Degenerate Perturbation Theory

12 Key excerpts on "Degenerate Perturbation Theory"

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  Quantum Mechanics

  • Arjun Berera, Luigi Del Debbio(Authors)
  • 2021(Publication Date)
  • Cambridge University Press

    (Publisher)

  Our purpose here was only to highlight to the reader the characteristics of the perturbation expansion that is generally encountered in quantum mechanics. 16.2 Degenerate Perturbation Theory We saw in the nondegenerate case discussed in the previous section that the perturbation can be thought of as inducing a small mixing of the unperturbed states. The perturbed states {| n} are in one-to-one correspondence with the unperturbed states {| n (0)  and go over smoothly to them in the limit λ → 0. Recall that degeneracy means there are several eigenstates with the same energy eigenvalue. If there are g such degenerate states, we say there is a g-fold degeneracy. In the presence of degeneracy, it is not at all clear to which of the infinitely many degenerate unperturbed states the perturbed states return as λ → 0. Suppose there is a g-fold degeneracy. It then means any linear combination of the corresponding g unper- turbed eigenstates is still an eigenstate with the same energy eigenvalue. Thus there is a g-dimensional subspace in which we can define any g orthogonal directions and they are just as good as any other g orthogonal directions in being the eigenstates of ˆ H 0 . The question then is for a given perturbation ˆ H  , which are the correct directions in this subspace from which to perturb. In other words, out of the infinitely many directions, which are the directions the perturbed states return to as λ → 0. Degenerate Perturbation Theory addresses this problem. The basic idea is within the g-dimensional degenerate subspace of states, to diagonalise the perturbation Hamiltonian ˆ H  . Since all directions of this g-dimensional subspace are equally good eigenstates of ˆ H 0 , diagonalising ˆ H  selects particular g-directions, which now up to first order in perturbation theory are eigenstates.

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  Quantum Mechanics

  • Alastair I. M. Rae, Jim Napolitano(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  8 a 3 0 . Because the upper bound can be evaluated using only the ground-state eigenfunction and the difference between the energies of the two lowest states, (10.31) can be usefully applied to the calculation of polarizabilities of other systems, even where the full set of matrix elements and energy differences are not known and expressions such as (10.30) cannot be evaluated. Moreover, as we shall see later in this chapter, the variational principle can often be used to generate a lower bound to α so that by combining both methods, a theoretical estimate can be made with a precision that is rigorously known. 10.2 PERTURBATION THEORY FOR DEGENERATE ENERGY LEVELS The application of perturbation theory to degenerate systems can often lead to powerful insights into the physics of the systems considered. This is because the perturbation often “lifts” the degeneracy of the energy levels, leading to additional structure in the line spectrum. We saw examples of this in Chapter 9 where the inclusion of a spin-orbit coupling term split the previously degenerate energy levels (Figure 9.3) and further splitting resulted from the application of magnetic fields (Figure 9.5). We shall now develop a general method for extending perturbation theory to such degenerate systems. The mathematical reason why we cannot apply the theory in the form developed so far to the degenerate case follows from the fact that, if one or more of the energy levels E 0 k in (10.15) is equal to E 0 n , then at least one of the denominators ( E 0 n - Time-independent Perturbation Theory and the Variational Principle 229 E 0 k ) in (10.15) would be zero, leading to an infinite value for the corresponding term in the series. To understand the reason why this infinity arises, we consider the case of twofold degeneracy and take u 01 and u 02 to be two eigenfunctions of the unperturbed Hamiltonian ˆ H 0 with eigenvalue E 01 .

