Kato META Notes

Kato Meta Notes PhL 3.26.05 We are interested in the stationary-state perturbation problem for a group of N degenerate states of Ho. How exactly do you handle this situation to arbitrary order in λn ? The Kato method provides an answer, and also has some application to the non-degenerate case as well. This subject was amazingly complex on first reading, but now seems less so. All my raw notes are in three files called Kato Round 1,2,3, and this Meta Notes is the fourth and last file. You can see in the earlier rounds how I was confused and gradually understood things after several passes through the material. I never looked up the actual 1949 Kato paper and that probably would have made this job easier, but I would have had to trudge over to the U to get it. Below we have first a contents of these Meta notes, then a Meta Meta review of them, then the actual Meta notes themselves. Page numbers here refer to the meta meta review. __________________________________________________________________________________ 1. The Kato Method and Degeneracy. 1 2. The Kato Method Applied to a non-degenerate Ea0 2 A. Comparison of the Kato state expansion with the Messiah special state expansion 3 B. Comments of Messiah at the start of his ga = 1 section page 717. 3 C. Computing the energy corrections 3 1. The Kato Method and Degeneracy 4 Levels k = 1, k = 2, k = 3. 9 2. The Kato Method Applied to a non-degenerate Ea0 11 A. Comparison of the Kato state expansion with the Messiah special state expansion 12 B. Comments of Messiah at the start of his ga = 1 section page 717. 16 C. Computing the energy corrections 16 Appendix A 19 __________________________________________________________________________________ META META REVIEW (about 4 pages long) 1. The Kato Method and Degeneracy. Here I review the Kato general method and show how it is used to resolve all situations involving degeneracy, including multiple levels of nested degeneracy. It is a complex problem. Kato was 1949. I am sure there are better methods on the market nowadays, but I think this was a great pioneering effort. (0) The subspaces and projection operators associated with this problem are described. For H one has Si and Pi, while for H0 one has S0i and P0i. (1) The special dim=N subspaces Ea0 and Ea are highlighted. Kets of Ea0 are |ψi ;0> = |φ>. Projectors which project onto these subspaces are called P0 and P. Each of these operators is 1:to:1 in the limited sense P: Ea0→ Ea and P0: Ea→ Ea0. (2) Any projector Pi for H can be represented as an integral around the pole at Ei: Pi = (2πi)-1 Γi dz (z-H)-1 P = (2πi)-1 ∫Γa dz (z-H)-1 On the right, the contour Γa encircles all the eigenenergy poles of space Ea. (3) A series simple expansion is developed for (z-H)-1: (z-H)-1= (z-H0)-1 + (z-H0)-1λV(z-H0)-1 + (z-H0)-1λV(z-H0)-1λV(z-H0)-1 + ... When installed in the above integral P and duly evaluated, we get a series expansion of P: P = P0 + λ (P0VQ0a + Q0aVP0 ) + λ2(six terms) + λ3 (lots of terms) + .... (4) This section discusses the method used to evaluate the contour integrals for the above series. (5) A similar expansion is developed for B ≡ (H-Ea0)P. We arrive at an EV equation from the SE: B|φα> = (Eα – Ea0) P|φα> with B = Σi=1 λnB(n) and P = Σi=0 λn P(n) (6) To approximately solve the equation, wee keep only terms through some order λk. (7) The problem is moved from Ea to Ea0 by applying P0 to each side, P0BP0 |φα> = (Eα – Ea0) P0PP0|φα> // both sides are kets in Ea0 which results in a reduction of the number of terms. If you use only k terms, the eigenkets of the EV equation are called |φα : k>, and vary (slightly) as you change k. (8) For k=1 one gets V |φα : 1> = ε1 |φα : 1> and det[V - ε1] = 0 determines the N values of ε1. (9) For k=2 one gets VQa0V |φα : 2> = ε2 |φα : 2> and det[VQa0V - ε2] = 0. (10) For k=3 one gets a true "generalized EV equation" and with an approximation one gets {–VQ0a2VP0V – VP0VQ0a2V + VQ0aVQ0aV + VQ0a2V} |φα : 3> = ε3|φα : 3> This shows how to handle all situations in our usual "picture" of nested degeneracy. In general, one has to solve a generalized EV problem through order λk and this delivers the eigenenergies ε12..k of H through that order, as well as the eigenkets |φα : k>. (11) Messiah only goes to k=2 in his non-Kato degeneracy analysis, results agree with Kato. 2. The Kato Method Applied to a non-degenerate Ea0 The basic EV equation P0BP0 |φ> = (Eα – Ea0) P0PP0|φ> is reviewed, and