Quantum Mechanics for Mathematicians

and
̃u(a, 1 ) =e−ia·P

so the Lie algebra representation is given by the usualPoperator. This action
of the translations is easily seen to commute withσ·Pand thus act on the
solutions to 34.6. It is the action of rotations that requires a more complicated
discussion than in the single-component case.
In chapter 19 we saw thatR∈SO(3) acts on single-component momentum
space solutions of the Schr ̈odinger equation by

ψ ̃E(p)→u ̃(0,R)ψ ̃E(p) =ψ ̃E(R−^1 p)

This takes solutions to solutions since the operator ̃u(0,R) commutes with the
Casimir operator|P|^2

̃u(0,R)|P|^2 =|P|^2 ̃u(0,R)⇐⇒ ̃u(0,R)|P|^2 u ̃(0,R)−^1 =|P|^2

This is true since

̃u(0,R)|P|^2 ̃u(0,R)−^1 ψ ̃(p) = ̃u(0,R)|P|^2 ψ ̃(Rp)

=|R−^1 P|^2 ψ ̃(R−^1 Rp) =|P|^2 ψ ̃(p)

To get a representation on two-component wavefunctions that commutes
with the operatorσ·Pwe need to change the action of rotations to

ψ ̃E,±(p)→ ̃u(0,Ω)ψ ̃E,±(p) = Ωψ ̃E,±(R−^1 p)

With this action on solutions we have

u ̃(0,Ω)(σ·P)u ̃(0,Ω)−^1 ψ ̃E,±(p) = ̃u(0,Ω)(σ·P)Ω−^1 ψ ̃E,±(Rp)

=Ω(σ·R−^1 P)Ω−^1 ψ ̃E,±(R−^1 Rp)

=(σ·P)ψ ̃E,±(p)

where we have used equation 6.5 to show

Ω(σ·R−^1 P)Ω−^1 =σ·RR−^1 P=σ·P

TheSU(2) part of the group acts by a product of two commuting different
actions on the two factors of the tensor product 34.2. These are:

1. The same action on the momentum coordinates as in the one-component
   case, just usingR= Φ(Ω), theSO(3) rotation corresponding to theSU(2)
   group element Ω. For example, for a rotation about thex-axis by angleφ
   we have
   ψ ̃E,±(p)→ψ ̃E,±(R(φ,e 1 )−^1 p)
   Recall that the operator that does this ise−iφL^1 where

−iL 1 =−i(Q 2 P 3 −Q 3 P 2 ) =−

(

q 2

∂

∂q 3

−q 3

∂

∂q 2

)