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  Quantum Mechanics

  A Paradigms Approach

  • David H. McIntyre(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  However, this equation is limited to only the two states comprising the degenerate energies, with the perturbation Hamilto- nian playing the role of the Hamiltonian, while the energy eigenvalue is the first-order correction that we are seeking. We commonly refer to this isolated subspace of the whole system as the degenerate subspace. We solve Eq. (10.107) by the standard procedure of diagonalization for eigenvalue equations, which yields two eigenvalues and two eigenstates. The eigenvalues are the two corrections E (1) 2 to the zeroth-order energy E (0) 2 . If we label the two energy solutions E (1) 2a and E (1) 2b , then the energies correct to first order are E 2a = E (0) 2 + E (1) 2a E 2b = E (0) 2 + E (1) 2b . (10.108) The two sets of a and b coefficients from solving Eq. (10.107) give the two eigenstate solutions @ 2 a 9 and @ 2 b 9 that form a new basis in the degenerate subspace that was originally defined by @ 2 (0) 9 and @ 3 (0) 9. In this new basis, the perturbation Hamiltonian is diagonal. Now let’s generalize our specific solution. The result of the twofold degenerate example was the eigenvalue equation (10.107) for the perturbation Hamiltonian within the degenerate subspace. The perturbation corrections are found by solving that eigenvalue equation through the diagonalization procedure we use throughout quantum mechanics. So there is no silver bullet formula for Degenerate Perturbation Theory. There is just the mantra: Diagonalize the perturbation Hamiltonian in the degenerate subspace. That’s it! Let’s see how it works. Example 10.4 An electron is bound to move on the surface of a sphere. Find the energy correc- tions caused by a perturbing magnetic field B = B 1 x n , limiting your consideration to the first two energy levels. 340 Perturbation Theory The zeroth-order Hamiltonian of a particle on a sphere is the kinetic energy, as we found in Section 7.6: H sphere = L 2 2I .

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  Lectures on Quantum Mechanics

  • Ashok Das(Author)
  • 2012(Publication Date)
  • WSPC

    (Publisher)

  001. This gives a solution which is accurate up to one part in 10 9 ( x = 2 is an exact root.) and shows the power of pertur-bation theory. 12.1 Non-degenerate perturbation We now discuss a perturbation approach for physical systems whose Hamiltonians are independent of time. This is why it is also known as stationary perturbation theory. This method is applicable when the complete Hamiltonian is independent of time and can be written as a sum of two parts H = H 0 + H 1 , (12.16) where H 0 is the Hamiltonian which we are able to diagonalize. In other words, we can determine the eigenstates and the eigenvalues of H 0 . The second term in the Hamiltonian, H 1 , is an additional Hamil-tonian which is assumed to be small. We would see what smallness means later – for the present, we simply note that the matrix elements of H 1 should be smaller than the differences in the energy levels of the Hamiltonian H 0 . Let | n 0 angbracketright denote the eigenstates of H 0 with the eigenvalues E (0) n . Thus, H 0 | n 0 angbracketright = E (0) n | n 0 angbracketright . (12.17) Furthermore, we assume that all the states are discrete and non-degenerate. The degenerate case has to be discussed separately. Be-cause of the perturbing Hamiltonian H 1 , the total Hamiltonian H 326 12 Stationary perturbation theory would now have a new set of eigenstates | n angbracketright with eigenvalues E n . If the perturbing Hamiltonian is small, then these eigenstates and eigenvalues would be very close to the unperturbed states and eigen-values. Thus, we can expand these quantities in powers of the effect of the perturbing Hamiltonian. However, since the Hamiltonian is an operator we cannot use it as an expansion parameter and, for book keeping purposes, we introduce a parameter λ to write H = H 0 + λH 1 .

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  Quantum Processes Systems, and Information

  • Benjamin Schumacher, Michael Westmoreland(Authors)
  • 2010(Publication Date)
  • Cambridge University Press

    (Publisher)

