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Degenerate perturbation theory (Sakurai's textbook)

  1. Jan 14, 2014 #1
    In the theory of degenerate perturbation in Sakurai’s textbook, Modern Quantum Mechanics Chapter 5, the perturbed Hamiltonian is [itex] H|l\rangle=(H_0 +\lambda V) |l\rangle =E|l\rangle [/itex] which is written as [itex]0=(E-H_0-\lambda V) |l\rangle [/itex](the formula (5.2.2)). By projecting [itex]P_1[/itex] from the left ([itex]P_1=1-P_0[/itex] and [itex]P_0[/itex] is projection operator onto the degenerate subspace):

    [itex]-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V)P_1|l\rangle=0 [/itex] (5.2.4)

    Then from this, the formula below is obtained:

    [itex]P_1|l\rangle =P_1 \frac{\lambda}{E-H_0-\lambda P_1 V P_1}P_1 V P_0|l\rangle [/itex] (5.2.5)

    But I never can reach to (5.2.5) from (5.2.4). Could anyone please help me?
     
  2. jcsd
  3. Jan 14, 2014 #2

    dextercioby

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    Multiply (5.2.4) by (E−H0−λP1V)-1, provided it exists.
     
  4. Jan 14, 2014 #3
    Thanks, But it gives
    [itex]P_1|l\rangle =\frac{\lambda P_1 V P_0}{(E-H_0-\lambda P_1 V)}|l\rangle[/itex] which is not the same as (5.2.5). Could you please guide me completely?
     
  5. Jan 14, 2014 #4

    dextercioby

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    Put now P_1 on both sides to the left and use that this is a projector (idempotent).
     
  6. Jan 15, 2014 #5
    Right, Thanks. But all my problem is with the extra [itex]P_1[/itex]in the denominator of (5.2.5). Where does it come from? In my idea, it seems to be a mistyped mistake. Also I think the formula (5.2.15) is mistyped because the sum hasn't to be over the degenerate space! However I am not confident about my idea (I have also seen exactly those formula in the new version of the book, 2011).
     
    Last edited: Jan 15, 2014
  7. Jan 16, 2014 #6

    Avodyne

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    [itex]-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V)P_1|l\rangle=0 [/itex] (5.2.4)

    This is equivalent to

    [itex]-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V P_1)P_1|l\rangle=0 [/itex]

    because

    [itex]P_1^2=P_1[/itex]

    since it is a projection operator.
     
  8. Jan 17, 2014 #7
    Thank you and dextercioby. It still remains another question. Why don't we regard the relation as [itex]P_1|l\rangle =\frac{\lambda P_1 V P_0}{(E-H_0-\lambda P_1 V)}|l\rangle[/itex]? Is it necessary to include the extra [itex]P_1[/itex]?
     
  9. Jan 17, 2014 #8

    Avodyne

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    This is not a valid expression, because [itex]P_1 V P_0[/itex] and [itex](E-H_0-\lambda P_1 V)^{-1}[/itex] do not commute. They must be written in a definite order.

    Strictly speaking, it's not necessary. However, it is helpful, because [itex]P_1 V P_1[/itex] is hermitian, and clearly acts only in the subspace projected by [itex]P_1[/itex].
     
  10. Jan 19, 2014 #9
    Excuse me. I don't understand your above sentence. Do you mean that if we use extra[itex]P_1[/itex] in the denominator, then[itex]P_1 V P_0[/itex] and [itex](E-H_0-\lambda P_1 V)^{-1}[/itex] would commute?
     
  11. Jan 20, 2014 #10

    Avodyne

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    No, they don't commute whether or not you include the extra [itex]P_1[/itex], so they must be written in a particular order.
     
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