6.2. FOUNDATIONS OF QUANTUM PHYSICS 345
Notice that
x 2 ∂ 2 −∂ 2 x 2 =x 3 ∂ 3 −∂ 3 x 3 =− 1.
Hence we get
(6.2.53) HˆLˆ 1 −Lˆ 1 Hˆ=h ̄^2 c(α^3 ∂ 2 −α^2 ∂ 3 ).
Similarly we have
(6.2.54)
HˆLˆ 2 −Lˆ 2 Hˆ= ̄h^2 c(α^1 ∂ 3 −α^3 ∂ 1 ),
HˆLˆ 3 −Lˆ 3 Hˆ= ̄h^2 c(α^2 ∂ 1 −α^1 ∂ 2 ).On the other hand, we infer from (6.2.49) and (6.2.52) that
HˆSˆj−SˆjHˆ=−ih ̄^2 cγ^5[
∂k(σkσj−σjσk)]
=−ih ̄^2 cγ^5 ( 2 iεk jlσl)∂k= 2 h ̄^2 cεk jlαl∂k,whereγ^5 is defined by
γ^5 =iγ^0 γ^1 γ^2 γ^3 =(
0 I
I 0
)
.
Hence we have(6.2.55)
HˆSˆ 1 −Sˆ 1 Hˆ=− 2 ̄h^2 c(α^3 ∂ 2 −α^2 ∂ 3 ),HˆSˆ 2 −Sˆ 2 Hˆ=− (^2) ̄h^2 c(α^1 ∂ 3 −α^3 ∂ 1 ),
HˆSˆ 3 −Sˆ 3 Hˆ=− 2 ̄h^2 c(α^2 ∂ 1 −α^1 ∂ 2 ).
ForJˆ=Lˆ+sSˆ, we derive from (6.2.53)-(6.2.55) that
HˆJˆ−JˆHˆ= 0 ⇐⇒ spins=^1
2
(6.2.56).
When fermions move on a straight line,Hˆ=cα^3 p 3 , Lˆ= 0.In this case, by (6.2.53)-(6.2.54), for straight line motion,
(6.2.57) HˆJˆ−JˆHˆ= 0 for anys.
Thus, by the conservation law (6.2.48), the assertion of Angular Momentum Rule for fermions
follows from (6.2.56) and (6.2.57).
- Bosons. Now, consider bosons which bey the Klein-Gordon equation inthe form
(6.2.43). It is known that the spinsJof bosons depend on the types of Klein-Gordon fields
(Ψ,Φ):
(6.2.58)
(
Ψ
Φ
)
=
a scalar field ⇒ J= 0 ,
a 4-vector field ⇒ J= 1 ,
a 2nd-order tensor field ⇒ J= 2 ,
a real valued field ⇒ neutral bosons,
a complex valued field ⇒ charged bosons.