QMGreensite_merged

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)