Mathematical Principles of Theoretical Physics

344 CHAPTER 6. QUANTUM PHYSICS

The formulas (6.2.46)-(6.2.48) are the conservation laws of the classical quantum mechanics.
Hence, the conservation laws in (6.2.45) are the generalization to the classical quantum
mechanics, and are applicable to all conservative quantum systems, including the Klein-
Gordon systems and nonlinear systems.

Angular momentum rule

From the conservation laws (6.2.45) and (6.2.48) we can deduce the following Angular
Momentum Rule.

Angular Momentum Rule 6.14.Only the fermions with spin J=^12 and the bosons with
J= 0 can rotate around a center with zero moment of force. The particles with J 6 = 0 ,^12 will
move on a straight line unless there is a nonzero moment of force present.

In the following we give a mathematical derivation of the Angular Momentum Rule.
1.Fermions. Consider fermions which obey the Dirac equations as (6.2.46) with the
Hamiltonian

(6.2.49) Hˆ=−ihc ̄(αk∂k)+mc^2 α^0 +V(r),

whereVis the potential energy of a central field, andα^0 ,αk( 1 ≤k≤ 3 )are the Dirac matrices

(6.2.50) α^0 =

(

I 0

0 −I

)

, αk=

(

0 σk

σk 0

)

for 1≤k≤ 3 ,

andσkare the Pauli matrices.
The total angular momentumJˆof a particle is

Jˆ=Lˆ+sSˆ,

wheresis the spin,Lˆis the orbital angular momentum

(6.2.51)

Lˆ= (Lˆ 1 ,Lˆ 2 ,Lˆ 3 ) =rˆ×pˆ, pˆ=−i ̄h∇,

Lˆ 1 =−ih ̄(x 2 ∂ 3 −x 3 ∂ 2 ),

Lˆ 2 =−ih ̄(x 3 ∂ 1 −x 1 ∂ 3 ),

Lˆ 3 =−ih ̄(x 1 ∂ 2 −x 2 ∂ 1 ),

andSˆis the spin operator

(6.2.52) Sˆ= (Sˆ 1 ,Sˆ 2 ,Sˆ 3 ), Sˆk=h ̄

(

σk 0

0 σk

)

for 1≤k≤ 3.

By (6.2.49)-(6.2.52), we see that

HˆLˆ 1 −Lˆ 1 Hˆ=h ̄^2 c[(x 2 ∂ 3 −x 3 ∂ 2 )(α^2 ∂ 2 +α^3 ∂ 3 )−(α^2 ∂ 2 +α^3 ∂ 3 )(x 2 ∂ 3 −x 3 ∂ 2 )]

=h ̄^2 c[α^2 ∂ 3 (x 2 ∂ 2 −∂ 2 x 2 )−α^3 ∂ 2 (x 3 ∂ 3 −∂ 3 x 3 )].