Degenerate perturbation theory (Sakurai's textbook)

[hokhani]

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?

 

[dextercioby]

Multiply (5.2.4) by (E−H0−λP1V)^-1, provided it exists.

 

[hokhani]

Multiply (5.2.4) by (E−H0−λP1V)^-1, provided it exists.

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?

 

[dextercioby]

Put now P_1 on both sides to the left and use that this is a projector (idempotent).

 

[hokhani]

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).

 

[Avodyne]

[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.

 

[hokhani]

[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.

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]?

 

[Avodyne]

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]?

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.

Is it necessary to include the extra [itex]P_1[/itex]?

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].

 

[hokhani]

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.

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?

 

[Avodyne]

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.