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arXiv:quant-ph/9606011

Algebraic Formulation of the Operatorial Perturbation Theory. Part I

Ary W. Espinosa Müller, Adelio Matamala-V�squez

Published Jun 12, 1996• 0 citations
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Abstract

A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his formalism. In this regard, it has been introduced a new set of superoperators featured by their algebraic structure. Also, two lemmas and one theorem have been developed in order to algebraically reformulate the theory on more rigorous grounds. Finally, we have been able to build a coherent and self-contained formalism independent on any matricial representation , removing in this way the degeneracy problem .

DOI
10.48550/arxiv.quant-ph/9606011

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Algebraic Formulation of the Operatorial Perturbation Theory. Part 2. Aplications

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The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of $\hat{a}^\dagger$, $\hat{a}$, $\hat{1}$ operators, the second one has been based in the angular momemtum algebra of $\hat{L}_+$, $\hat{L}_-$ and $\hat{L}_0$ operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.

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Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.

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Formal perturbation theory

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Background There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development , carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development , where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory. Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways. Classical derivation of Rayleigh–Schrödinger perturbation theory The perturbation Ansatz We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).

Cambridge University Press eBooks•Aug 6, 2009•0 citations

Self-Adjoint Ladder Operators. III

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Nondegenerate perturbation theory: A solution to the eigenvalue problem by the adiabatic-theorem formalism

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A nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors \ensuremath{\Vert}\ensuremath{\alpha}〉 and energies ${E}_{\ensuremath{\alpha}}$ which are analytically expressed with respect to two matrix series ${\mathrm{scrV}}^{(n)}$ and ${E}^{(n)}$. The nth term of each series (${\mathrm{scrV}}^{(n)}$ and ${E}^{(n)}$) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence, ${\mathrm{scrV}}^{(n)}$=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$), ${E}^{(n)}$=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.

