56 CHAPTER 2. FUNDAMENTAL PRINCIPLES OF PHYSICS
2.2.6 Relativistic quantum mechanics
Quantum physics is based on several fundamental principles, also called basic postulates of
quantum mechanics, which will be introduced systematically in Chapter 6.
For our purpose, we introduce hereafter two of these basic postulates.
Basic Postulate 2.22.An observable physical quantity can be represented by a Hermitian
linear operator. In particular, the energy E, momentum~P, scalar-valued momentum P are
represented by the operators given by
E=ih ̄∂
∂t
(2.2.45) , P~=−i ̄h∇,
P 0 =i ̄h(~σ·∇) for massless fermions,
P 1 =−ih ̄(~α·∇) for massive fermions,(2.2.46)
whereh is the Plank constant, ̄ ~σ= (σ 1 ,σ 2 ,σ 3 ),~α= (α 1 ,α 2 ,α 3 ),σk( 1 ≤k≤ 3 )are the
Pauli matrices defined by
(2.2.47) σ 1 =
(
0 1
1 0
)
, σ 2 =(
0 −i
i 0)
, σ 3 =(
1 0
0 − 1
)
,
andαk( 1 ≤k≤ 3 )are 4-th order matrices given by
(2.2.48) α 1 =
(
0 σ 1
σ 1 0)
, α 2 =(
0 σ 2
σ 2 0)
, α 3 =(
0 σ 3
σ 3 0)
.
Basic Postulate 2.23.For a quantum system with observable Hermitian operators L 1 ,···,Lm,
if the physical quantities lkcorresponding to Lk( 1 ≤k≤m)satisfy a relation
(2.2.49) H(l 1 ,···,lm) = 0 ,
then the following equation induced by (2.2.49)
(2.2.50) H(L 1 ,···,Lm)ψ= 0
may give a model describing this system provided the operator H(L 1 ,···,Lm)in (2.2.49) is
Hermitian.
Remark 2.24.If the operatorH(L 1 ,···,Lm)in (2.2.49) is irreducible, then (2.2.49) must
describe the system.
Based on Postulates2.22and2.23, we derive three basic equations of relativistic quantum
mechanics: the Klein-Gordon equation describing the bosons, the Weyl equations describing
massless free fermions, and the Dirac equations describingmassive fermions.
1.Klein-Gordon equation. By the Einstein energymomentum relation for energyE,
momentumPand the rest massm:
E^2 −c^2 ~P^2 −m^2 c^4 = 0 ,