QMGreensite_merged

199

Inatomicphysics,theintegervaluesofthelquantumnumberareassignedletters,
namely:

l = 0 S

l = 1 P

l = 2 D

l = 3 F (12.43)

Beyond l = 3, the assignment is alphabetical, i.e. l = 3 , 4 , 5 , 6 ...correspondto
”F,G,H,I...” respectively(the reasons forthese letter assignments goback to the
historyofspectroscopy,andneednotconcernushere.)^2 Thusthe{nlm}={ 100 }
stateisreferredtoasthe”1S”state;the{nlm}={ 21 m}statesarereferredtoas
the”2P”states,andsoon,asindicatedinFig. [12.1].
Wenowusetheformulasineq. (12.36)tocalculatethe 1 Sand 2 Swavefunctions
explicitly.

• The1SGroundState Sincejs=n−l− 1 ≥0,thesmallestpossiblevaluefor
  nisn=1,whichimpliesthatl=m=0.Thisisthelowestenergystate,or”ground
  state”ofthe hydrogenatom; itisthestatewherethe bindingenergy isgreatest.
  Accordingto(12.36)wehave

φ 100 = N

u 10 (2r/a 0 )

r

Y 00

=

N

√

4 π

u 10 (2r/a 0 )

r

(12.44)

where

u 10 (ρ) = e−ρ/^2 ρF 10 (ρ)

= e−ρ/^2 ρ

1 −∑ 0 − 1

j=0

cjρj

= e−ρ/^2 ρ (12.45)

Therefore

φ 100 =

N

√

4 π

2

a 0

e−r/a^0 (12.46)

(^2) ”S”standsforthe”sharp”seriesofspectrallines,”P”standsforthe”principal”series,”D”for
”diffuse”and”F”for”fundamental.”