and
ρ=√
− 8 μE
̄h^2r.Theprinciple quantum numbernis an integer from 1 to infinity.
n= 1, 2 , 3 ,...This principle quantum number is actually the sum of the radial quantum number plusℓplus 1.
n=nr+ℓ+ 1and therefore, the total angular momentum quantum numberℓmust be less thann.
ℓ= 0, 1 , 2 ,...,n− 1This unusual way of labeling the states comes about because a radial excitation has the same energy
as an angular excitation for Hydrogen. This is often referred to asanaccidental degeneracy.
16.1 The Radial Wavefunction Solutions
Defining theBohr radius
a 0 =
̄h
αmc,
we can compute the radial wave functions (See section 16.3.2) Hereis a list of the first several radial
wave functionsRnℓ(r).
R 10 = 2
(
Z
a 0)^32
e−Zr/a^0R 21 =
1
√
3
(
Z
2 a 0) 32 (
Zr
a 0)
e−Zr/^2 a^0R 20 = 2
(
Z
2 a 0) 32 (
1 −
Zr
2 a 0)
e−Zr/^2 a^0R 32 =
2
√
2
27
√
5
(
Z
3 a 0)^32 (
Zr
a 0) 2
e−Zr/^3 a^0R 31 =
4
√
2
3
(
Z
3 a 0) 32 (
Zr
a 0)(
1 −
Zr
6 a 0)
e−Zr/^3 a^0R 30 = 2
(
Z
3 a 0) 32 (
1 −
2 Zr
3 a 0+
2 (Zr)^2
27 a^20)
e−Zr/^3 a^0For a given principle quantum numbern,the largestℓradial wavefunction is given by
Rn,n− 1 ∝rn−^1 e−Zr/na^0