it is noted that Ea0 has one state |φ>, and Ea has one state |ψ>. If φ is normalized then it turns out that the action of our two basic projectors may be written as P0|ψ> = |φ> and P|φ> = k |ψ> where k = <φ|P|φ> = 1 + λ2C + λ3E + ... As noted above, P = P0 + λ (P0VQ0a + Q0aVP0 ) + λ2(six terms) + λ3 (lots of terms) + .... and this causes there to be no order λ term in k. The various constants like C and E can be evaluated by inserting the full expansion P = Σn=0 λnP(n). It happens that <ψ|ψ> = 1/k . A. Comparison of the Kato state expansion with the Messiah special state expansion The Kato method provides a natural state correction expansion |ψ> = Σn=0 λnP(n)|φ> = |φ> + λ P(1)|φ> + ... This expansion is very much like my initial "personal notes" expansion which had this form, ψmi = { umi + ΣjBij umj } Σk≠m Bik uk ψmi H(m) H where Bij = Σn=0 λnBij,n |φ> and same for Bik , where |φ> = umi. In my expansion, it is to be expected that some of the perturbation correction is proportional to umi. But this is a different expansion than the one which appears in Schiff and Messiah where we have to renormalize things to get ψ'mi = { umi + Σj≠iB'ij umj } Σk≠m B'ik uk where now none of the "corrections" are in the direction of umi. If we want to recast the Kato expansion so it looks like the Schiff/Messiah expansion, we have to do some rearrangement work on the series, and that is what this section is about. If the S/M expansion correction states are |ψ ;n>, then we find |ψ ;n> = coefficient of λn in { <φ| Σm=1 λm P(m)|φ>-1 [ Σk=1 λk P'(k)|φ> ] } where we have to expand both sums sufficiently to extract the correct λn term. The prime indicates that subterms are thrown out which begin with P0. We compare this state expansion to the iterative expansion given earlier by Messiah. B. Comments of Messiah at the start of his ga = 1 section page 717. These comments seemed very mysterious on first reading, but are now clear, and here we document this simple fact. C. Computing the energy corrections The EV equation becomes P0BP0|φ> = k(Eα – Ea0)|φ> since P0PP0|φ> = k|φ>. Thus. Σn=1λnεn = (Eα – Ea0) = k-1<φ|B|φ> = <φ| (Σm=1 λm P(m))-1|φ> [ Σk=1 λk B(k)|φ> ] } where again we have to extract a power in this sense εn = coefficient of λn in { <φ| (Σm=1 λm P(m))-1|φ> [ Σk=1 λk B(k)|φ> ] } For the lowest terms we find that ε1 = <φ| B(1)|φ>, ε2 = <φ| B(2)|φ>, but ε3 is where we first see the mixing of the two power series in λ ε3 = <φ| B(3)|φ> – C<φ| B(1)|φ> C = – <φ| VQ0a2V|φ> Here, C is the first constant in k = <φ|P|φ> = 1 + λ2C + λ3E + ... , so we then have ε3 = <φ| B(3)|φ> + <φ| VQ0a2V|φ><φ| B(1)|φ> = <φ| B(3)|φ> + <φ| VQ0a2V|φ><φ|V|φ> = <φ| B(3)|φ> + <φ| VQ0a2VP0|V|φ> This extra term is exactly what is needed to make this give the right answer for ε3 as quoted in Messiah page 717 (74). So this is some evidence that the expression above is correct, though it does not appear in Messiah. It is doubtless easier to compute the εn using the non-Kato method Messiah develops, which is this: εn = tr(B(n)) = <φ| B'(n)|φ> where prime means we knock out any one P0 factor in each subterm of B(n). This is a more direct way to compute the energy corrections to arbitrary order. A clean and simple prescription. In Appendix A, I took a quick (but not successful) shot at proving directly that these two methods for computing εn give the same results. Assuming I made no mistakes above, I know they are both correct. And I did show that they give the same results for ε1, ε2 and ε3 where the last was non-trivial. __________________________________________________________________________ META REVIEW: 1. The Kato Method and Degeneracy (0) The solution space of the full H can be partitioned into a set of subspaces Si each of which has a projection operator Pi. Suppose there are m such subspaces (m could be infinite). We can then say H = i=1,m Si 1 = Σi=1,m Pi We have H = H0 + λV and we could write a similar partition for H0 as follows H0 = i=1,r S0i 1 = Σi=1,r P0i and we will use this idea a few subsections below. Here r is the number of subspaces S0i which partition H0 and we would in general expect to find r ≤ m since the perturbation tends to break degeneracy. (1) In the Kato method for treating a degenerate situation, the N unperturbed states form a space called Ea0 where "a" just labels this group of states, since there are other eigenstates of H0 including possible other degenerate groups. The space Ea0 is spanned by N