356 CHAPTER 6. QUANTUM PHYSICS
In addition, all experiments measuring neutrino velocity had found no violation to the
speed of light.
3.Infinite number of eigenvalues and eigenstates. The neutrino oscillation theory faces
the problem of the existing of infinite number of eigenvalues. In the massive model (6.3.34),
the wave functions are the Dirac spinors
φ= (φ^1 ,φ^2 ,φ^3 ,φ^4 )t.For free neutrinos moving on a straight line,φdepends only onz. Thus the eigenvalue
equations in (6.3.34) become
(6.3.40)
−i ̄hcσ 3d
dz(
φ^3
φ^4)
+mc^2(
φ^1
φ^2)
=λ(
φ^1
φ^2)
,
−i ̄hcσ 3d
dz(
φ^1
φ^2)
−mc^2(
φ^3
φ^4)
=λ(
φ^3
φ^4)
,
where
(6.3.41) σ 3 =
(
1 0
0 − 1
)
.
The equations (6.3.40) possess infinite number of eigenvalues
(6.3.42) λ=
√
m^2 c^4 +4 π^2 n^2 h ̄^2 c^2
l^2, ∀l> 0 ,n= 0 , 1 , 2 ,···,and each eigenvalue has two eigenstates
(6.3.43)
φ 1 =ei^2 πnz/l
√
2 l^3 /^2
√
1 +mc^2 /λ
√^0
1 −mc^2 /λ
0
,
φ 2 =ei^2 πnz/l
√
2 l^3 /^2
√^0
1 +mc^2 /λ
0
−√
1 −mc^2 /λ
.
The problem is that which eigenvalues and eigenstates in (6.3.42) and (6.3.43) are the ones
in the neutrino oscillation model (6.3.34), and why only three of (6.3.42)-(6.3.43) stand for
6.3.1 Discrepancy of the solar neutrinos.
The Weyl equations (6.2.13) can replace the Dirac equations to describe the neutrino
oscillation, which we call massless neutrino oscillation model, expressed as follows
(6.3.44)
νk=φk(x)e−iλkt/h ̄,
ihc ̄(~σ·∇)φk=λkφk fork= 1 , 2 , 3 ,
νe
νμ
ντ
=A
ν 1
ν 2
ν 3
, Ais as in (6.3.26),