  The degeneracy of the unperturbed situation is due to a symmetry – a kind of rotational symmetry in this case – that is “broken” by the perturbation. We choose our unperturbed basis states with this broken symmetry in mind. Let us consider a more general situation, but suppose that the spectrum of H 0 has a two-fold degeneracy. That is, there is a two-dimensional H 0 eigenspace spanned by states | a and | b for which ε a = ε b = ε . Consider the matrix W = W aa W ab W ba W bb = a | H P | a a | H P | b b | H P | a b | H P | b . (17.18) (This is a submatrix of a matrix representation of H P .) If the | a and | b basis states are the “right” ones, then W is diagonal, and the diagonal elements are the first-order energy corrections W aa = E ( 1 ) a and W bb = E ( 1 ) b . But what if the basis we chose happens to be a “wrong” one? It is still true that the first-order energy corrections are the eigenvalues of the matrix W . We can find these by solving the characteristic equation: 0 = det W aa − E ( 1 ) W ab W ba W bb − E ( 1 ) = ( W aa − E ( 1 ) )( W bb − E ( 1 ) ) − W ab W ba . (17.19) The quadratic formula gives us the two first-order energy corrections: E ( 1 ) ± = 1 2 W aa + W bb ± ( W aa + W bb ) 2 + 4 | W ab | 2 . (17.20) Problems 17.6 and 17.7 explore these ideas in a simple situation with two-fold degeneracy. 17.3 Perturbing the dynamics Once again, we suppose our unperturbed Hamiltonian H 0 is independent of time and has eigenstates | n with energies ε n . Now, however, the perturbation H P is not assumed to be time-independent. Since the total Hamiltonian may now be varying with time, it may not have stationary states and fixed energy levels. In this situation, we will seek to find out how the perturbation affects the time evolution of quantum states of the system. The 354 Perturbation theory perturbation may induce transitions between the unperturbed eigenstates | n – states that would be entirely stationary in the absence of H P ( t ) .

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  Quantum Chemistry

  • John Lowe(Author)
  • 2012(Publication Date)
  • Academic Press

    (Publisher)

  The example we have described in this section illustrates that, even though perturbation theory was derived in a framework of eigenfunctions and eigen- values of an unperturbed hamiltonian, we are not limited to cases where these are known. The Hückel MOs and orbital energies are certainly not eigenfunc- tions and eigenvalues on the strictest sense since they result from variational calculations using a very limited basis and a hamiltonian that cannot be defined so as to enable one to evaluate Η { . However, the HMOs and their orbital energies are eigenvectors and eigenvalues of a matrix equation, and this suffices to make them amenable to treatment by perturbation theory. 12-6 Perturbation Theory for a Degenerate State Equations (12-19), (12-21), and (12-24) have denominators containing Ei — Ej. If ι and are degenerate, this leads to difficulty, alerting us to the fact that our earlier derivations do not take proper account of states of interest that are degenerate. The problem results from our initial expansion of as a power series in . The leading term in the expansion, \ Ό \ is the wavefunction that ι becomes in the limit when = 0. In nondegenerate systems there is no ambiguity; $ 0 ) has to be { . But if 0, is degenerate with 3 ·, any linear combina- tion of them is also an eigenfunction. We need a method to determine how and should be mixed together to form the correct zeroth-order functions \ 0) and ψ\ To find the conditions that \ 0) and $ 0 ) must satisfy, we return to the first- order perturbation equation (12-11) but with $ 0 ) in place of { : {H r - WFW' + {H 0 - EM» = 0 (12-73) Multiplying from the left by $ 0) * and integrating gives (taking $ 0 ) and $ 0 ) to be orthogonal and H Q hermitian) <# 0) |#W>> + &> ~ ^ ^Ί^) = o (12-74)

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  Electronic Structure Modeling

  Connections Between Theory and Software

  • Carl Trindle, Donald Shillady(Authors)
  • 2008(Publication Date)
  • CRC Press

    (Publisher)