kets |ψi ;0> = |φ>. The N kets |ψ> are the corresponding exact states of the full H and they span a space called Ea. Kato's method does not deal with any "state expansions", but we could if we liked write one down, such as |ψ> = |φ> + Σn=1,∞ λn |ψi ;n> = Σn=0,∞ λn |ψi ;n> and this subject is addressed in a later section below. The space Ea is the union of some of the subspaces Si of H each of which has a projector Pi. There could be anywhere from 1 to N subspaces Si in Ea, depending on the degeneracy structure of the states in Ea. Let's imagine there are n such subspaces. Then we can say [ where 1 ≤ n ≤ N ] Ea = i=1,n Si P ≡ Σi=1,nPi The projection operator P thus projects out of any ket that part belonging to the space Ea, so we really have P:H→ Ea . If we restrict the domain to Ea0, we can talk about operator P: Ea0→ Ea and this thing is 1-to-1 and therefore invertible with respect to this pair of spaces. Thus, for example, we can write any eigenstate |ψ> of H in the space Ea in the form |ψ> = P |φ>, where |φ> lies in Ea0 . Specifically, we obtain |φ> as |φ> = P-1 |ψ>. This simple fact is crucial to the Kato development below. The same argument can be made concerning P0: Ea→ Ea0 being 1-to-1 with respect to these two spaces and thus being invertible. This fact will also be used below. [In general we do not have P-1 = P0 or vice versa.] There is certainly nothing mysterious about P's definition as the projector onto Ea. The projector for the subspace Ea0 is called P0 and will be used later. ( that is, P0 ≡ P0a using the notation shown in (0) above). (2) Kato goes on to produce a strange expansion of P in powers of λ, and this is much harder to understand. His first step is to express P as a contour integral. The following initial sequence of steps seems pretty clear: [ book uses G(z) = (z-H)-1 but we just write it all out here ] (z-H)-1Pi = (z-Ei)-1Pi => (z-H)-1 = Σi (z-Ei)-1Pi => Pi = (2πi)-1 Γi dz (z-H)-1 The Ei are the eigenenergies of full H. Variable z is that of a "complex energy plane". So the rightmost equation tells us Pi as a function only of H and the eigenenergy Ei of Si. Let's show that Pi given in this form really works: Pi|j> = (2πi)-1 Γi dz (z-H)-1|j> = (2πi)-1 Γi dz (z-Ej)-1|j> The integrand has a pole at Ej. If |j> Si, the integral is 1 since the contour captures the pole. If |j> is in some other Sj. the contour integral gives 0. So this in itself is really a derivation of our formula for Pi. If we make the contour Γa encircle just the eigenenergies of the space Ea (but also the point Ea0), we get a sum of the Pi for the subspaces Si whose union is the space Ea, and we have already defined this to be P, so we get P = (2πi)-1 ∫Γa dz (z-H)-1 and here is a picture of this contour (3) The next step is to expand (z-H)-1 in powers of λ, and this will then provide an expansion of P in powers of λ which is our goal. The expansion of (z-H)-1 starts out this way: 1 = (z-H0)-1 (z-H0) = (z-H0)-1 (z- H +H - H0) = (z-H0)-1 (z-H) + (z-H0)-1 (H - H0) (z-H)-1 = (z-H0)-1 + (z-H0)-1(H - H0) (z-H)-1 = (z-H0)-1 + (z-H0)-1λV (z-H)-1 (*) The first line is trivial, and to get the second we right multiply the first line by (z-H)-1. We thus end up with an integral equation of sorts for (z-H)-1 which has the following formal series solution: (z-H)-1 = Σn=0∞ λn {(z-H0)-1V(z-H0)-1V.... V(z-H0)-1} // n factors of V = (z-H0)-1 + (z-H0)-1λV(z-H0)-1 + (z-H0)-1λV(z-H0)-1λV(z-H0)-1 + ... where the second form makes it clear this is a solution of (*). I think this kind of solution has some formal name like a Neumann series, but I forget right now. If we then insert this expansion of (z-H)-1 into our integral representation of P, we get our expansion of P: P = P0 + λ (P0VQ0a + Q0aVP0 ) + λ2(six terms) + λ3 (lots of terms) + .... (4) How were these terms evaluated? Looking at the first term, it must be true that (2πi)-1 ∫Γa dz (z-H0)-1λV(z-H0)-1 = λ (P0VQ0a + Q0aVP0 ) The way this evaluation is done is based on the following. First we can write each (z-H0)-1 factor as (z-H0)-1 = (z-Ea0)-1P0 + Σi≠a (z-Ei0)-1P0i // sum over other Si0 which is just a restatement of (z-H)-1 = Σi (z-Ei)-1Pi above when λ=0. The poles at z = Ei0 lie outside the contour Γa as our picture shows. The only poles inside our contour are at z = Ea0, so we may shrink our fancy contour to a simple circulation about the point z = Ea0. Kato then uses a Taylor series to expand the functions (z-Ei0)-1 about the point z = Ea0 and the