n1, 2, and 3
nl ml () () R(r) (r, , )
10 0 er^ a^0 er^ a^0
20 0 2 er^2 a^0 2 er^2 a^0
21 0 cos er^2 a^0 er^2 a^0 cos
21 1 e^ i sin er^2 a^0 er^2 a^0 sin e^ i
30 0 27 18 (^2) er^3 a^0 27 18 (^2) er^3 a^0
31 0 cos 6 er^3 a^0 6 er^3 a^0 cos
31 1 e^ i sin 6 er^3 a^0 6 er^3 a^0 sin e^ i
3 2 0 (3 cos^2 1) er^3 a^0 er^3 a^0 (3 cos^2 1)
32 1 e^ i sin cos er^3 a^0 er^3 a^0 sin cos e^ i
32 2 e^2 i sin^2 er^3 a^0 er^3 a^0 sin^2 e^2 i
The quantity a 0 4 0 ^2 /me^2 5.292 10 ^11 m is equal to the radius of the innermost Bohr orbit.
r^2
a^20
1
162 a 03 2
r^2
a^20
4
81 30 a 03 2
^15
4
1
2
r^2
a^20
1
81 a 03 2
r^2
a^20
4
81 30 a 03 2
^15
2
1
2
r^2
a^20
1
81 6 a 03 2
r^2
a^20
4
81 30 a 03 2
^10
4
1
2
r
a 0
r
a 0
1
81 a 03 2
r
a 0
r
a 0
4
81 6 a 03 2
^3
2
1
2
r
a 0
r
a 0
2
81 a 03 2
r
a 0
r
a 0
4
81 6 a 03 2
^6
2
1
2
r^2
a^20
r
a 0
1
81 3 a 03 2
r^2
a^20
r
a 0
2
81 3 a 03 2
1
2
1
2
r
a 0
^1
8 a 03 2
r
a 0
1
2 6 a 03 2
3
2
1
2
r
a 0
1
4 2 a 03 2
r
a 0
1
2 6 a 03 2
6
2
1
2
r
a 0
1
4 2 a 03 2
r
a 0
1
2 2 a 03 2
1
2
1
2
1
a 03 2
^2
a 03 2
1
2
1
2
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