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 $H|l\rangle=(H_0 +\lambda V) |l\rangle =E|l\rangle$ which is written as $0=(E-H_0-\lambda V) |l\rangle$(the formula (5.2.2)). By projecting $P_1$ from the left ($P_1=1-P_0$ and $P_0$ is projection operator onto the degenerate subspace):

$-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V)P_1|l\rangle=0$ (5.2.4)

Then from this, the formula below is obtained:

$P_1|l\rangle =P_1 \frac{\lambda}{E-H_0-\lambda P_1 V P_1}P_1 V P_0|l\rangle$ (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
$P_1|l\rangle =\frac{\lambda P_1 V P_0}{(E-H_0-\lambda P_1 V)}|l\rangle$ 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 $P_1$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]

$-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V)P_1|l\rangle=0$ (5.2.4)

This is equivalent to

$-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V P_1)P_1|l\rangle=0$

because

$P_1^2=P_1$

since it is a projection operator.

 

[hokhani]

$-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V)P_1|l\rangle=0$ (5.2.4)

This is equivalent to

$-\lambda P_1 V P_0|l\rangle +(E-H_0-\lambda P_1 V P_1)P_1|l\rangle=0$

because

$P_1^2=P_1$

since it is a projection operator.

Thank you and dextercioby. It still remains another question. Why don't we regard the relation as $P_1|l\rangle =\frac{\lambda P_1 V P_0}{(E-H_0-\lambda P_1 V)}|l\rangle$? Is it necessary to include the extra $P_1$?

 

[Avodyne]

Why don't we regard the relation as $P_1|l\rangle =\frac{\lambda P_1 V P_0}{(E-H_0-\lambda P_1 V)}|l\rangle$?

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

Is it necessary to include the extra $P_1$?

Strictly speaking, it's not necessary. However, it is helpful, because $P_1 V P_1$ is hermitian, and clearly acts only in the subspace projected by $P_1$.

 

[hokhani]

This is not a valid expression, because $P_1 V P_0$ and $(E-H_0-\lambda P_1 V)^{-1}$ 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$P_1$ in the denominator, then$P_1 V P_0$ and $(E-H_0-\lambda P_1 V)^{-1}$ would commute?

 

[Avodyne]

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