Quantum Mechanics for Mathematicians

This is the orthogonal transformation ofR^3 given by

v=





v 1

v 2

v 3



→





cos 2θ −sin 2θ 0

sin 2θ cos 2θ 0

0 0 1









v 1

v 2

v 3



 (6.2)

The same calculation can readily be done for the case ofe 1 , then use the
Euler angle parametrization of equation 6.1 to show that a generalu(θ,w) can
be written as a product of the cases already worked out.

Notice that asθgoes from 0 to 2π,u(θ,w) traces out a circle inSp(1). The
homomorphism Φ takes this to a circle inSO(3), one that gets traced out twice
asθgoes from 0 to 2π, explicitly showing the nature of the double covering
above that particular circle inSO(3).
The derivative of the map Φ will be a Lie algebra homomorphism, a linear
map
Φ′:sp(1)→so(3)

It takes the Lie algebrasp(1) of pure imaginary quaternions to the Lie algebra
so(3) of 3 by 3 antisymmetric real matrices. One can compute it easily on basis
vectors, using for instance equation 6.2 above to find for the case~w=k

Φ′(k) =

d

dθ

Φ(cosθ+ksinθ)|θ=0

=





−2 sin 2θ −2 cos 2θ 0

2 cos 2θ −2 sin 2θ 0

0 0 0





|θ=0

=





0 −2 0

2 0 0

0 0 0



= 2l 3

Repeating this on other basis vectors one finds that

Φ′(i) = 2l 1 ,Φ′(j) = 2l 2 ,Φ′(k) = 2l 3

Thus Φ′is an isomorphism ofsp(1) andso(3) identifying the bases

i

2

,

j

2

,

k

2

and l 1 ,l 2 ,l 3

Note that it is thei 2 ,j 2 ,k 2 that satisfy simple commutation relations

[

i

2

,

j

2

]

=

k

2

,

[

j

2

,

k

2

]

=

i

2

,

[

k

2

,

i

2

]

=

j

2

6.2.4 The spin group andSU(2)

Instead of doing calculations using quaternions with their non-commutativity
and special multiplication laws, it is more conventional to choose an isomorphism