above becomes (z-H0)-1 = P0/ (z-Ea0) + Σn=1,∞ (-1)n-1 (z- Ea0)n-1 [ Σi≠a (Ea0- Ei0)-n Pi0] = P0/ (z-Ea0) + Σn=1,∞ (-1)n-1 (z - Ea0)n-1 (Q0a)n where now the operator Q0a suddenly appears! We know that Q0a and Q0 involve sums over states that are in the perp space of Ea0 and the perp space is the union of all the Si0 being summed by Σi≠a, so this result is not too shocking. Details are given in the Round Three notes. This expression for (z-H0)-1 is certainly valid near the point z = Ea0. The evaluation of all the Kato type integrals is now just a question of examining the interaction between the pole at z = Ea0 and the various zeros at the same point. (5) So, we have concocted the following series expansion for P: P = P0 + λ (P0VQ0a + Q0aV P0 ) + λ2(six terms) + λ3 (lots of terms) + .... = Σn=0 λnP(n) A term in the set of terms for λn always has the form xVxVxV..Vx where there are N factors of V, and the x are either P0 or Q0a. In a similar fashion, Kato provides an expansion for B ≡ (H-Ea0)P which is this B = (H-Ea0)P = 0 + λ(P0VP0) + λ2(three terms) + λ3 (lots of terms) + .... = Σn=1 λnB(n) Now we said above that we can write any eigenstate |ψ> of H in the space Ea in the form |ψ> = P |φ>, where unique |φ> lies in Ea0. Thus consider an eigenstate of H called |ψα>. We can say H|ψα> = Eα|ψα> // both sides are kets in Ea and we know there exists a unique |φα> in Ea0 such that HP|φα> = EαP|φα> We can then subtract EαP|φα> from both sides to get (HP– Ea0P) |φα> = (Eα – Ea0) P|φα> B|φα> = (Eα – Ea0) P|φα> // both sides are kets in Ea We recognize (Eα – Ea0) as the energy correction to all orders to the unperturbed energy Ea0 and we know this is going to be order λ or higher. Now we can insert our Kato expansions for B and P on the two sides to get: Σi=1 λnB(n) |φα> = (Eα – Ea0) Σi=0 λn P(n) |φα> (6) The Kato approximation method is to truncate the left side sum at some order i = k and the right side sum at some order k-1 so that both sides will have terms through λk : [ λB(1) + λ2B(2) + .. λkB(k)] |φα> = (Eα – Ea0)[ 1 + λP(1) + ... + λk-1 P(k-1)] |φα> where we used P(0) = P0 and P0 |φα> = |φα>. Since there is some operator on the right between the eigenvalue and the state, this is a "generalized eigenvalue equation" which I assume is solvable by people who know about these things (I do not). (7) Kato actually uses a slightly different presentation. Due to the 1-1 connection between Ea and Ea0 provided by the projector P0: Ea → Ea0, we can apply P0 from the left to both sides of all our equations above and then our kets will all be in Ea0 instead of in Ea. At the same time, we are free to extract a factor of P0 on the right from the state |φα>. This leads to the alternative presentation: P0BP0 |φα> = (Eα – Ea0) P0PP0|φα> // both sides are kets in Ea0 B"|φα> = (Eα – Ea0) P"|φα> // both sides are kets in Ea where we use a double prime to indicate that we have sandwiched P0 around B and P. Our approximation above then becomes P0 [ λB(1) + λ2B(2) + .. λkB(k)] P0 |φα> = (Eα – Ea0) P0 [ 1 + λP(1) + ... + λk-1 P(k-1)] P0 |φα> This tends to simplify things because the leading (and also the trailing) P0 allows for various terms to be removed since we have P0Q0a = Q0aP0 = 0. In particular we have P'(1) = P0 P(1)P0 = P0 (P0VQ0a + Q0aV P0)P0 = 0 and various other terms simplify, so we can write out the first few terms on each side: P0 [ λV + λ2VQa0V + .. λkB(k)] P0 |φα : k> = (Eα – Ea0) P0 [ 1 – λ2VQa0V + ... + λk-1 P(k-1)] P0 |φα :k> where our notation |φα : k> reminds us that this eigenket solves this degree k problem. Do not confuse this with the eigenstate correction concept mentioned earlier. Here I used a colon separator. For example, suppose you set k = 4 and suppose you somehow manage to solve this generalized eigenvalue equation for the N eigenkets |φα :4> and for the eigenenergies (Eα – Ea0). You then know these energies to order λ4 (I think). You could then compute the eigenkets in Ea as |ψα> = P|φα> where you would use the expansion for P through λ4, and the solved-for eigenkets |φα>. Levels k = 1, k = 2, k = 3. (8) For k = 1 the approximation takes this form P0 [ λV] P0 |φα : 1> = (Eα – Ea0) |φα : 1> or just: λV |φα : 1> = (Eα – Ea0) |φα : 1> This is in the N dimensional space Ea0 and the problem is then to diagonalize V within this space (only). The eigenvalues of this problem then tell us the eigenvalues of Eα to order λ. We usually write this equation as V |φα : 1> = ε1 |φα : 1> det[V - ε1] = 0 determines the N values of ε1 (9) For k = 2 the approximation takes this form [ λV+ λ2VQa0V] |φα : 2> = (Eα – Ea0) |φα : 2> = (λε1 + λ2ε2) |φα : 2> ≡ ε12 |φα : 2> Although we have a messier operator to diagonalize, we still have a non-generalized EV problem which we think we know how to solve. The secular equation would be det [V+ VQa0V - ε12 ] = 0 which we could solve for our N values of ε12, and then know ε2 = ε12 - ε1. If we restrict this N dimensional problem to a M<N dimensional problem of a group of kets which remained degenerate in first order, we could approximate λV with the single eigenvalue of that first order problem, call it [Eα(1) – Ea0], then the above becomes [λ2VQa0V] |φα : 2> = (Eα – Ea0 – [Eα(1) – Ea0]) |φα : 2> = (Eα – Eα(1)) |φα : 2> and then solving this EV problem would give us Eα through second order λ2 , and would give us some new states |φα : 2> which are 'better' than those of the first order solution. We normally write this last equation then as VQa0V |φα : 2> = ε2 |φα : 2> det[VQa0V - ε2] = 0 determines the M values of ε2 In the picture below, we show an example where M is written as N1 Of course this would also apply this approximation in the case M = 1 which would be our state "n" above, and in this case our equation is simply a 1x1 equation VQa0V |φ : 2> = ε2 |φ : 2> and we then find that ε2 = <φ : 2|VQa0V |φ : 2> which is the traditional non-degenerate state result Messiah p 694 (26) also shown as 717 (74). So to this order, for a state like n that started out degenerate and became solo at the first level, the ε2 correction is that same as if he started out solo, provided you use the correct eigenstate. (10) For k = 3 and higher, we have to worry about the operator on the RHS and we are in the realm of the generalized eigenvalue problem which is a little hazy to me. Here is what I got in my Round Three notes { λV + λ2VQ0aV –λ3VQ0a2VP0V – λ3VP0VQ0a2V + λ3VQ0aVQ0aV } |φα : 3> = (Eα – Ea0){ 1- λ2VQ0a2V} |φα : 3> Perhaps you could move the right side operator to the left saying { 1- λ2VQ0a2V}-1 ≈ { 1+ λ2VQ0a2 + O(λ3) } to get this new LHS { 1+ λ2VQ0a2 + O(λ3) }{ λV + λ2VQ0aV –λ3VQ0a2VP0V – λ3VP0VQ0a2V + λ3VQ0aVQ0aV } = { λV + λ2VQ0aV –λ3VQ0a2VP0V – λ3VP0VQ0a2V + λ3VQ0aVQ0aV + λ3 VQ0a2V} and then we have a regular eigenvalue equation { λV + λ2VQ0aV –λ3VQ0a2VP0V – λ3VP0VQ0a2V + λ3VQ0aVQ0aV + λ3 VQ0a2V} |φα : 3> = (Eα – Ea0) |φα : 3> = ε123 |φα : 3> and we could write the secular equation as: det [V + VQ0aV –VQ0a2VP0V – VP0VQ0a2V + VQ0aVQ0aV + VQ0a2V – ε123 ] = 0 to get all N values of the energy corrections through third order, one for each of the N states. If we still had a degenerate subspace J ≤ M ≤ N, we could approximate λV + λ2VQ0aV ≈ λε1 + λ2ε2 to get this equation: {–λ3VQ0a2VP0V – λ3VP0VQ0a2V + λ3VQ0aVQ0aV + λ3 VQ0a2V} |φα : 3> = λ3ε3|φα : 3> or {–VQ0a2VP0V – VP0VQ0a2V + VQ0aVQ0aV + VQ0a2V} |φα : 3> = ε3|φα : 3> with a simpler secular equation det [–VQ0a2VP0V – VP0VQ0a2V + VQ0aVQ0aV + VQ0a2V – ε3 ] = 0 In terms of the above picture, this last calculation would be applied to the group of states still degenerate in the ε2 rightmost column. Think of these three bands as N ≥ M ≥ J . We could also apply this to solo state "m" in which case we get the result (ie, no need for a determinant) ε3 = <φ : 3| –VQ0a2VP0V – VP0VQ0a2V + VQ0aVQ0aV + VQ0a2V |φ : 3> This result differs a little bit from the ε3 correction of a state which begins life non-degenerate in the left column. The difference is that for that non-degenerate problem, the last term cancels the second term and we get Messiah page 717 (74). This is discussed below. [ I might be wrong here ] (11) How does all this compare with Messiah's traditional degeneracy approach pp 698-700 ? He only discusses up to the λ2 term. He uses the projection operator P(1) to reduce our N dimensional space to a smaller M dimensional space as we just did above. He ends up with P(1)VQa0V] |φα > = ε2 P(1)|φα > which is the same as our k=2 result above. 