  Now recalling that the polarized functions are normalized and inserting the expressions for the coefficients a nm E (2) n ¼ X a C n ab C n ab h c (0) n a j V j c (0) k 0 ih c (0) k 0 j V j c (0) n a i E (0) k 0 E (0) n A simple example of a degenerate reference system is provided by the bending modes of a linear molecule. The displacements of the central atom in the x and y directions, which are symmetry equivalent, define the potential V ¼ 1 2 k ( x 2 þ y 2 ) 1 2 kr 2 The energy levels are E ( n , m ) ¼ ¯h v 0 ( n þ m þ 1) The degeneracy is considerable: for energy of n quanta the index is n . Total energy in quanta 1 2 3 4 States 0,0 1,0; 0,1 2,0; 1,1; 0,2 3,0; 2,1; 1,2; 0,3 A perturbation proportional to a coordinate product Axy has selection rules for each of the two vibrational modes; D n ¼ 1 and D m ¼ 1. Individual integrals for scaled functions are h j j q j j þ 1 i ¼ [( j þ 1) = 2] 1 = 2 The set of degenerate states with energy 2, subjected to this kind of poten-tial, is described by the matrix h 1,0 j Axy j 1,0 i l h 1,0 j Axy j 0,1 i h 0,1 j Axy j 1,0 i h 0,1 j Axy h 0,1 j l ¼ l C C l ¼ 0 Perturbation Theory 195 The roots are A and the combinations are ( j 1,0 i j 0,1 i ) = ffiffi 2 p The selection rules limit mixing of {1,0; 0,1} to the members 2,1 and 1,2 of the degenerate state with energy 4. The second-order energy effect is E (2) 2 j ¼ h 1,0 j Ayx j 2,1 ih 2,1 j Axy j 1 : 0 i 2 h 0,1 j Ayx j 1,2 ih 2,1 j Axy j 1 : 0 i 2 The value of the second-order stabilization is A 2 . Perturbation Theory in Approximate MO Theory Perturbation theory is invaluable in chemical spectroscopy since the field effects are typically small, but is also valuable as a means of discussions of interactions and structures of molecules even when the central presump-tions of the theory, that only small disturbances are present, cannot be defended. The argument is based on some simple outcomes of perturbation theory already developed.

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  Lectures on Quantum Mechanics

  • Steven Weinberg(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

  (5.1.8) A problem arises in the case of degeneracy. Suppose there are two states  b  =  a for which E b = E a . Then Eq. (5.1.8) is inconsistent unless   b , δ H  a  vanishes, which need not be the case. But we can always avoid this problem by a judicious choice of the degenerate unperturbed states. Suppose there are a number of states  a1 ,  a2 , etc., all with the same energy E a . The quantities   ar , δ H  as  form an Hermitian matrix, so according to a general theorem of matrix algebra the vector space on which this matrix acts is spanned by a set of orthonormal eigenvectors u rn of this matrix, such that  r   as , δ H  ar  u rn =  n u sn . (5.1.9) 5.1 First-Order Perturbation Theory 171 (See footnote 7 in Section 3.3.) We can define eigenstates of H 0 with the same energy E a :  an ≡  r u rn  ar , (5.1.10) for which   am , δ H  an  =  rs u ∗ sm u rn   as , δ H  ar  =  s u ∗ sm u sn  n = δ nm  n , (5.1.11) in which we have used the orthonormality relation ∑ s u ∗ sm u sn = δ nm . For these states the off-diagonal matrix elements of the perturbation all vanish, so we avoid the problem of inconsistency with Eq. (5.1.8) if we start with the s instead of the  s. If we stubbornly insist on taking one of the  ar as our unperturbed state, where some   as , δ H  ar  for s  = r do not vanish, then perturbation theory doesn’t work; even a tiny perturbation causes a very large change in the state vector. For instance, suppose that H 0 is rotationally invariant, and we add a per- turbation δ H =  · v, where v is some vector operator. As we saw in the previous chapter, because H 0 is rotationally invariant, there are 2 j +1 states with the same unperturbed energy and the same eigenvalue  2 j ( j +1) of J 2 . If our unperturbed state is an eigenstate of J 3 , but  is not in the 3-direction, then no matter how small  is, there will be a large correction to the state vector.

