Presentation Softic Klein Gordon

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Relativistic wave equation for spin − 0 particles Klein-Gordon equation

Hajreta Softi c´

Institute for Theoretical Physics

15.5.2008

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

1

Klein-Gordon equation Derivation Free solution Interpretation

2

Non-relativistic limit

3

Summery

4

Literature

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Schr¨odinger equation E=

~p 2 + V (x) 2m0

∂ Eˆ = i~ ∂t ~ pˆ = −i~∇   ∂ψ(x, t) ~2 ~ 2 → i~ = − ∇ + V (x) ψ(x, t) ∂t 2m0

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Relativistic energy-momentum relation E2 − ~p · ~p = m02 c 2 c2 c . . . the velocity of light in vacuum p µ pµ =

m0 . . . the rest mass of the particles

 2  1 ∂2 ∂ ∂2 ∂2 − + + c 2 ∂t 2 ∂x 2 ∂y 2 ∂z 2   1 ∂2 2 2 ≡ −~  = −~ −∆ c 2 ∂t 2

∂ ∂ pˆµ pˆµ = −~ = −~2 ∂xµ ∂x µ 2



...the four-dimensional d’Alembertian Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Klein-Gordon equation → pˆµ pˆµ ψ = m02 c 2 ψ     m02 c 2 1 ∂2 ∂2 ∂2 ∂2 m02 c 2 + 2 ψ= − − − + 2 ψ=0 ~ c 2 ∂t 2 ∂x 2 ∂y 2 ∂z 2 ~ Differential equation of second order in space and time 



 pˆµ pˆµ + m0 2 c 2  | {z } | {z } invariant

invariant

ψ |{z}

=0

wavefunction

→ covariant equation

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Free solution     i i µ ψ = exp − pµ x = exp + (~p · ~x − Et) ~ ~ µ

pˆµ pˆ ψ =

m02 c 2 ψ



i → pµ p exp − pµ x µ ~ µ

 =

m02 c 2 exp

  i µ − pµ x ~

Energy eigenvalue E2 − ~p · ~p = m02 c 2 c2 q E = ± m02 c 2 + ~p 2

→

→ Solutions with positive and negative energies Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Continuity equation

pˆµ pˆµ − m02 c 2 ψ = 0 ⇒ pˆµ pˆµ − m02 c 2 ψ ∗ = 0 {z } | {z } | ·(ψ ∗ )



·(ψ)





 −ψ ∗ ~2 ∇~µ ∇~µ + m02 c 2 ψ + ψ ~2 ∇~µ ∇~µ + m02 c 2 ψ ∗ = 0 Separation:

   i~ ∂ψ ∗ ∂ ∗ ∂ψ ψ − ψ ∂t 2m0 c 2 ∂t ∂t  h i i~ ~ ~ ∗) = 0 +div − ψ ∗ (∇ψ) − ψ(∇ψ 2m0

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Continuity equation ∂ρ + div~j = 0 ∂t  i~  ∗ ~ ~ µψ∗ ψ ∇µ ψ − ψ ∇ jµ = 2m0   i~ ∂ψ ∗ ∗ ∂ψ ρ= ψ −ψ 2m0 c 2 ∂t ∂t ρ . . . density j . . . current density

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Derivation Free solution Interpretation

Remarks: ρ can get smaller than zero → Can not be intepreted as a probality density There exist solutions for negative energy → Are connected with the existence of antiparticles

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Non-relativistic limit of the Klein-Gordon equation Ekin = E − m0 c 2  m0 c 2 ∂ϕ i~ ≈ Ekin ϕ  m0 c 2 ϕ ∂t The ansatz: 

i ψ(r , t) = ϕ(r , t) exp − m0 c 2 t ~

Hajreta

Softi´ c



Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Calculations ∂ψ = ∂t

   ∂ϕ m0 c 2 i 2 −i ϕ exp − m0 c t ∂t ~ ~   m0 c 2 i 2 ≈ −i ϕ exp − m0 c t ~ ~ 

    ∂2ψ ∂ ∂ϕ m0 c 2 i 2 = −i ϕ exp − m0 c t ∂t 2 ∂t ∂t ~ ~     2m0 c 2 ∂ϕ m0 2 c 4 i 2 =− i + ϕ exp − m0 c t ~ ∂t ~2 ~

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

    1 2m0 c 2 ∂ϕ m0 2 c 4 i 2 + m c t = i ϕ exp − 0 c2 ~ ∂t ~2 ~  2    ∂ ∂2 ∂2 m0 2 c 2 i 2 + + − ϕ exp − m0 c t ∂x 2 ∂y 2 ∂z 2 ~2 ~ −

Schr¨odinger equation for free particles  2  ∂ϕ ~2 ∂ ∂2 ∂2 ~2 =− + + ϕ = − ∆ϕ. i~ ∂t 2m0 ∂x 2 ∂y 2 ∂z 2 2m0

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Summery Relativistic energy momentum relation → Klein-Gorden equation Solutions with positive and negative energies Density not positive definite Non-relativistic limit → Schr¨ odinger equation

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles

Outline Klein-Gordon equation Non-relativistic limit Summery Literature

Literature Walter Greiner: Relativistic Quantum Mechanics - Wave Equations

Hajreta

Softi´ c

Relativistic wave equation for spin − 0 particles