2. The Kato Method Applied to a non-degenerate Ea0 Everything done to this point allows for degeneracy N of the space Ea0. In the case that N = 1, there are certain dramatic simplifications. First, go back to our result above which said B"|φα> = (Eα – Ea0) P"|φα> // both sides are kets in Ea If there is only one state |φ> in Ea0, then there is also only one state |ψ> in Ea. Recall in deriving the above equation we claimed that there exists a unique |φ> in Ea0 such that |ψ> = P |φ>. No claim was made that either |ψ> or |φ> is normalized to unity. In the Messiah/Schiff (M/S) normalization scheme, we normalize ψ in a certain manner, and we want |φ> to have unit normalization. In this case, where we might be scaling both ψ and φ separately, we are really claiming that there exists a normalized |φ> and a specially normalized |ψ> such that k |ψ> = P|φ> where k is some constant. This constant cancels on both sides of the above equation so we don't have to worry about it there. Before examining the above eigenvalue equation and finding the eigenenergies, we digress first to talk about the "correction state expansion", something Messiah does not mention in the Kato section. A. Comparison of the Kato state expansion with the Messiah special state expansion Consider this expansion which we will call "the Kato state expansion", where I assume that the state |φ> is unit-normalized, k|ψ> = P|φ> = Σn=0 λnP(n)|φ> = |φ> + λ Q0aV|φ> + λ2 { (Q0aVQ0aV) – (Q0a2VP0V + P0VQ0a2V)} |φ> + λ3 (..)|φ> + ... where we just insert the P(n) from above and simplify wherever possible (ie, delete right-side P0 factors, and delete terms with Q0a on the right). One might be tempted to equate this expansion term by term with the usual Messiah/Schiff (M/S) state expansion which is this |ψ> = |φ> + Σn=1,∞ λn |ψ ;n> but there is a problem in doing this. In the M/S expansion, we are supposed to assume the normalization conditions that <φ|ψ ;n> = 0 for n>0, and also <φ|φ> = 1. In other words, none of the "correction states" |ψ ;n> is supposed to have a component in |φ>. Suppose we partition the eigenstates of Ho into our non-degenerate state of interest |φ> and all the other states we will just call |k>. Then in the M/S expansion we are supposed to have correction states |ψ ;n> which are linear combinations only of the states |k>. But this, it turns out, is not true for the P(n) expansion above. For example, we see a term – λ2 P0VQ0a2V |φ> appearing in this expansion. Due to the leftmost P0, this correction state really has the form λ2 C |φ> where C is some constant, so this term is some multiple of |φ>. We can compute: C = – <φ| VQ0a2V|φ> = – Σk (Ea0- Ek)-2 |Vk0|2 So the Kato expansion becomes k|ψ> = ( 1 + λ2C) |φ> + λ Q0aV|φ> + λ2 { (Q0aVQ0aV) – (Q0a2VP0V )} |φ> + λ3 (..)|φ> + ... Any term that begins with a Q0a cannot have a component in |φ>. For example, let's look at the first λ2 term and compute its projection onto <φ|. <φ| λ2 (Q0aVQ0aV) |φ> = λ2<φ| Q0(Ea0- H0)-1Q0VQ0aV) |φ> = 0 because <φ| Q0 = 0, since Q0 projects only onto the |k> states. So, imagine that we write out the entire Kato expansion, and we compute all terms that begin with P0 and find what multiple such terms are of |φ>, and then we gather all these multiples up where we put the λ2C shown above. Then we have this situation: ( the state |φ> is normalized to 1) k|ψ> = ( 1 + λ2C + λ3C' + λ4C'' + .... ) |φ> + λ Q0aV|φ> + λ2 { (Q0aVQ0aV) – (Q0a2VP0V )} |φ> + λ3 (..)|φ> + ... where in the main body of the expansion, there are no longer any terms beginning with P0. Thus, all the above expansion terms respect the normalization condition <φ|term> = 0. Let's then identify k = ( 1 + λ2C + λ3C' + λ4C'' + .... ) ________________________________________________________________________ Theorem: k = <φ| P |φ> and <ψ|ψ> = 1/k; Messiah's use of the word "norm". Note injected: Consider again P|φ> = Σn=0 λnP(n)|φ> = |φ> + Σn=1 λnP(n)|φ> This tells us that <φ| P |φ> = 1 + <φ| Σn=1 λnP(n)|φ> = ( 1 + λ2C + λ3C' + λ4C'' + .... ) = k Thus, we can represent k in this manner k = <φ| P |φ> and we can then say |ψ> = k-1P|φ> => <ψ|ψ> = k-2<φ|P P|φ> = k-2<φ| P|φ> = k-1 = 1/<φ| P |φ> In Stakgold, the "norm" is defined as ||x|| = <x,x>1/2 top of page 108 volume 1. But Messiah on page 165 defines his norm to be N = <ψ|ψ>. Schiff and Saxon don't really use the word norm very much. So Messiah uses the word "norm" to be the length of a vector squared! I never noticed this before, and I think this is sort of a language misuse by Messiah. But given Messiah's definition of the word "norm", it would be correct to say that the norm of the state |ψ> is 1/<φ| P |φ> as he says on page 717. ______________________________________________________________________ And we then have |ψ> = |φ> + λk-1 Q0aV|φ> + λ2 k-1 { (Q0aVQ0aV) – (Q0a2VP0V )} |φ> + λ3 k-1 (..)