Pauli matrices

Spin is an intrinsic angular momentum of any 
rotating fundamental particles as an electron within 
an atom.


S = Sx i + Sy j + Sz k 
S2 = S2x + S2y + S2z

The two possible spin states are |s m>  are 
|1/2 1/2>  and |1/2 -1/2> .


We use the following notations:
|0> or |↑> for |1/2 1/2>, and 
|1> or |↓> for |1/2 -1/2>


|0> and |1> are simultaneous eigenvectors 
of S2 and Sz. Therefore 
S2|0>  = (1/2) (1/2 + 1)ℏ2|0> 
S2|1>  = (1/2) (1/2 + 1)ℏ2|1>  
Sz|0>  = (1/2) ℏ|0>  
Sz|1>  = (-1/2) ℏ|1>   

S2|0>  = (3/4)ℏ2|0> 
S2|1>  = (3/4)ℏ2|1>  
Sz|0>  = (1/2)ℏ|0>  
Sz|1>  = (-1/2)ℏ|1>   

The matrix for Sz in ℏ/2 units is:
|1   0|
|0  -1|

We have also:
Sx = (1/2)(S+ + S-)
Sy = (1/2i)(S+ - S-)
So
S+ |s m> = [s(s+1) -m(m+1)]1/2ℏ |s m+1>
S= |s m> = [s(s+1) -m(m-1)]1/2ℏ |s m-1>
Thus:
S+ |0> = 0
S+ |1> = ℏ |0>
S- |0> = ℏ |1>
S- |1> = 0

Therefore:
Sx11 =  <0|Sx|0> = 
<0|(1/2)(S+ + S-)|0> = (1/2)<0|S+|0>  + (1/2)<0|S-|0> 
(1/2)<1|S+|1>  + (1/2)<1|S-|1>  = 0 + 0 = 0

Sx22 = <1|Sx|1> = 
<1|(1/2)(S+ + S-)|1> = (1/2)<1|S+|1>  + (1/2)<1|S-|1> 
(1/2)<1|S+|1>  + (1/2)<1|S-|1>  = 0 + 0 = 0

Sx12 = <0|Sx|1> = 
<0|(1/2)(S+ + S-)|1> = (1/2)<0|S+|1>  + (1/2)<0|S-|1> 
(1/2)<0|S+|1>  + (1/2)<0|S-|1>  =ℏ/2 + 0 = ℏ/2

Sx21 = <1|Sx|0> = 
<1|(1/2)(S+ + S-)|0> = (1/2)<1|S+|0>  + (1/2)<1|S-|0> 
(1/2)<1|S+|1>  + (1/2)<1|S-|1>  = 0 + ℏ/2 = ℏ/2

The matrix for Sx in ℏ/2 units is:
|Sx11   Sx12| 
|Sx21   Sx22|

That is:
|0   1|
|1   0|

For a similar calculation, we have:

Sy11 =  <0|Sy|0> = 
<0|(1/2i)(S+ - S-)|0> = (1/2i)<0|S+|0>  - (1/2)<0|S-|0> 
(1/i)<1|S+|1>  + (1/2i)<1|S-|1>  = 0 - 0 = 0

Sy22 = <1|Sy|1> = 
<1|(1/2i)(S+ - S-)|1> = (1/2i)<1|S+|1>  - (1/2i)<1|S-|1> 
(1/2i)<1|S+|1>  - (1/2i)<1|S-|1>  = 0 - 0 = 0

Sy12 = <0|Sy|1> = 
<0|(1/2)(S+ - S-)|1> = (1/2)<0|S+|1>  - (1/2)<0|S-|1> 
(1/2i)<0|S+|1>  - (1/2i)<0|S-|1>  =ℏ/2i - 0 = - i ℏ/2

Sy21 = <1|Sy|0> = 
<1|(1/2i)(S+ - S-)|0> = (1/2i)<1|S+|0> - (1/2i)<1|S-|0> 
(1/2i)<1|S+|1>  -(1/2i)<1|S-|1>  = 0 - ℏ/2i = +i ℏ/2i

The matrix for Sy in ℏ/2 units is:
|Sy11   Sy12| 
|Sy21   Sy22|

That is:
|0  -i|
|i   0|