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  Heisenberg's Quantum Mechanics

  • Mohsen Razavy(Author)
  • 2011(Publication Date)
  • World Scientific

    (Publisher)

  Degenerate Perturbation Theory 323 m = -1 respectively. Since the perturbation potential in this case is V ( z ) = e E z and this perturbation commutes with L z , [ L z , e E z ] = 0 , (11.106) therefore L z is a conserved quantity. Taking the matrix element of this commu-tator between h 2 P -1 | and | 2 P 1 i we obtain h 2 P -1 | [ e E z, L z ] | 2 P 1 i = h 2 P -1 | e E z ( L z | 2 P 1 i ) -( h 2 P -1 | L z ) e E z | 2 P 1 i = 2¯ h h 2 P -1 | e E z | 2 P 1 i = 0 . (11.107) If we examine other matrix elements of V = e E z we find that the only nonva-nishing off-diagonal matrix elements are h 2 S 0 | e E z | 2 P 0 i and h 2 P 0 | e E z | 2 S 0 i , i.e. the two states with m = 0. These matrix elements are equal and their value can be evaluated with the help of the hydrogen atom wave function h 2 S 0 | e E z | 2 P 0 i = -3 e E a 0 , (11.108) where a 0 is the Bohr radius. For the H -atom because of the symmetry under the parity operation the diagonal elements of V are zero. Thus the matrix (11.99) reduces to 0 -3 e E a 0 -3 e E a 0 0 . (11.109) For instance if the H atom which is originally in the | 2 S 0 i state is placed in a uniform electric field then the spectral line will split and we get two perturbed eigenstates 1 √ 2 ( | 2 S 0 i + | 2 P 0 i ) , and 1 √ 2 ( | 2 S 0 i -| 2 P 0 i ) , (11.110) and the energies of these two states are -e 2 2 a 0 1 4 + 6 E e a 2 0 ! , and -e 2 2 a 0 1 4 -6 E e a 2 0 ! , (11.111) respectively. 11.3 Almost Degenerate Perturbation Theory Now let us consider the effect of a small perturbation on almost degenerate levels. By “almost” we mean that the level spacing E (0) 2 -E (0) 1 for the two levels 324 Heisenberg’s Quantum Mechanics 1.2 1.1 1 0 0.2 0.4 0.6 0.8 1 0.9 0.8 Figure 11.2: Plot of the energy levels E + and E -as functions of the strength of the perturbation λ . 1 and 2 is comparable with the energy shift produced by the perturbation.

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  Quantum Mechanics

  • Richard Fitzpatrick(Author)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Chapter 7

  Time-Independent Perturbation Theory

  7.1Introduction

  We have developed techniques by which the general energy eigenvalue problem can be reduced to a set of coupled partial differential equations involving various wavefunctions. Unfortunately, the number of such problems that yield exactly soluble systems of equations is comparatively small. It is, therefore, necessary to develop techniques for finding approximate solutions to otherwise intractable problems.

  Consider the following problem, which is very common. The Hamiltonian of some quantum mechanical system is written

  Here, H 0 is a simple Hamiltonian for which we know the exact eigenvalues and eigenstates. H 1 introduces some interesting additional physics into the problem, but it is sufficiently complicated that, when we add it to H 0 , we can no longer find the exact energy eigenvalues and eigenstates. However, H 1 can, in some sense (which we shall specify more exactly later on), be regarded as small compared to H 0 . Let us try to find approximate eigenvalues and eigenstates of the modified Hamiltonian, H 0 + H 1 , by performing a perturbation expansion about the eigenvalues and eigenstates of the original Hamiltonian, H 0 .

  We shall start, in this chapter, by considering time-independent perturbation theory [Schrödinger (1926a)], in which the modification to the Hamiltonian, H 1 , has no explicit dependence on time. It is usually assumed that the unperturbed Hamiltonian, H 0 , is also time independent.

