Mathematical Principles of Theoretical Physics

332 CHAPTER 6. QUANTUM PHYSICS

In the Planck classical quantum assumption that the energy is discrete for a fixed frequency,

and the frequency is continuous. Our results are different in two aspects. One is that the

energy levels have an upper bound. Two is that the frequencies are also discrete and finite.

Outline of Chapter 6

Section6.2presents the basic postulate and facts from quantum physicsfrom the Hamil-

tonian dynamics and Lagrangian dynamics points of view. We prove also a new angular

momentum rule, which is useful for describing the weaktons constituents of charged leptons

and quarks.

Section6.3presents new alternative approaches for solar neutrino problem, and is based

on the recent work of authors (Ma and Wang,2014f).

Section6.4introduces energy levels of subatomic particles, and is based on (Ma and Wang,

2014g).

The field theory and basic postulates for interacting multi-particle systems are established

in Section6.5, which is based entirely on (Ma and Wang,2014d).

6.2 Foundations of Quantum Physics

6.2.1 Basic postulates

The main components of quantum physics include quantum mechanics and quantum field

theory, which are based on the following basic postulates.

Postulate 6.1.A quantum system consists of some micro-particles, which are described by

a set of complex value functionsψ= (ψ 1 ,···,ψN)T, called wave functions. In other words,

each quantum system is identified by a set of wave functionsψ:

(6.2.1) a quantum system=ψ,

which contain all quantum information of this system.

Postulate 6.2.For a single particle system described by a wave functionψ, its modular

square

|ψ(x,t)|^2

represents the probability density of the particle being observed at point x∈R^3 and at time
t. Hence,ψsatisfies that ∫

R^3

|ψ|^2 dx= 1.

Postulate 6.3.Each observable physical quantity L corresponds to an Hermitian operator

L, and the values of the physical quantity L are given by eigenˆ valuesλofL:ˆ

Lˆψλ=λ ψλ,

and the eigenfunctionψλis the state function in which the physical quantity L takes valueλ.

In particular, the Hermitian operators corresponding to position x, momentum p and energy