50 magnitude of magnetic dipole moment near from
e
m e
2
. It is predicted that the magnetic dipole moment is from electron.
Goudsmit and Uhlenbeck studied spectral lines of hydrogen and alkaline metal. They found that certain lines in optical spectrum of hydrogen and alkaline metal show
that it is found line pairs. This phenomenon is called fine structure. Sommerfeld can explaine the fine structure using Bohr atomic model. Fine structure in hydrogen atom
occurred caused by the change of electron mass which move in high speed. This assumption can not explaine fine structure in optical spectrum of alkaline atom. The
truth of Sommerfeld assumption is questioned.
To explain fine structure Goudsmit and Uhlenbeck in 1925 suggested an assumtion
:” Electron has angular momentum and magnetic dipole moment intrinsically, which has z component which stated by spin magnetic quantum number m
s
which only has 2 value of +12 and -12
”. If the assumption accepted, then electron in hydrogen atom need four quantum
number as follow: n : principal quantum number
l : orbital quantum number m: magnetic quantum number usually has symbol of m
l
m
s
: spin magnetic quantum number +12 dan -12
C. Spin Elektron dan Nilai Eigen Atom H
Analog with angular momentum, magnitude of spin angular momentum of electron
S
can be written:
1
s s
S
6.2 Component of
S
in z direction is:
s z
m S
6.3
In this case s related with m
s
, for m
s
has 2 values which has difference of 1 1
s
m ,
mean while its value is laid between –s and +s.
m
s
= -12, +12 s = 12
The result of experiment show that 1
s s
m g
, then it is obtained gyration factor or spin factor has value of :
51 2
s
g Further more if spin motion is applied in hydrogen atom system, the motion
must be represented in a function of
ms
, and operator
op
S
and
zop
S
operate with rule as follow:
ms ms
ms op
s s
S
2 2
2
4 3
1
6.4
ms s
ms zop
m S
6.5 There are 2 spin functions:
for m
s
= +12, and
for m
s
= -12 Operator
op
S
and
zop
S
only work in function of
and
and does mot work in coordinat function
, ,
r
. Also differential operator does not work in a function of
ms
. Space quantification of spin angular momentum is presented in Figure 6.2. Wave function of hydrogen atom can be notated:
s l
m m
l n
m m
l n
s l
, ,
,
6.6
By this model the relation of its eigen value fulfilled:
s l
n s
l op
m m
l n
E m
m l
n H
6.7
s l
s l
op
m m
l n
l l
m m
l n
L
2 2
1
6.8
s l
l s
l Zop
m m
l n
m m
m l
n L
6.9
s l
s l
op
m m
l n
s s
m m
l n
S
2 2
1
6.10
s l
s s
l Zop
m m
l n
m m
m l
n S
6.11
52
Figure 6.2. Space Quantification of Spin Angular Momentum
For normalized wave function:
1
s l
s l
m m
l n
m m
l n
6.12
1
6.14
1
6.15
6.16
D. Hydrogen Atom with Spin in External Magnetic Field Magnetic dipole moment of hydrogen atom is:
S L
H
where H hydrogen 6.17 Then it can be determined:
S g
L g
m e
S m
eg L
m eg
s l
e H
e s
e l
H
2 2
2
6.18 z
S
z
2
1
2
1
S
4 3
spin-up
4 3
spin-down
53 It have been known that
1
l
g and
2
s
g ; then magnetic dipole moment of hydrogen
atom fullfilled:
S L
m e
e H
2
2
6.19
Potential energy of hydrogen atom in extdernal magnetic field:
B S
B L
m e
B V
e H
B
. 2
. 2
.
z z
e B
S L
m eB
V 2
2
direction of
B
is parallel with z axis 6.20
The attendance of external magnetic field
B
cause the shift of total energy in hydrogen atom:
s l
B s
l e
m m
B m
m m
eB E
2 2
2
6.21
E. Diagram of Energy States in Hydrogen Atom
