§3.4 Dirac equation in the central force field
§3.4 Dirac equation in the central force field
§3.4 Dirac equation in the central force field
Noting: up to the order Normalization condition must be ensured
Thomas term
Darwin term
§3.4 Dirac equation in the central force field
§3.4 Dirac equation in the central force field
Quantum number K
§3.4 Dirac equation in the central force field
r r cσ ⋅ p χ= ϕ 2 2mc + E '− V
§3.4 Dirac equation in the central force field
In non-relativistic approximation
E '− V 2mc 2
r r cσ ⋅ p x= ϕ 2 2mc + E '− V
§3.1 Klein – Gordon equation
Lorentz transormation the same weight K – G equation time, space are of
§3.1 Klein – Gordon equation
§3.1 Klein – Gordon equation
§3.4 Dirac equation in the central force field
1 E '− V r r χ≈ 1 − 2mc 2 σ ⋅ pϕ 2mc
1 r r r r E' r r r r (σ ⋅ p )(σ ⋅ p )ϕ − (σ ⋅ p )(σ ⋅ p )ϕ 2 2 2m 4m c 1 r r r r + (σ ⋅ p )V (σ ⋅ p )ϕ + V ϕ = E 'ϕ 2 2 4m c
§3.4 Dirac equation in the central force field
By using
§3.4 Dirac equation in the central force field
Relativistic correction of kinetic energy
§3.4 Dirac equation in the central force field
ih ∂ψ ' imc 2 ∂ψ ' imc 2 ρ= [ψ '⋅ ( ψ ') −ψ '* ⋅ ( ψ ')] − + 2 2mc ∂t h ∂t h ≈ ψ '* ψ ' = ψ *ψ
§3.1 Klein – Gordon equation
With electromagnetic field
§3.1 Klein – Gordon equation
Chapter 3 Relativistic Quantum Mechanics
Introduction
Non-relativistic quantum mechanics relativistic quantum mechanics Schrödinger equation Klein-Gordon equation S ~ integer Dirac equation S ~ half integer Spin is automatically contained in Dirac equation
p2 Eϕ s = (1 + 2 2 ) × 8m c p2 E ' p2 1 p2 r r r r V + 2m − 4m 2 c 2 + 4m 2 c 2 (σ ⋅ p )V (σ ⋅ p ) (1 − 8m 2 c 2 )ϕ s
§3.4 Dirac equation in the central force field
§3.3 solutions of the free particle
Dirac hole theory Dirac sea Hole: (+Ep>0, +m>0, +e>0) (positron) 1932, Anderson discovered positron from cosmic ray using cloud chamber
{
}
r r r dL = c(α × p) dt
§3.3 solutions of the free particle
Spin angular momentum
r σ 0 Σ= r 0 σ r
σ i 0 Σi = (i = x, y, z ) 0 σi
Or
§3.3 solutions of the free particle
§3.1 Klein – Gordon equation
§3.1 Klein – Gordon equation
Non-relativistic limit: K-G eq Sch eq
§3.1 Klein – Gordon equation
§3.1 Klein – Gordon equation
§3.1 Klein – Gordon equation
Covariant form
§3.1 Klein – Gordon equation
§3.1 Klein – Gordon equation
§3.1 Klein – Gordon equation
§3.2 Dirac equation
§3.3 solutions of the free particle
r If p = {0, 0, p} ,we find
Eigenvalues:
±h / 2
§3.3 solutions of the free particle
Eigenstates:
§3.3 solutions of the free particle
How to overcome the negative probability difficulty
§3.2 Dirac equation
§3.2 Dirac equation
§3.2 Dirac equation
§3.2 Dirac equation
The condition for α and β
§3.2 Dirac equation
3 ∂ ∂ ih (ψψ ) = −ihc ∑ k (ψα kψ ) ∂t i =1 ∂x
ρ = ψ †ψ ,
j k = cψ †α kψ
§3.3 soபைடு நூலகம்utions of the free particle
§3.3 solutions of the free particle
§3.4 Dirac equation in the central force field
Equation in non-relativistic limit
§3.4 Dirac equation in the central force field
§3.4 Dirac equation in the central force field
r r h [ Σ, H ] = −ihc(α × p ) 2
§3.3 solutions of the free particle
§3.3 solutions of the free particle
Helicity operator
§3.3 solutions of the free particle
1) They must follow the relation
E =c p +m c
2 2 2
2 4
2) Operator H must be Hermitian 3) Lorentz invariance
§3.2 Dirac equation
§3.2 Dirac equation
§3.2 Dirac equation
dLx 1 = [ Lx , H ] dt ih c = Lx , α x px + α y p y + α z pz ih c = α x [ Lx , px ] + α x Ly , p y + α z [ Lx , pz ] ih r r = c α y pz − α z p y = c(α × p ) x ≠ 0
§3.3 solutions of the free particle
§3.3 solutions of the free particle
§3.3 solutions of the free particle
§3.3 solutions of the free particle
§3.3 solutions of the free particle
§3.4 Dirac equation in the central force field
§3.4 Dirac equation in the central force field
h2 p2 ϕ s = (1 − 2 2 ∇ 2 )ϕ = (1 + 2 2 )ϕ 8m c 8m c
§3.4 Dirac equation in the central force field
§3.2 Dirac equation