Relativistic quantum mechanics - Relativistic quantum mechanics Quantum mechanics 2 - Lecture 11 Igor

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  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Relativistic quantum mechanics Quantum mechanics 2 - Lecture 11

    Igor Lukačević

    UJJS, Dept. of Physics, Osijek

    January 15, 2013

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    1 Klein-Gordon equation

    2 Dirac equation

    3 Free-electron solution of Dirac equation

    4 Electron magnetic moment

    5 Literature

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Contents

    1 Klein-Gordon equation

    2 Dirac equation

    3 Free-electron solution of Dirac equation

    4 Electron magnetic moment

    5 Literature

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Non-relativistic physics

    E = p2

    2m 99K

     E 7→ Ê = i~ ∂ ∂t

    ~p 7→ p̂ = −i~∇

     =⇒ i~∂ψ∂t = − ~22m∆ψ ↓

    free-particle S.E.

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Non-relativistic physics

    E = p2

    2m 99K

     E 7→ Ê = i~ ∂ ∂t

    ~p 7→ p̂ = −i~∇

     =⇒ i~∂ψ∂t = − ~22m∆ψ ↓

    free-particle S.E.

    Statistical interpretation of ψ(r, t):

    ρ(r, t) = |ψ(r, t)|2

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Relativistic physics

    E 2 = m2c4 + c2p2 99K

     E 7→ Ê = i~ ∂ ∂t

    ~p 7→ p̂ = −i~∇

     =⇒

    ( i~ ∂ ∂t

    )2 ψ =

    [ m2c4 + c2 (−i~∇)2

    ] ψ

    ↓ relativistic S.E. (Fock’s equation)

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Relativistic physics

    E 2 = m2c4 + c2p2 99K

     E 7→ Ê = i~ ∂ ∂t

    ~p 7→ p̂ = −i~∇

     =⇒

    ( i~ ∂ ∂t

    )2 ψ =

    [ m2c4 + c2 (−i~∇)2

    ] ψ

    =⇒ ( �− κ2

    ) ψ = 0 Klein-Gordon equation

    � = ∆− 1 c2

    ∂2

    ∂t2

    κ = mc

    ~ ,

    1

    κ =

    ~ mc reduced Compton wavelength [3]

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    A question

    What is Compton wavelength for an electron?

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Statistical interpretation of ψ in Schrödinger theory:

    S.E. ⇒ equation of continuity

    ∂ρ

    ∂t + divj = 0

    ρ = ψ∗ψ probability density

    j = − ~ 2im

    [ψ∗∇ψ − (∇ψ∗)ψ]

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Statistical interpretation of ψ in Klein-Gordon theory:

    K-G equation ⇒ equation of continuity

    ∂ρ

    ∂t + divj = 0

    ρ = ψ∗ψ̇ − ψ̇∗ψ j = −c2 [ψ∗∇ψ − (∇ψ∗)ψ]

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Statistical interpretation of ψ in Klein-Gordon theory:

    K-G equation ⇒ equation of continuity

    ∂ρ

    ∂t + divj = 0

    ρ = ψ∗ψ̇ − ψ̇∗ψ R 0 problem!

    j = −c2 [ψ∗∇ψ − (∇ψ∗)ψ]

    Problem

    ρ depends on the initial conditions: ψ(0) and ψ̇(0)

    ρ cannot be interpreted as the probability density

    Good side

    K-G equation describes well the spinless bosons, like π-mesons.

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Contents

    1 Klein-Gordon equation

    2 Dirac equation

    3 Free-electron solution of Dirac equation

    4 Electron magnetic moment

    5 Literature

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Requirements for the relativistic wave equation

    1 keep the statistical interpretation of ψ

    2 must be relativistically invariant

    3 must be of the 1st order in time variable

    4 agrees with the K-G equation in the limit of large quantum numbers

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Requirements for the relativistic wave equation

    1 keep the statistical interpretation of ψ

    2 must be relativistically invariant

    3 must be of the 1st order in time variable

    4 agrees with the K-G equation in the limit of large quantum numbers

    (2) ⇒ symmetrical in spatial and time derivatives

    (3) in analogy with S.E.

    ⇒ must be linear in spatial derivatives: Ĥ = cα · p + βmc2

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Factorisation of K-G equation gives [5]( Ê − cα · p− βmc2

    )( Ê + cα · p + βmc2

    ) ψ = 0

    Comparisson with K-G equation imposes

    β2 = 1 , αkβ + βαk = 0 , (1) α2x = 1 , αxαy + αyαx = 0 , (2) α2y = 1 , αyαz + αzαy = 0 , (3) α2z = 1 , αzαx + αxαz = 0 (4)

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    A task

    1 Write down conditions (1)-(4) using anticommutators. (Hint: consult Ref. [1].)

    2 Pauli matrices

    σx =

    [ 0 1 1 0

    ] , σy =

    [ 0 −i i 0

    ] , σx =

    [ 1 0 0 −1

    ] ,

    satisfy conditions (2)-(4). Please, verify if they satisfy condition (1).

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Dirac’s matrices

    αi =

    [ 0 σi σi 0

    ] , β =

    [ I 0 0 −I

    ]

    αx =

     0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0

     , αy = 

    0 0 0 −i 0 0 i 0 0 −i 0 0 i 0 0 0

     ,

    αz =

     0 0 1 0 0 0 0 −1 1 0 0 0 0 −1 0 0

     , β = 

    1 0 0 0 0 1 0 0 0 0 −1 0 0 0 0 −1

    

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    A task

    Please, verify if Dirac’s matrices satisfy condition (1).

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Dirac’s equation ( i~ ∂ ∂t − cα · p− βmc2

    ) ψ = 0

    Solution is a four-component column matrix (spinor)

    ψ(r, t) =

     ψ1(r, t) ψ2(r, t) ψ3(r, t) ψ4(r, t)

    

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation Free-electron solution of Dirac equation

    Electron magnetic moment Literature

    Interpretation of ψ in Dirac equation

    Dirac equation ⇒ equation of continuity

    ∂ρ

    ∂t + divj = 0

    ρ = ψ†ψ = |ψ1|2 + |ψ2|2 + |ψ3|2 + |ψ4|2 ≥ 0 probability density j = −cψ†αψ probability density

    current

    requirement (1) is satisfied

    Igor Lukačević Relativistic quantum mechanics

  • Contents Klein-Gordon equation

    Dirac equation