200 CHAPTER12. THEHYDROGENATOM
WemustnormalizethewavefunctiontodetermineN:
1 =
∫
r^2 drdΩ|φ 100 |^2
= N^2
∫
drr^2 u 10 (2r/a 0 )^2
= N^2
(
2
a 0
) 2 ∫∞
0
drr^2 e−^2 r/a^0
= N^2 a 0 (12.47)
So,finally,theground-statewavefunctionis
φ 100 (r,θ,φ)=
2
√
4 πa^30
e−r/a^0 (12.48)
- The 2SExcitedState Then= 2 statesareknownasthe”firstexcited”
states,sincetheycorrespondtothelowestenergyexcitationabovethegroundstate.
Thel= 0 stateisalwaysreferredtoastheS-state;sothe{nlm}={ 200 }isknown
asthe2Sstate,inspectroscopicnotation.Again,usingeq. (12.36)
φ 200 (r,θ,φ) = N
u 20 (r/a 0 )
r
Y 00
u 200 (ρ) = e−ρ/^2 ρF 20 (ρ)
= e−ρ/^2 ρ
2 −∑ 0 − 1
j=0
cjρj
= e−ρ/^2 ρ(c 0 +c 1 ρ)
c 1 =
0 +l+ 1 −n
(0+1)(0+ 2 l+2)
c 0
= −
1
2
(12.49)
sothat
φ 200 =
N
√
4 π
(1−
1
2
ρ)
ρ
r
e−ρ/^2
=
N
a 0
√
4 π
(
1 −
r
2 a 0
)
e−r/^2 a^0 (12.50)