The Chemistry Maths Book, Second Edition

18.3 Matrix algebra 511

where and A1= 1 h 22 π where his Planck’s constant. The commutation

properties of these matrices are

[S

x

, S

y

] 1 = 1 S

x

S

y

1 − 1 S

y

S

x

and similarly for the other pairs. Therefore

[S

x

, S

y

] 1 = 1 iAS

z

,[S

y

, S

z

] 1 = 1 iAS

x

,[S

z

, S

x

] 1 = 1 iAS

y

(18.34)

In addition,

and similarly forS

y

2

andS

z

2

. Therefore

(18.35)

represents the square of spin angular momentum. The quantity is the square of

the magnitude of the spin angular momentum of an electron, whose (total) spin

quantum number is ,

(18.36)

0 Exercise 42

Multiplication by a unit matrix

If Ais an m 1 × 1 nmatrix and if I

m

and I

n

are the unit matrices of orders mand n,

respectively, then

I

m

A 1 = 1 A 1 = 1 AI

n

(18.37)

ss()+=, 1 s=

3

4

1

2

22

for

s=

1

2

3

4

2



SSS I

xyz

222 2

3

4

++=

SSS

xxx

22 2

1

4

01

10

01

10

1

4

1

==

































=

00

01

1

4

2

















= I

=

−

















=

1

2

10

01

2

ii

z

S

=

−

















−

−

































1

4

0

0

0

0

2



i

i

i

i

=

















−

















−

−















1

4

01

10

0

0

0

0

2



i

i

i

i



































01

10

i=− 1 ,