Transitions which changeℓby one unit are strongly preferred, as we will later learn.
16.3 Derivations and Calculations
16.3.1 Solution of Hydrogen Radial Equation*.
The differential equation we wish to solve is.
(
d^2
dr^2+
2
rd
dr)
REℓ(r) +2 μ
̄h^2(
E+
Ze^2
r−
ℓ(ℓ+ 1) ̄h^2
2 μr^2)
REℓ(r) = 0First we change to adimensionless variableρ,
ρ=√
− 8 μE
̄h^2r,giving the differential equation
d^2 R
dρ^2+
2
ρdR
dρ−
ℓ(ℓ+ 1)
ρ^2R+
(
λ
ρ−
1
4
)
R= 0,
where the constant
λ=Ze^2
̄h√
−μ
2 E=Zα√
−μc^2
2 E