of 43/43
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 Lukaˇ cevi´ c UJJS, Dept. of Physics, Osijek January 15, 2013 Igor Lukaˇ cevi´ c Relativistic quantum mechanics

Relativistic quantum mechanics - UNIOSfizika.unios.hr/.../qm2/Lecture_11_Relativistic_quantum_mechanics.pdf · Relativistic quantum mechanics Quantum mechanics 2 - Lecture 11 Igor

  • View
    54

  • Download
    14

Embed Size (px)

Text of Relativistic quantum mechanics -...

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Relativistic quantum mechanicsQuantum mechanics 2 - Lecture 11

    Igor Lukačević

    UJJS, Dept. of Physics, Osijek

    January 15, 2013

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Non-relativistic physics

    E =p2

    2m99K

    E 7→ Ê = i~∂∂t

    ~p 7→ p̂ = −i~∇

    =⇒ i~∂ψ∂t = − ~22m∆ψ↓

    free-particle S.E.

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Non-relativistic physics

    E =p2

    2m99K

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Relativistic physics

    E 2 = m2c4 + c2p2 99K

    E 7→ Ê = i~∂∂t

    ~p 7→ p̂ = −i~∇

    =⇒

    (i~ ∂∂t

    )2ψ =

    [m2c4 + c2 (−i~∇)2

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

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Relativistic physics

    E 2 = m2c4 + c2p2 99K

    E 7→ Ê = i~∂∂t

    ~p 7→ p̂ = −i~∇

    =⇒

    (i~ ∂∂t

    )2ψ =

    [m2c4 + c2 (−i~∇)2

    =⇒(�− κ2

    )ψ = 0 Klein-Gordon equation

    � = ∆− 1c2

    ∂2

    ∂t2

    κ =mc

    ~,

    1

    κ=

    ~mc reduced Compton wavelength [3]

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    A question

    What is Compton wavelength for an electron?

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Statistical interpretation of ψ in Klein-Gordon theory:

    K-G equation ⇒ equation of continuity

    ∂ρ

    ∂t+ divj = 0

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

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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 spatialand time derivatives

    (3) in analogy with S.E.

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

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    A task

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

    2 Pauli matrices

    σx =

    [0 11 0

    ], σy =

    [0 −ii 0

    ], σx =

    [1 00 −1

    ],

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

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Dirac’s matrices

    αi =

    [0 σiσi 0

    ], β =

    [I 00 −I

    ]

    αx =

    0 0 0 10 0 1 00 1 0 01 0 0 0

    , αy =

    0 0 0 −i0 0 i 00 −i 0 0i 0 0 0

    ,

    αz =

    0 0 1 00 0 0 −11 0 0 00 −1 0 0

    , β =

    1 0 0 00 1 0 00 0 −1 00 0 0 −1

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    A task

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

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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 densityj = −cψ†αψ probability density

    current

    requirement (1) is satisfied

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Dirac equation (E − cα · p− βmc2

    )ψ = 0

    Suppose a plane wave solution

    ψ(r, t) = uei~ (p·r−Et) , E =

    p2

    2m

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Dirac equation (E − cα · p− βmc2

    )ψ = 0

    Suppose a plane wave solution

    ψ(r, t) = uei~ (p·r−Et) , E =

    p2

    2m

    =⇒(Eu − cα · pu − βmc2u

    )= 0

    E −mc2 0 0 00 E −mc2 0 00 0 E +mc2 00 0 0 E +mc2

    u1u2u3

    u4

    −c

    0 0 pz px − ipy0 0 px + ipy −pzpz px − ipy 0 0px + ipy −pz 0 0

    u1u2u3

    u4

    = 0Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    ∣∣∣∣∣∣∣∣E −mc2 0 −c(pz) −c(px − ipy )

    0 E −mc2 −c(px + ipy ) c(pz)−c(pz) −c(px − ipy ) E + mc2 0

    −c(px + ipy ) c(pz) 0 E + mc2

    ∣∣∣∣∣∣∣∣ = 0

    E 2 = c2p2 + m2c4

    E = ±√

    c2p2 + m2c4

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    1 E+ = +√

    c2p2 + m2c4

    u(+)↑ = N

    10cpz

    E+ + mc2c(px + ipy )

    E+ + mc2

    , u(+)↓ = N

    01

    c(px − ipy )E+ + mc2−cpzE+ + mc2

    2 E− = −

    √c2p2 + m2c4

    u(−)↑ = N

    cpz

    E− −mc2c(px + ipy )

