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Quantum Mechanics Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

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  • Quantum Mechanics 9 Time Independent Perturbations 9-1

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    9 Time Independent Perturbations 9.1 Introduction 9.2 Non-Degenerate Perturbation Theory 9.3 Applications 9.4 Degenerate Perturbation Theory 9.5 Application - Relativistic Correction 9.6 Application - Zeeman Effect 9.7 9.8

    9.1 Introduction eigen energy eigenstate rotation nuclear spin, finite potential well, harmonic potential, central potential 3 Hamiltonian eigen equation

    n n nH E = eigen energy { }nE eigenstate { }n

    Quantum Well

  • Quantum Mechanics 9 Time Independent Perturbations 9-2

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    quantum well finite square well 1 model probability amplitude quantum well EK

    GaAlAs GaAs

    x

    2a

    quantum wellEK

    model

    x

    2a

    ( )V x

    GaAlAs GaAs

    x

    2a

    quantum wellEK

    model

    x

    2a

    ( )V x

    electrostatic q e=

    ( )electric ( ) ( )V x q x e x a= = EK ( )x electrical potential = EK K square well

    0

    well

    0

    2( ) 0 2 0

    0

    V x aV x a x

    V x

    < = <

  • Quantum Mechanics 9 Time Independent Perturbations 9-3

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    hydrogen atom ground state EK z

    Coulomb potential 2 2

    00

    2 4p e ZHm r=

    EK

    hydrogen atom

    z

    spectrum hydrogen atom

    EK

    hydrogen atom

    z

    spectrum hydrogen atom

    (9.1) hydrogen atom 1 H e z= E

    K Hamiltonian

    2 20 1

    0

    2 4p e ZH H H e zm r

    = + = EK

    Hamiltonian eigen energy eigenstate spectrum Hamiltonian H eigen energy eigenstate perturbation theory

  • Quantum Mechanics 9 Time Independent Perturbations 9-4

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    Interaction 2 () 1H "perturbations" ( )

    r

    En

    0( )V r

    1E

    2E3E

    atom

    fn

    in

    0 2 coseH H t= K K

    r

    En

    0( )V r

    1E

    2E3E

    atom

    fn

    in

    0 2 coseH H t= K K perturbation 1H 1 1 ( )H H t=

    f iE E E = perturbation eigenstate time-dependent perturbation theory time dependent perturbation theory

    Formal Notations perturbation theory

  • Quantum Mechanics 9 Time Independent Perturbations 9-5

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    Hamiltonian

    0 1 H H H= + ______________________ (9.1) eigenstate { }n eigen energy { }nE

    n n nH E = ______________________ (9.2) Hamiltonian H 2 1) 0H Hamiltonian eigenstate eigen energy { }(0)n { }(0)nE

    (0) (0) (0)0 n n nH E = ______________________ (9.3)

    superscript (0) Hamiltonian 0H (9.2) 2) 1H Hamiltonian "perturbations" Hamiltonian 0 1 H H H= + 1H 0H 1H perturbing Hamiltonian 0H unperturbed Hamiltonian

    perturbations theory eigenstate { }n eigen energy { }nE Hamiltonian 0 1 H H H= + { }(0)n

    { }(0)nE 1H 0H

  • Quantum Mechanics 9 Time Independent Perturbations 9-6

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    9.2 Non-Degenerate Perturbation Theory perturbing Hamiltonian 1H "time independent perturbation" Erwin Schrdinger Lord Rayleigh derive eigenstate { }n eigen energy { }nE Hamiltonian (9.1)

    0 1 H H H= + ______________________ (9.4) 0 1 perturbing Hamiltonian 1)

    1H perturbation Hamiltonian 0H 2) perturbation theory eigen energy nE eigenstate n summation

    (0) (1) 2 (2) 3 (3)n n n n nE E E E E = + + + +" ___________ (9.5)

    0 1 (0) (1) 2 (2)n n nE E E < < "

    (0)nE eigen energy unperturbed Hamiltonian 0H (1)nE 1st order correction term (0)nE nE (2)nE 2nd order correction term

  • Quantum Mechanics 9 Time Independent Perturbations 9-7

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    nE (0)nE (1) (2), ,n nE E "

    1st order 2nd order Perturbations eigen energy nE (9.5) eigenstate n

    (0) (1) 2 (2) 3 (3)n n n n n = + + + +" ___________ (9.6)

    Hamiltonian H (9.4) , eigen energy (9.5) , eigenstate (9.6) (9.2)

    ( )( )( )( )

