QMGreensite_merged

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

= 0 (12.13)