    E− −mc210

    , u(−)↓ = N

    c(px − ipy )E− −mc2−cpz

    E− −mc201

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    HW

    Calculate the normalization constant N. (Solution can be found, for example, inRef. [1])

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Why are there ↑ and ↓ arrows in the subscripts?

    u(+) relate to nonrelativistic limit

    cpzE+ + mc2

    c(px + ipy )

    E+ + mc2

    ∼v

    cv�c−−−→ 0

    u(+)↑ (r, t) ∼

    [10

    ]e

    i~ (p·r−Et) ,

    u(+)↓ (r, t) ∼

    [01

    ]e

    i~ (p·r−Et)

    free spin 1/2 particles

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Interpretation of E+ and E−

    Positron - experimentaldiscovery:

    D. Skobelstyn (1929).

    C.-Y. Chao (1929).

    C. D. Anderson (1932). -Nobel Prize (1936).

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Consider an electron in an emg field:

    H = cα ·(

    p− qc

    A)

    + βmc2 + qφ(r)

    D.E. ⇒ [cα ·

    (p− q

    cA)

    + βmc2 + qφ(r)]ψ = Eψ

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Consider an electron in an emg field:

    H = cα ·(

    p− qc

    A)

    + βmc2 + qφ(r)

    D.E. ⇒ [cα ·

    (p− q

    cA)

    + βmc2 + qφ(r)]ψ = Eψ

    α symmetry ⇒[c(

    p− qc

    A)· σ]

    +(mc2 + qφ(r)

    )W = EW[

    c(

    p− qc

    A)· σ]−(mc2 − qφ(r)

    )V = EV

    where

    W =

    [ψ(1)

    ψ(2)

    ], V =

    [ψ(3)

    ψ(4)

    ]

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    V =

    [c (p− (q/c)A) · σE − qφ+ mc2

    ]W

    c2[(

    p− qc

    A)· σ(E − qφ+ mc2

    )−1(p− (q/c)A) · σ

    ]W

    +(mc2 + qφ(r)

    )W = EW

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    V =

    [c (p− (q/c)A) · σE − qφ+ mc2

    ]W

    c2[(

    p− qc

    A)· σ(E − qφ+ mc2

    )−1(p− (q/c)A) · σ

    ]W

    +(mc2 + qφ(r)

    )W = EW

    v/c → 0⇒

    ‖V ‖‖W ‖ → 0

    E ′ = E −mc2 , ‖qφ‖ � mc2

    ⇒(E ′ − qφ+ 2mc2

    )−1=

    1

    2mc2

    (1− E

    ′ − qφ2mc2

    + · · ·)

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    D.E. in nonrelativistic limit[1

    2m

    (p− q

    cA)2− q

    mcS · B + qφ

    ]W = E ′W

    where

    S =~2σ

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    D.E. in nonrelativistic limit[1

    2m

    (p− q

    cA)2− q

    mcS · B︸ ︷︷ ︸

    µ · B

    +qφ

    ]W = E ′W

    where

    S =~2σ

    ⇒ µ = qmc

    S magnetic moment of an electron

    Igor Lukačević Relativistic quantum mechanics

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    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

  • ContentsKlein-Gordon equation

    Dirac equationFree-electron solution of Dirac equation

    Electron magnetic momentLiterature

    Literature

    1 R. L. Liboff, Introductory Quantum Mechanics, Addison Wesley, SanFrancisco, 2003.

    2 I. Supek, Teorijska fizika i struktura materije, II. dio, Školska knjiga,Zagreb, 1989.

    3 Compton wavelength

    4 P.A.M. Dirac - life & interesting facts

    5 P. A. M. Dirac, ”The Quantum Theory of the Electron”, Proceedings ofthe Royal Society A: Mathematical, Physical and Engineering Sciences 117(778): 610 (1928).

    6 C. D. Anderson - Nobel lecture about the dicovery of positron

    Igor Lukačević Relativistic quantum mechanics

    http://math.ucr.edu/home/baez/lengths.htmlhttp://www.dirac.ch/PaulDirac.htmlhttp://www.nobelprize.org/nobel_prizes/physics/laureates/1936/anderson-lecture.html

    ContentsKlein-Gordon equationDirac equationFree-electron solution of Dirac equationElectron magnetic momentLiterature