194 CHAPTER12. THEHYDROGENATOM
whichreduces,asshowninthepreviouslecture,tothe”radialequation”forRkl(r)
d^2 Rkl
dr^2
+
2
r
dRkl
dr
+
[
2 m
̄h^2
{Ekl+
e^2
r
}−
l(l+1)
r^2
]
Rkl(r)= 0 (12.5)
or [
1
r
d^2
dr^2
r−
l(l+1)
r^2
+
2 me^2
̄h^2
1
r
+
2 mE
̄h^2
]
R(r)= 0 (12.6)
Weareinterestedinthespectrumofatomichydrogen,sothismeanswewould
liketosolvefortheboundstates, i.e. thosestatesforwhichE<0. Fromnowon,
wetakeE=−|E|<0.Multiplytheaboveequationbyrontheleft,anddefine
u(r)=rR(r) (12.7)
sothat [
d^2
dr^2
−
l(l+1)
r^2
+
2 me^2
̄h^2
1
r
−
2 m|E|
̄h^2
]
u(r)= 0 (12.8)
Thisequationinvolvestwoconstants,namely
2 mE
̄h^2
and
2 me^2
̄h^2
(12.9)
whichwecanreducetooneconstantbyrescalingr. Define
k^2 =
2 m|E|
̄h^2
and r=
ρ
2 k
(12.10)
Substituteinto(12.8),andwefind
d^2 u
dρ^2
−
l(l+1)
ρ^2
u+
(
λ
ρ
−
1
4
)
u= 0 (12.11)
wherewehavedefined
λ=
me^2
k ̄h^2
=
1
ka 0
(12.12)
(a 0 isthe Bohrradius ̄h^2 /me^2 ). Theproblemisreduced to finding values λand
functionsu(ρ)whichsatisfy(12.11).
Thestrategyforsolving(12.11)istofirstsolvetheequationintheasymptotic
limitsρ→∞andρ→0.Withthesolutionsforverylargeandverysmallρinhand,
wethenlookforasolutionwhichinterpolatesbetweenthesetworegions.
Beginwithρ→∞. Thentermsproportionalto 1 /ρ^2 and 1 /ρcanbedropped,
andtheequationbecomes
d^2 u
dρ^2
−
u
4