6.4. ENERGY LEVELS OF SUBATOMIC PARTICLES 373
Each energy levelEkcan be expressed as
(6.4.60) Ek=E 0 +λk ( 1 ≤k≤N),
whereE 0 is the intrinsic energy of mediators.
It follows from (6.4.59) and (6.4.60) that the frequencies of mediators are finite and dis-
crete
(6.4.61) ωk=
1
h ̄Ek ( 1 ≤k≤N),and the difference of two adjacent frequencies is
(6.4.62) ∆ωk=ωk+ 1 −ωk=
1
h ̄(λk+ 1 −λk).Remark 6.20.In the classical quantum mechanics, the frequencies of particles are continu-
ously distributed. Here we derive from the energy spectrum theory that the frequencies are
finite and discrete. In fact, by the estimate (3.6.51) of the number of negative eigenvalues, we
can verify that the frequency difference (6.4.62) is too small to measure in the next subsection.
6.4.6 Discreteness of energy spectrum
Based on the spectral theory developed in Section3.6, the energy levels of all subatomic
particles are finite and discrete:
(6.4.63) 0 <E 1 <···<EN,
where the numberNof energy levels depends on the particle type. Each subatomic particle
lies in an energy state ofEk( 1 ≤k≤N)in (6.4.63). In traditional conception, the energy
distribution of all particles is infinite and continuous, i.e. a particle can lie in any state of
energyEwith 0<E<∞. Hence the energy level theory established here arrive at a very
different conclution:
Physical Conclusion 6.21.The energy distribution of subatomic particles is finite anddis-
crete.
Usually, we cannot observe the discreteness of energy because the numberNof energy
levels is so large that the average difference of adjacent energy level:
∆Ek=Ek+ 1 −Ek≃EN−E 1
N
≃ 0 ,
is very small. In the following discussion we shall show thispoint.
To this end, we first give the estimates of the numberNof energy levels for various types
of subatomic particles.
1.Energy level number of electrons and quarks.To consider the approximatively com-
putation of numberNof energy levels, we always ignore the magnetic effects. In this case,