Spectroscopy Cheat Sheet
Atomic Spectroscopy
• Energy of hydrogen atom
En=ÃZ2ħ2
2µa2
µ!1
n2(1)
aµ=4π²0ħ2
µe2=me
µ
a0(2)
• Mean radii of hydrogen atom
¿1
rÀ∝Z
n2(3)
〈r〉=aµn2(1 +(1 −l(l+1)/n2)/2) (4)
• Spin-orbit coupling arises due to electron spin magnetic moment in-
teracting with magnetic moment generated by nuclear charge orbiting
electron in electron rest frame. This effect is proportional to the mag-
netic interaction ∝1/r3®and the nuclear charge Z, so overall you
would expect SO coupling to be proportional to Z4. In fact
¿1
r3À=2
a3
µn3l(l+1)(2l+1) (5)
• Selection rules. For optically allowed transitions
∆n=anything (6)
∆l= ±1 from conservation of angular momentum and parity (7)
∆ml=0,±1 from conservation of angular momentum (8)
∆s=0 from conservation of spin (9)
∆ms=0 (10)
∆j=0,±1 (no j=0→0) conservation of angular momentum (11)
∆mj=0,±1 (12)
These are the same for many electron atoms with l,ml,j,mj,s,msre-
placed by L,ML,J,MJ,S,MS.
• Zeeman effect splits levels with angular momentum
EZeeman =µBgJBM J(13)
gJ=1+J(J+1) −L(L+1) +S(S+1)
2J(J+1) (14)
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