|φ> + ... We know that k = ( 1 + λ2C + λ3E ...) so that k-1 = 1 - λ2C + λ3D + ..., so we get |ψ> = |φ> + λ(1 - λ2C + λ3D + ..) Q0aV|φ> + + λ2 (1 - λ2C + λ3D + ..) { (Q0aVQ0aV) – (Q0a2VP0V )} |φ> + λ3 (1 - λ2C + λ3D + ..) (..)|φ> + ... and then we can collect powers of λ to get |ψ> = |φ> + λ Q0aV|φ> + + λ2 { (Q0aVQ0aV) – (Q0a2VP0V )} |φ> + λ3{ (..) – C Q0aV } |φ> + ... where we show a λ3 piece of the Q0aV|φ> term showing up now in the λ3 term. What we have learned here is that the Kato state expansion using P(n) and the M/S expansion are subtle rearrangements of each other, but that the λ0, λ1 and λ2 terms of both expansions are the same, provided we drop the P0-starting term in the λ2 term. Thus we can now identify from the M/S expansion , |ψ> = |φ> + Σn=1,∞ λn |ψ ;n> that we have |ψ ;0> = |φ> |ψ ;1> = λ Q0aV|φ> // agrees with Messiah p 680 (14) |ψ ;2> = λ2(Q0aVQ0aV – Q0a2VP0V )|φ> // agrees with Messiah p 694 (27) and we get agreement with the non-Kato method used earlier by Messiah. We could of course use the Kato expansion "as is", |ψ> = Σn=0 λnP(n)|φ> if we are willing to have the normalization of |ψ> come out as it may (so no k-1on the RHS), and if we are willing to have terms in the expansion which contain some |φ> component. We just have to keep in mind that this expansion of state corrections is then different from the fancy M/S expansion with its unusual requirements that <φ |ψ ;n> = 0. There is really something to be said in favor of the Kato expansion because it let's you see all the terms in each λn order that "appear", including terms proportional to |φ>. To summarize our Kato state correction method cast into the M/S convention |ψ> = |φ> + k-1 Σn=1 λnP'(n)|φ> = |φ> + <φ| Σm=1 λmP(m)|φ>-1 [ Σk=1 λk P'(k)|φ> ] where P'(n) means we delete any terms that begin with P0 in P(n). Thus we can say |ψ ;n> = coefficient of λn in { <φ| Σm=1 λmP(m)|φ>-1 [ Σk=1 λk P'(k)|φ> ] } Although this is well-defined, it is admittedly not very convenient (we did some examples above), and perhaps this is why Messiah did not mention this expansion method. Note that (1+x)-1 = 1 - x + x2 - x3 + ... so that (1+ aλ2 + bλ3 + cλ4)-1 = 1 -(aλ2 + bλ3 + cλ4) + (aλ2 + bλ3 + cλ4)2 + ... = 1 - aλ2 - bλ3 + (-c + a2) λ4 + .... so we could at least do it all if we had to. Messiah on page 688 describes the non-Kato method of getting the state corrections. You use this iterative scheme: |ψ ;n> = Q0a(V-ε1) |ψ ;n-1> - ε2 |ψ ;n-2> – ... – εn-1 |ψ ;1> and use this to get started: |ψ ;1> = Q0aV |ψ ;0> and |ψ ;0> = |φ> To use this iterative method, we need the εn. We can either get these from the method described below which says εn = tr(B(n)) = <φ| B'(n)|φ>, or we could use the traditional double iteration method which is this: |1> = Q0aV |0> 1 s = 1 ε1 = <0| V |0 > = Vnn // Messiah p 689 (12) 1 |2> = Q0aV |1> – ε1 Q0a |1> 2 s = 2 ε2 = <0| V |1> = <0|V Q0aV |0> // Messiah p 694 (26) 1 |3> = Q0aV |2> – ε1 Q0a |2> – ε2 Q0a |1> 5 s = 3 ε3 = <0| V |2> 2 |4> = Q0aV |3> – ε1 Q0a |3> – ε2 Q0a |2> – ε3 Q0a |1> 13 s = 4 ε4 = <0| V |3> 5 |5> = Q0aV |4> – ε1 Q0a |4> – ε2 Q0a |3> – ε3 Q0a |2> – ε4 Q0a |1> 34 s = 5 ε5 = <0| V |4> 13 where I show in the second last column the number of terms involved. B. Comments of Messiah at the start of his ga = 1 section page 717. When I first read these comments, I had no idea what he was talking about, but now they all make sense. k|ψ> = P|φ> => k2<ψ|ψ> = <φ|P P|φ> = <φ|P|φ> "the norm (Messiah's norm) of P|φ> is || P|φ>||2 = <φ|P|φ> " agreed! " insert P and you could compute <φ|P|φ>" agreed! " P|φ> is a multiple of the eigenvector |ψ> formed in Section I (that of the MS expansion) " The above I would associate with my claim that k|ψ> = P|φ>. agreed! "the norm (Messiah's norm)of the latter( |ψ>) is 1/<φ|P|φ>" agreed! C. Computing the energy corrections Our eigenvalue problem as noted above is this B"|φ> = (Eα – Ea0) P"|φ> // both sides are kets in Ea where |φ> is our unperturbed unit-normalized sole state in Ea0. We have already commented that P|φ> = k |ψ> where k = <φ|P|φ>. Our expansion above became k|ψ> = k |φ> + λ Q0aV|φ> + λ2 { (Q0aVQ0aV) – (Q0a2VP0V )} |φ> + λ3 (..)