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I am studying Perturbation Theory from J.J. Sakurai's textbook Modern Quantum Mechanics. I am having trouble proving formulas on page 299 (5.2.5) and (5.2.6) from the previous ones [mainly (5.2.4)]. Can anyone help?

This is a problem of working with perturbation theory when the non-perturbed state has degeneracy. To elaborate I do not get why $(E-H_0-\lambda P_1VP_1)$ is not singular.

More importantly is the equation below correct and if so why:

$$(E-\lambda P_1VP_1)|l \rangle = E_D^{(0)}|l \rangle $$

That is the core of my problem. If the above equation - in the framework of Sakurai's statement of the problem - is correct I can work out all the rest on my own. However I do not get why this equation would hold true.

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  • $\begingroup$ The reference to a book rather than explicitly stating your question is probably why this has a downvote right now. It's easier for us if you make the question self-contained. Writing out what you're having trouble with - what all the components of the problem are - may also help clarify your thoughts. $\endgroup$ Jun 14, 2012 at 23:59
  • $\begingroup$ Thanks for the help. The reason I referenced the book is because it was the most compact way to describe my problem. Stating the problem explicitly would require a lot of writing - if not outright copying from the book. But since you are right I ll edit the question and also add a rapidshare link to a pdf file that contains the three paragraphs that give all the context I d have to state for the question to be self contained. $\endgroup$
    – Karsus
    Jun 15, 2012 at 0:40
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    $\begingroup$ Outright copying from the book is fine (in limited quantities, which this would be) as long as you mark it as a quote. Linking to a scanned PDF is more questionable, but I don't know that we have an actual policy against that so I won't remove the link. $\endgroup$
    – David Z
    Jun 15, 2012 at 3:37

2 Answers 2

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I think there's a typo. In the line after (5.2.4), the expression

$$P_1(E - H_0 - \lambda P_1 V P_1)$$

should have the second parenthesis moved to read

$$P_1(E - H_0 - \lambda P_1 V) P_1$$

which is equivalent to

$$E P_1 - P_1 H_0 P_1 - \lambda P_1 V P_1$$

Saying that this is not singular is the same as saying it doesn't map anything to zero. It doesn't, for the reason the text says: $E$ and the eigenvalues of $P_1 H_0 P_1$ cannot be the same since $E$ is just the perturbed value of $E_0$ while $H_0$ is acting on states that are orthogonal to the subspace $\{|m^{(0)}\rangle\}$.

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  • $\begingroup$ Thanks for the reply. Couldn't, however, $-\lambda P_1VP_1$ give an eigenvalue that is equivelant to the difference of $E-P_1H_0P_1$ making the whole expression singular? The text does not adress the term $-\lambda P_1VP_1$ and I don't see why this happens. $\endgroup$
    – Karsus
    Jun 15, 2012 at 7:31
  • $\begingroup$ That term is proportional to $\lambda$, which is the perturbation and therefore small. $\endgroup$ Jun 15, 2012 at 13:34
  • $\begingroup$ Hm, so subtle but so essential. I see now why the term is not singular. Thanks for the help. $\endgroup$
    – Karsus
    Jun 15, 2012 at 13:45
  • $\begingroup$ Now regarding the calculation of equation (5.2.6) from (5.2.5) I still cannot do it. I have worked out the math and I can prove it if the equation $(E-\lambda P_1VP_1)|l>=E_D^{(0)}|l>$ is correct. Qualitatively I think of it as the action of the perturbation canceling itself outside the degenrate space shifting the energy back to the unperturbed value. But that is not as strict and rigorous as I would like to be sure it is correct instead of wishful thinking. $\endgroup$
    – Karsus
    Jun 15, 2012 at 13:50
  • $\begingroup$ You might be able to refute the assertion $(E-\lambda P_1VP_1)|l\rangle=E_D^{(0)}|l\rangle$ by considering operation on both sides from the left by $P_0$. Since $P_{0}P_{1}=0$, you find the false statement $EP_{0}|l\rangle=E_D^{(0)}P_{0}|l\rangle$. $\endgroup$ Sep 20, 2016 at 0:19
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The equation as stated by you $E- \lambda P_1 V P_1 |l\rangle = E^{(0)}_D |l\rangle $ seems correct to me. This is so because $\lambda$ is small. Hence, $E$ is a small departure from the unperturbed energies in the degenerate subspace, i.e. $E_D^{(0)}$. So in the limit $\lambda \rightarrow 0$, the following two limits are approached: $E \rightarrow$ $E_D^{(0)}$ and $\lambda P_1 V P_1 \rightarrow 0$.

BTW @Mark Eichenlaub, I don't think there is any typo in the book (as suggested by you), primarily because I did not find any wrong statement/equations in the book. If you found out any, please point it out.

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