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Physical Review A•Jan 1, 1991•87 citations
A new totally algebraic formalism based on general, abstract ladder operators has been proposed. This approach heavily grounds in the superoperator formalism of Primas. However it is necessary to introduce many improvements in his formalism. In this regard, it has been introduced a new set of superoperators featured by their algebraic structure. Also, two lemmas and one theorem have been developed in order to algebraically reformulate the theory on more rigorous grounds. Finally, we have been able to build a coherent and self-contained formalism independent on any matricial representation , removing in this way the degeneracy problem .
The algebraic approach to operator perturbation method has been applied to two quantum--mechanical systems ``The Stark Effect in the Harmonic Oscillator'' and ``The Generalized Zeeman Effect''. To that end, two realizations of the superoperators involved in the formalism have been carried out. The first of them has been based on the Heisenberg--Dirac algebra of a^†\hat{a}^\daggera^†, a^\hat{a}a^, 1^\hat{1}1^ operators, the second one has been based in the angular momemtum algebra of L^+\hat{L}_+L^+​, L^−\hat{L}_-L^−​ and L^0\hat{L}_0L^0​ operators. The successful results achieved in predicting the discrete spectra of both systems have put in evidence the reliability and accuracy of the theory.
Abstract We have already mentioned that Schriöidinger’s formulation of quantum mechanics was slightly preceded by a totally different algebraic formulation proposed by Heisenberg. Although Schriöidinger’s methods were so much simpler that they were generally preferred, both approaches seemed to give the same answers, so it was natural to ask how they were related. This was particularly important because some features of quantum theory were easier to understand in Heisenberg’s formalism. In fact, as we have already noted, Schriöidinger was able to show that the two methods were essentially equivalent. Soon mathematicians like Hermann Weyland John von Neumann were able to find a general mathematical scheme that encompassed both approaches, and it is this that we shall now describe.
Background There are two stages in the study of perturbation theory and related techniques (although they are mixed intimately in most derivations in the literature). The first is the formal development , carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave function), without attempt to express anything in terms of one- and two-body quantities (components of Ĥ, orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of Ĥ. The second is the many-body development , where all expressions are obtained in terms of orbitals (one-electron states) and oneand two-electron integrals. We shall try to keep these separate for a while and begin with a consideration of formal perturbation theory. Another aspect of the study of many-body techniques is the large variety of approaches, notations and derivations that have been used. Each different approach has contributed to the lore and the language of many-body theory, and each tends to illuminate some aspects better than the other approaches. If we want to be able to read the literature in this field, we should be familiar with several alternative formulations. Therefore, we shall occasionally derive some results in more than one way and, in particular, we shall derive the basic perturbation-theory equations and their many-body representations in several complementary ways. Classical derivation of Rayleigh–Schrödinger perturbation theory The perturbation Ansatz We begin with a classical textbook derivation of formal Rayleigh–Schrödinger perturbation theory (RSPT).
The method of self-adjoint ladder operators, developed in Parts I and II, is applied to the solution of the generalized angular-momentum problem. This reveals many interesting aspects of this approach to eigenvalue problems and, in particular, its relationship to addition of angular momentum. The complete set of irreducible unitary representations of the underlying algebra is obtained and also the corresponding Clebsch-Gordan (Wigner) coefficients for the addition of spin and angular momentum in a space of arbitrary dimension.
A nondegenerate perturbation theory is studied using the adiabatic-theorem formalism. A new factorization of the adiabatic time-evolution operator analogous to that obtained by Morita in his derivation of the linked-cluster theorem is given by carrying out a series of intermediate representations. This leads to a new expansion for the perturbed eigenvectors \ensuremath{\Vert}\ensuremath{\alpha}〉 and energies ${E}_{\ensuremath{\alpha}}$ which are analytically expressed with respect to two matrix series scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n). The nth term of each series (scrV(n){\mathrm{scrV}}^{(n)}scrV(n) and E(n){E}^{(n)}E(n)) includes perturbational contributions of power p, p\ensuremath{\ge}n. An explicit recurrence, scrV(n){\mathrm{scrV}}^{(n)}scrV(n)=V(${\mathrm{scrV}}^{n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$), E(n){E}^{(n)}E(n)=E(${\mathrm{scrV}}^{(n\mathrm{\ensuremath{-}}1)}$, ${E}^{(n\mathrm{\ensuremath{-}}1)}$) is derived and allows us to calculate these matrices to an arbitrary iteration order (n). This is a central feature of our approach since the eigenvectors and eigenvalues can be generated, to this arbitrary order, by a simple numerical program.
The Wigner-Weyl (WW) phase-space formulation of quantum mechanics is discussed within the Liouville-space formalism, where quantum operators A^ are viewed as vectors, represented by L kets \ensuremath{\Vert}A^>>, on which act ``superoperators''; the scalar product is A^\ensuremath{\Vert}B^>>=TrA${\mathrm{^}}^{\mathrm{\ifmmode^\circ\else\textdegree\fi{}}}$B^. With every operator A^, we associate commutation and anticommutation superoperators A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and A${\mathrm{^}}^{+}$, defined by their actions on any operator B^ as A${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$B^=${\mathrm{\ensuremath{\Elzxh}}}^{\mathrm{\ensuremath{-}}1}$[A^,B^], A${\mathrm{^}}^{+}$B^=1/2(A^B^+B^A^). The WW representation corresponds to the choice of a special basis in Liouville space, namely, the eigenbasis of the position and momentum anticommutation superoperators q${\mathrm{^}}^{+}$ and p${\mathrm{^}}^{+}$ (where [q^,p^]=i\ensuremath{\Elzxh}). These, together with the commutation superoperators q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$ and p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$, form a canonical set of superoperators, [q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$]=[q${\mathrm{^}}^{\mathrm{\ensuremath{-}}}$,p${\mathrm{^}}^{+}$]=i (the other commutators vanishing), as functions of which all other super- operators can be expressed. Weyl ordering is expressed as f(q^,p^)Weyl{)}_{\mathrm{Weyl}})Weyl​ ordering=f(q${\mathrm{^}}^{+}$,p${\mathrm{^}}^{+}$)1^. A generalization of Ehrenfest's theorem is obtained.