Mathematical Principles of Theoretical Physics

342 CHAPTER 6. QUANTUM PHYSICS

1) If a particleAconsists ofNmore fundamental particlesAi( 1 ≤i≤N), in a small ball

with radiusr,

A=A 1 +···+AN,

then the momentumpof each particleAiis at least

p≥h ̄/ 4 r.

Hence, the smaller the composite particle, the greater the bounding energy is needed to

hold its constituents together.

2) Most particles are unstable, and their lifetimeτis very short. In addition, the energy

distribution of each particle is in a range, called the energy widthΓ. By the uncertainty

relation,τandΓsatisfy that

τΓ≥h ̄/ 2.

This relation is very important in experiments, because thewidthΓis observable which

determines the lifetime of a particle by the uncertainty relationτ≃ ̄h/ 2 Γ.

3) Uncertainty relations (6.2.38) also imply that the energy and momentum conservations

may be violated in a small scale of time and space. Both conservations are only the

averaged results in larger scale ranges of time and space.

Pauli exclusion principle

We recall that particles are classified two types:

fermions = particles with spinJ=

n

2

for oddn,

bosons = particles with spinJ=nfor integern.

Fermions and bosons display very different characteristics. The fermions do not like to live
together with the same fermions, but bosons are sociable particles. This difference is charac-
terized by the Pauli exclusion principle.

Pauli Exclusion Principle 6.13.In a quantum system, there are no two or more fermions
living in the same quantum states, i.e. possessing entirelythe same quantum numbers.

6.2.4 Angular momentum rule

In Section5.3.2we introduced the Angular Momentum Rule, which was first proved in
(Ma and Wang,2015b). It says that only the fermions with spinJ=^12 can rotate around
a central force field. In fact, this rule can be generalized toscalar bosons, i.e. particles with
spinJ=0. In this subsection we shall discuss the rule in more details. To this end, we first
introduce conservation laws of quantum systems based on Principle6.10.

Conservation laws based on quantum Hamiltonian dynamics (QHD)