Quantum Mechanics for Mathematicians

where
(
ψ 1 (q)
ψ 2 (q)

)

=

1

(2π)

(^32)

∫

R^3

δ(|p|^2 − 2 mE)(ψ ̃E,+(p) +ψ ̃E,−(p))eip·qd^3 p

=

1

(2π)

(^32)

∫

R^3

δ(|p|^2 − 2 mE)(αE,+(p)u+(p) +αE,−(p)u−(p))eip·qd^3 p

(34.10)

34.3 TheE ̃(3)-invariant inner product

One can parametrize solutions to the Pauli equation and write anE ̃(3)-invariant
inner product on the space of solutions in several different ways. Three different
parametrizations of solutions that can be considered are:

• Using the initial data at a fixed time
  (
  ψ 1 (q)
  ψ 2 (q)

)

Here theE ̃(3)-invariant inner product is

〈(

ψ 1 (q)

ψ 2 (q)

)

,

(

ψ 1 ′(q)

ψ 2 ′(q)

)〉

=

∫

R^3

(

ψ 1 (q)

ψ 2 (q)

)†(

ψ′ 1 (q)

ψ′ 2 (q)

)

d^3 q

This parametrization does not make visible the decomposition into irre-

ducible representations ofE ̃(3).

• Using the Fourier transforms
  (
  ψ ̃ 1 (p)
  ψ ̃ 2 (p)

)

to parametrize solutions, the invariant inner product is

〈(

ψ ̃ 1 (p)

ψ ̃ 2 (p)

)

,

(

̃ψ′

1 (p)

ψ ̃′ 2 (p)

)〉

=

∫

R^3

(

ψ ̃ 1 (p)

ψ ̃ 2 (p)

)†(

ψ ̃′ 1 (p)

ψ ̃′ 2 (p)

)

d^3 p

The decomposition of equation 34.5 can be used to express solutions of

energyEin terms of two-component functionsψ ̃E(p), with an invariant

inner product on the space of such solutions given by

〈ψ ̃E(p),ψ ̃′E(p)〉=

1

4 π

∫

S^2

ψ ̃E(p)†ψ ̃′

E(p) sin(φ)dφdθ

where (p,φ,θ) are spherical coordinates on momentum space andS^2 is

the sphere of radius

√

2 mE.

Theψ ̃E(p) parametrize not a single irreducible representation ofE ̃(3) but

two of them, including both helicities.