  7.2Two-State System

  Consider the simplest non-trivial system, in which there are only two independent eigenkets of the unperturbed Hamiltonian. These are denoted

  It is assumed that these states, and their associated eigenvalues, are known. Because H 0

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  Solid State Physics

  Essential Concepts

  • David W. Snoke(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  One way to treat a strongly interacting system is to find the proper eigenstates of the whole system. Often, however, it is not possible to find the exact eigenstates, because the mathematics are too complicated. Many-body perturbation theory gives us methods of saying meaningful things about the system even when we do not know the exact eigenstates. 8.1 Higher-Order Time-Dependent Perturbation Theory In Section 4.7, we deduced a general result of time-dependent perturbation theory, |ψ (t) = S(t, 0)|ψ (0) = e −(i/)  t t 0 V int (t  )dt  / |ψ (0) (8.1.1) =  1 + 1 i  t 0 dt  V int (t  ) +  1 i  2  t 0 dt   t  0 dt  V int (t  )V int (t  ) + · · ·  |ψ (0), where |ψ (t) is the state of the system at time t in the interaction representation. (The full state at time t as determined by the Schrödinger equation is written in our notation as 426 427 8.1 Higher-Order Time-Dependent Perturbation Theory |ψ t , and is related to the interaction representation by the relation |ψ t  = e −iH 0 t/ |ψ (t).) We used the result (8.1.1) to deduce Fermi’s golden rule, restricting our attention to the first-order term. Suppose that we want to include higher orders of perturbation theory. Once again, fol- lowing the dicussion of Section 4.7, we assume that the Hamiltonian consists of a main part H 0 plus a perturbation term V int , H = H 0 + V int , (8.1.2) in which we assume that the eigenstates of H 0 are known, defined by H 0 |n = E n |n. Recalling that V int (t) = e iH 0 t/ V int e −iH 0 t/ , and writing the initial state at t 0 = 0 as |i, we then have n|ψ (t) = n|i + 1 i  t 0 n|V int |ie (i/)(E n −E i )t  dt  +  1 i  2  t 0 dt   t  0 dt   m n|V int |mm|V int |i × e (i/)(E n −E m )t  e (i/)(E m −E i )t  + · · · . (8.1.3) Here we have inserted a sum over the complete set of eigenstates, ∑ |mm| = 1. Depletion of the initial state. The case when |n = |i requires special attention.

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  Introduction to Quantum Mechanics

  • David J. Griffiths, Darrell F. Schroeter(Authors)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  When this happens (and it’s not uncommon), you need to use second-order Degenerate Perturbation Theory (see Problem 7.40). Problem 7.40 If it happens that the square root in Equation 7.33 vanishes, then E 1 + = E 1 − ; the degeneracy is not lifted at first order. In this case, diagonalizing the W matrix puts no restriction on α and β and you still don’t know what the “good” states are. If you need to determine the “good” states—for example to calculate higher-order corrections—you need to use second-order Degenerate Perturbation Theory. (a) Show that, for the two-fold degeneracy studied in Section 7.2.1, the first-order correction to the wave function in Degenerate Perturbation Theory is ψ 1 =  m =a,b α V ma + β V mb E 0 − E 0 m ψ 0 m . (b) Consider the terms of order λ 2 (corresponding to Equation 7.8 in the nondegenerate case) to show that α and β are determined by finding the eigenvectors of the matrix W 2 (the superscript denotes second order, not W squared) where  W 2  i j =  m =a,b  ψ 0 i   H    ψ 0 m  ψ 0 m   H    ψ 0 j  E 0 − E 0 m and that the eigenvalues of this matrix correspond to the second-order energies E 2 . (c) Show that second-order Degenerate Perturbation Theory, developed in (b), gives the correct energies to second order for the three-state Hamiltonian in Problem 7.39. ∗∗ Problem 7.41 A free particle of mass m is confined to a ring of circumference L such that ψ(x + L ) = ψ(x ). The unperturbed Hamiltonian is 318 CHAPTER 7 Time-Independent Perturbation Theory H 0 = −  2 2 m d 2 dx 2 , to which we add a perturbation H  = V 0 cos  2 π x L  . (a) Show that the unperturbed states may be written ψ 0 n (x ) = 1 √ L e i 2 π n x /L for n = 0, ±1, ±2 and that, apart from n = 0, all of these states are two-fold degenerate. (b) Find a general expression for the matrix elements of the perturbation: H  mn =  ψ 0 m    H     ψ 0 n  . (c) Consider the degenerate pair of states with n = ±1.

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