|φ> + ... where no terms on the RHS other than the first have any component in |φ>. Therefore, we may conclude that P0|ψ> = |φ>. Of course to make this work, we have to delete any terms on the RHS that begin with P0. Assume we do this. Then we have for our two projectors: P0|ψ> = |φ> and P|φ> = k |ψ> P0|φ> = |φ> and P|ψ> = |ψ> k = <φ| P |φ> = 1 + <φ| Σn=1 λnP(n)|φ> = ( 1 + λ2C + λ3C' + λ4C'' + .... ) The first line shows our 1-to-1 pair of mappings in the case each space has only one state! The ket |φ> is normalized to unity, but |ψ> is not so normalized. In fact, we know <ψ|ψ> = 1/k . Our equation above says P0BP0|φ> = (Eα – Ea0) P0PP0|φ> But look at the RHS operator P0PP0|φ> = P0P|φ> = k P0 |ψ>= k |φ> so we end up with P0BP0|φ> = k (Eα – Ea0) |φ> // all terms in B are included here, φ is normalized and thus, since <φ|φ> = 1, k (Eα – Ea0) = <φ|B|φ> = Σn=1 λn<φ| B(n)|φ> Thus we get Eα = Ea0 + k-1 Σn=1 λn<φ| B(n)|φ> = Ea0 + k-1 Σn=1 λn<φ| B(n)|φ> = Ea0 + (1 - λ2C + λ3D + ..) Σn=1 λn<φ| B(n)|φ> = Ea0 + (1 - λ2C + λ3D + ..) (λ<φ| B(1)|φ> + λ2<φ| B(2)|φ> + ...) = Ea0 + λ<φ| B(1)|φ> + λ2<φ| B(2)|φ> + λ3{ <φ| B(3)|φ> – C<φ| B(1)|φ> } + ... Notice this correction we get on the λ3 term, we come back to this below. In theory, we could compute our constant k = <φ|P|φ> to any desired order, and then we could write out the above energy corrections to any order. It turns out that there is an easier way to do this, but we hold off on that for a moment and continue along here to show that things really work right. To compute the first two terms above, we look up B(1) = P0VP0 ≈ V B(2) = (Q0aVP0VP0 + 2 perms) ≈ P0VQ0aVP0 ≈ VQ0aV In order to compute the λ3 term, we look up the following facts: C = – <φ| VQ0a2V|φ> B(3) = – (Q0a2VP0VP0VP0 + 3 perms) + (Q0aVQ0aV P0VP0 +5 perms) ≈ – (P0VQ0a2VP0VP0 + P0V P0 VQ0a2VP0) + P0VQ0aVQ0aV P0 ≈ VQ0aVQ0aV – (VQ0a2VP0V + V P0 VQ0a2V) B(1) = P0VP0 ≈ V Thus, the λ3 term is given by λ3{ <φ| B(3)|φ> – C<φ| B(1)|φ> } = λ3 <φ| {B(3) – C B(1)} |φ> = λ3 <φ| { VQ0aVQ0aV – (VQ0a2VP0V + V P0 VQ0a2V) + <φ| VQ0a2V|φ>V} |φ> The last term can be written: λ3<φ| <φ| VQ0a2V|φ>V} |φ> = λ3<φ| V |φ><φ| VQ0a2V|φ> = λ3<φ| VP0VQ0a2V|φ> so that the sum is now = λ3 <φ| { VQ0aVQ0aV – (VQ0a2VP0V + V P0VQ0a2V) + VP0VQ0a2V } |φ> = λ3 <φ| { VQ0aVQ0aV – VQ0a2VP0V } |φ> since the last two terms cancel. So we may now summarize the first three energy correction terms: Eα = Ea0 + λ<φ| B(1)|φ> + λ2<φ| B(2)|φ> + λ3{ <φ| B(3)|φ> – C<φ| B(1)|φ> } + ... = Ea0 + λ<φ| V |φ> + λ2 <φ| VQ0aV |φ> + λ3 <φ| { VQ0aVQ0aV – VQ0a2VP0V } |φ> and this agrees exactly with Messiah page 717 (74), although he used a different method to arrive at this result. We can compare his method and our method for finding the eigenenergies: Our method here: k (Eα – Ea0) = <φ|B|φ> = Σn=1 λn<φ| B(n)|φ> => εn = <φ| B(n)|φ> = <φ| B(n)|φ>/ <φ|P|φ> Messiah's method used earlier was this (see Round Three notes p ~ 15), εn = tr(B(n)) = <φ| B'(n)|φ> where prime means we knock out any one P0 factor in each subterm of B(n). This is a more direct way to compute the energy corrections to arbitrary order. A clean and simple prescription. Comparing our methods, the following interesting fact must be true <φ| B(n)|φ> = <φ| B'(n)|φ> <φ|P|φ> = <φ| B'(n)P0P|φ> // definition of P0 I tried proving this going back and forth between the trace and <φ|...|φ> forms, but that does not seem to do the trick! This is some kind of more global theorem. See appendix A below for more comments. ____________________________________________________________________________ Appendix A Another approach is to write out the explicit form of B(n) B(n) = Σ(n-1) [ RiV RiV ...... V Ri] // sum on all indices that sum to n-1 P(n) = Σ(n) [ RiV RiV ...... V Ri] where R0 ≡ P0 and Rk ≡ (-1)k-1 (Q0a)k k>0 Then the theorem must state that <φ| Σ(n-1)[ RiV RiV ...... V Ri] |φ> = <φ| Σ(n-1)[ RiV RiV ...... V Ri], one P0 knocked out |φ> * <φ| Σn λn Σ(n) [ RiV RiV ...... V Ri] |φ> = Σn λn <φ| Σ(n-1)[ RiV RiV ...... V Ri], one P0 knocked out * P0 Σ(n) [ RiV RiV ...... V Ri] |φ> I have to say this theorem is very non-obvious when viewed in this manner! But in fact we have already proven it, so I will formally declare a theorem Theorem: <φ| B(n)|φ> = <φ| B'(n)|φ> <φ|P|φ> = <φ| B'(n)P0P|φ> Proof: We compute εn (our energy correction in nth order) by two different valid methods, and comparing the results proves the theorem. It looks like renormalization. One more try: the above also says <φ| Σ(n-1)[ RiV RiV ...... V Ri] |φ> = <φ| Σn λn Σ(n) [ RiV RiV ...... V Ri] |φ> * <φ| Σ(n-1)[ RiV RiV ...... V Ri], one P0 knocked out |φ> = Σn λn Σ(n,i) <φ| [ RiV RiV ...... V Ri] P0 Σ(n-1,j)[ RjV RjV ...... V Rj], one P0 knocked out |φ> = Σn λn Σ(n,i) Σ(n-1,j) <φ| [ RiV RiV ...... V Ri] P0[ RjV RjV ...... V Rj]one P0 knocked out |φ> This is a combinatoric problem which I don't know how to solve. Let's call it quits in this!