    (0) (1) 2 (2)0 1

    (0) (1) 2 (2) (0) (1) 2 (2)

    n n n

    n n n n n n

    H H

    E E E

    + + + += + + + + + +

    "

    " "

    ( )( )( )( )

    (0) (1) 2 (2)0 1

    (0) (1) 2 (2) (0) (1) 2 (2)

    0 n n n

    n n n n n n

    H H

    E E E

    = + + + + + + + + + +

    "

    " "

    ___________________ (9.7) 0 :

    { }(0) (0) (0) 00 n n nH E 1 :

  • Quantum Mechanics 9 Time Independent Perturbations 9-8

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    { }(1) (0) (0) (1) (1) (0) 10 1 n n n n n nH H E E + 2 :

    { }(2) (1) (0) (2) (1) (1) (2) (0) 20 1 n n n n n n n nH H E E E + (9.7)

    (0) (0) (0)0 n n nH E = _________ (9.8)

    (1) (0) (0) (1) (1) (0)0 1 n n n n n nH H E E + = + _________ (9.9)

    (2) (1) (0) (2) (1) (1) (2) (0)0 1 n n n n n n n nH H E E E + = + + ________ (9.10)

    1st order energy correction term (1)nE bra (0)n (9.9)

    ( )(0) (1) (0) (0) (0) (0) (1) (0) (1) (0)

    0 1

    (0) (1) (0) (0) (0) (0) (1) (1) (0) (0)10

    1

    n n n n n n n n n n

    n n n n n n n n n n

    H H E E

    H H E E

    =

    + = ++ = +

    Hamiltonian 0H bra (0)n adjoint operator 0H Hamiltonian Hermitian operator 00 H H= (0) (0) (0) (0)00 n n n nH H E = =

    (0) (0) (1)n n nE (0) (0) (0) (0) (1)1n n n n nH E + = (1) (0) (0)

    1

    n n nE =

    +

  • Quantum Mechanics 9 Time Independent Perturbations 9-9

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    (1) (0) (0)1n n nE H = ______________ (9.11)

    1st order correction eigen energy bra-ket operator 1H 1st order correction eigenstate (1)n eigen state { }(0)n complete linear superposition (1)n

    (0)(1)n k k

    k nc

    = (0) (1)k nkc =

    kc expansion summation k k n (9.6) (0)k n = linear superposition kc bra (0)k (9.9)

    (0) (0) (0) (0)(1) (0) (0) (1) (1) (0)0 1

    (0) (0) (0) (0) (0)(1) (0) (0) (1) (1) (0)1

    0

    n n n n n nk k k k

    n n n n n nk k k k k

    H H E E

    E H E E

    =

    + = ++ = +

    (0) (0)

    1(0) (1)(0)(0)

    nknk

    n k

    H

    E E

    = k n

  • Quantum Mechanics 9 Time Independent Perturbations 9-10

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    (0) (0)1 (0)(1)(0)(0)

    nkn k

    k n n k

    H

    E E

    = ______________ (9.12)

    perturbation derive 2nd order energy correction bra (0)n (9.10)

    (0) (2)0n nH (0) (1)1

    (0) (0) (2)

    n n

    n n n

    H

    E

    +

    = (0) (1) (1) (0) (2) (0)0

    n n n n n nE E =

    + +

    (2) (0) (1)

    1n n nE H = (1)n (9.12)

    (0) (0)1 (0)(2) (0)

    1(0)(0)

    (0) (0)(0) (0)1 1

    (0)(0)

    (0) (0)(0) (0)1 1(2)

    (0)(0)

    nkn n k

    k n n k

    n nk k

    k n n k

    n nk kn

    k n n k

    HE H

    E E

    H H

    E E

    H HE

    E E

    =

    =

    =

    2(0) (0)

    1(2)(0)(0)

    nkn

    k n n k

    HE

    E E

    = ______________ (9.13) (9.4) derive correction term

  • Quantum Mechanics 9 Time Independent Perturbations 9-11

    Dr. Teepanis Chachiyo teepanis@kku.ac.th Draft Oct 2009

    (9.11) , (9.12) , (9.13) perturbation theory 1 =

    (0) (1) (2) (3)n n n n nE E E E E= + + + +" ______________ (9.14)

    (0) (1) (2) (3)

    n n n n n = + + + +" ___________ (9.15)

    Simple Application perturbation theory eigen energy Hamiltonian

    22 2 2 1

    2 2pH m x bxm

    = + + ________________ (9.16) Hamiltonian eigen energy perturbation theory Hamiltonian H (9.16) eigen energy H

    22 2 2

    0 1

    0

    1 2 2

    H

    pH m x bx H Hm

    = + + = +

    _____________

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