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ELECTRON SPIN AND ANGULAR
MOMENTUM
Mohammad B
...

!

!
e
e!
µ =−
l µB =
(Bohr magneton)
2m
2m

•  The magne>c moment associated with the orbital
angular momentum is quan>sed
...
The electrons precess about the direc>on of
the field, called Larmor precession
...


Electron spin
•  Electron spin was introduced by Uhlenbeck
and Goudsmit in 1925
...
Four
quantum numbers: n, l, ml, ms
...

"
e !
| s |= s(s +1)! µ s = −gS
s
2m

Total Angular Momentum
•  If j and mj are quantum numbers for the single electron (hydrogen
atom)
...


•  The total angular momentum quantum number for the single electron
can only have the values

4

The total angular momentum &
magne>c moment
! ! !
j = l + s, µ = µl + µ s

j
l

µ j = µl cos( j, l) + µ s cos( j, s)
= −g j J µ B
ˆ2
ˆ
3 1 s −l
gj = + ( 2 )
ˆ
2 2
j

s

µS

µl

2

µ µj

Fine structure
•  All energy levels except the s states of oneelectron atoms are spit into two substates
...

•  It can not be explained by Coulomb interac>on
between the nucleus and the electrons
...


Spin-orbit coupling (I)
•  An electron revolving about a nucleus finds
itself in a magne>c field produced by the
nucleus which is circling about it in its own
frame of reference
...

S=1/2 could make some single state (n,l,m)
split into two substates
...
The quantum number j is the quantum
number of the total angular momentum
...


The symbolism for atomic states
•  The atomic state is denoted by a symbol, and
the orbital angular momentum is indicated by
upper case letters S,P,D,F
...

•  The multiplicity 2s+1 (the number of j values)
is indicated by a superscript to the left of the
letter
...

µj

µ j = − j( j +1)g j µ B

µ jz = −m j g j µ B g j =

µB

j( j +1)
•  For the orbit angular momentum, j=l, gl=1
...

•  For the total angular momentum, g factor
is given by:
2
ˆ2

ˆ
3 1 s −l
gj = + ( 2 )
ˆ
2 2
j

The alkali atoms
•  The alkali atoms have a
weakly bound outer
electron, the so-called
valence electron, and all
other (Z-1) electrons are
in closed shells
...
e
...
6nm
...

•  The sharp secondary series: nSà2P, Δl = ±1,
Δj = ±1, 0
doublet, two lines are separated
uniformly with n increased
...

2D à2P , 2D à 2P , 2D à 2P
5/2
3/2
3/2
3/2
3/2
1/2;
2D à2P (?)
5/2
1/2

Spectrum of alkali metals
First primary series

Sharp secondary series

Diffuse secondary series

Stern-Gerlach experiment (I)
•  Space quan>za>on was first explicitly
demonstrated by Stern & Gerlach in 1921
...

•  When the magne>c moment is parallel to B,
it moves in the direc>on of increasing field
strength, while an an>-parallel magne>c
moment moves towards lower field
strength
...

•  For hydrogen atoms in ground state, n=1,
l=0, j=s, mj=-1/2, +1/2, gj=2, mjgj =±1,

The Zeeman effect
•  The total angular momentum J is the vector
sum of orbital angular momentum and spin
angular momentum, J=L+S

J=

j( j +1)!

Jz = m j!

•  When there is no external magne>c field,the
total angular momentum is conserved
...

•  Selec>on rules:

Δm = 0, ±1

Total Angular Momentum
No external magne5c field:
•  Only Jz can be known
because the uncertainty
principle forbids Jx or Jy
from being known at the
same >me as Jz

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