234 CHAPTER14. THEADDITIONOFANGULARMOMENTUM
Then,usingthefactthatfortheHydrogenatomgroundstate
|ψ(r=0)|^2 =1
πa^30(14.83)
withBohrradiusa 0 =h ̄^2 /me^2 ,weget
∆E=
4 gp ̄h^4
3 mpm^2 ec^2 a^40{ 1
4 (triplet)
−^34 (singlet)(14.84)
Itisthenpossiblefortheelectron-protonsystemtomakeatransitionfromthe
higherenergytripletstate,tothelowerenergysingletstate,emittingaphoton,whose
energyisthedifferenceinenergiesbetweenthetriplet/singletstates. Accordingto
DeBroglierelation,thefrequencyofthisemittedphotonwillbe
f =∆Etriplet−∆Esinglet
h=4 gp ̄h^4
3 mpm^2 ec^2 a^4 h
= 1420 MHz (14.85)withacorrespondingwavelengthof 21 cm.
Atomichydrogenisthe mostcommonelementfoundininterstellar space, and
transitionsbackandforthbetweenthetripletandsingletstates(separatedbyonly
about 6 millionthsofanelectronvolt),occurveryeasily.Forthatreason,the 21 cm
spectrallineduetotransitionsbetweenthehydrogentripletandsingletstates,isone
ofthemostdominantfeaturesoftheinterstellarradiospectrum.