Physics
Perturbation Theory
Perturbation theory is a mathematical method used to approximate solutions to problems that cannot be solved exactly. It involves breaking down a complex problem into simpler parts and then adding small adjustments to the solution to account for the effects of the more complex parts. This method is commonly used in quantum mechanics and other areas of physics.
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10 Key excerpts on "Perturbation Theory"
eBook - PDF- William Paulsen(Author)
- 2013(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 7 Perturbation Theory Perturbation Theory is a collection of methods used to find the approximate solution to a problem for which the exact solution cannot be solved in closed form. The strategy is to introduce a parameter into the problem, so that the problem is solvable when = 0, or as approaches 0. Although we have made the problem more complicated by adding an additional parameter, we can determine a solution to the original problem as a power series in . y = f 0 ( x ) + f 1 ( x ) + 2 f 2 ( x ) + · · · . Perturbation Theory originated as a way to tackle the three body problem in celestial mechanics. Using Newtonian mechanics, the Earth would revolve about the sun in a perfect ellipse if it weren’t for the moon and other planets. However, the moon causes the motion of the Earth to be perturbed , so that the Earth wiggles slightly as it goes around the sun. Other planets also affect the Earth’s orbit by a tiny amount. Likewise, the moon’s orbit around the Earth is perturbed by the presence of the sun. Using Perturbation Theory, we could accurately approximate the deviation from the elliptical orbit, and this variance was verified by observation. Since then, Perturbation Theory has been used for algebraic equations, differential equations, and quantum mechanics. The techniques can be used in a variety of problems in applied mathematics. 7.1 Introduction to Perturbation Theory In order to introduce the technique of Perturbation Theory, let us consider solving polynomial equations. Example 7.1 Estimate the roots of the equation x 3 -1 . 01 x + . 03 = 0 . S OLUTION : This is currently not a perturbation problem, since there is no small parameter. However, it is clear that the problem is close to another 317 318 Asymptotic Analysis and Perturbation Theory problem that can be solved fairly easily: x 3 -x = 0. Thus, we can consider the polynomial equation to be a special case of a one-parameter family of polynomial equations x 3 -(1 + ) x + 3 = 0 .
- Francisco M. Fernandez(Author)
- 2000(Publication Date)
- CRC Press(Publisher)
Chapter 9 Perturbation Theory in Classical Mechanics 9.1 Introduction Although the title of this book refers only to quantum mechanics, in this chapter we outline the application of Perturbation Theory in classical mechanics. It is not our purpose to give a thorough account of the subject that may be found in other books [179, 180], but simply to show that some of the approaches developed in preceding chapters for quantum systems are suitable for classical ones with just slight modifications. We hope that this fact may facilitate a unified teaching of perturbation methods in undergraduate and graduate courses. We first consider the simplest perturbation expan-sion that applies when the amplitude of the motion is sufficiently small. This approach is based on the Taylor expansion of the nonlinear force (or the anharmonic potential-energy function) around the origin and resembles the polynomial approximation in quantum mechanics discussed in Chapter 7. We obtain the perturbation series for the trajectory of a particle moving in a one-dimensional space under the effect of an arbitrary nonlinear conservative force, and the perturbation series for the period of the motion. Straightforward integration of the perturbation equations gives rise to secular terms that one easily removes by appropriately scaling the frequency. We choose the simple pendulum as an illustrative example. Most of that discussion is based on an appropriate modification and adaptation of a recent pedagogical article on the subject [181]. Later, we concentrate on Hamilton’s equations of motion that allow the development of pertur-bation theory in operator form that is reminiscent of the interaction picture in quantum mechanics, already discussed in Section 1.3.1. We explicitly consider secular and canonical perturbation theories using one-dimensional anharmonic oscillators as illustrative examples.
eBook - PDFElectronic Structure Modeling
Connections Between Theory and Software
- Carl Trindle, Donald Shillady(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
8 Perturbation Theory It is often the case that we can consider properties of molecules to be defined by their response to a disturbance. For example, the dipole moment of a molecule is defined by a change in energy of the system as an electric field is applied. Much of spectroscopy is simply the study of responses of a molecu-lar system to a periodic perturbation provided by external fields. Scattering experiments similarly can be considered the study of the effects of molecu-lar collisions, momentary disturbances of the reference systems. Perturb-ation theory permits a systematic representation of the effects of small disturbances on a reference system, and is the natural language for many molecular phenomena. First-Order Correction to a Nondegenerate Reference System We may seek solution for H c ¼ E c . Assume we can solve a similar reference system H 0 c 0 n ¼ E 0 n c 0 n for which the Hamiltonian H 0 closely resembles H . H ¼ H (0) þ l V where lim l ! 0 H ¼ H 0 Here the term V is the perturbation. The key idea of Perturbation Theory is to represent the unknown wave function as the reference system’s wave func-tion with a series of corrections. The wave function is then c n ¼ c (0) n þ lc (1) n þ l 2 c (2) n þ l 3 c (3) n þ where again lim l ! 0 c n ¼ c (0) n Furthermore, the associated eigenvalues are each represented as the refer-ence system’s energies with a sequence of corrections. E n ¼ E (0) n þ l E (1) n þ l 2 E (2) n þ l 3 E (3) n þ 187 Note that, for instance, in l 3 E (3) n 3 is the power of l and in E (3) n 3 refers to the third-order correction to the n th energy value.
eBook - PDF- D. J. Thouless, H. S. W. Massey, Keith A. Brueckner(Authors)
- 2013(Publication Date)
- Academic Press(Publisher)
IV. Perturbation Theory 1. General discussion In principle, perturbation methods enable us to start with a soluble problem and to proceed systematically to the solution of a physical problem. In many-body problems, it is often convenient to start, with the problem of noninteracting particles, and to regard the interactions as a perturbation. Although it is quite easy to write down a series ex-pansion for the effect of the perturbation, it is usually very difficult to evaluate more than two or three of the terms in the expansion, or to determine whether or not the series is convergent. The usefulness of an expansion depends not so much on whether it is convergent, but on whether the leading terms give a good approximation. An example of an expansion in which the behaviour of the first few terms is misleading has been given by Bethe (1956). The Brillouin-Wigner perturbation expansion is usually regarded as an improvement on the Rayleigh-Schrödinger perturbation expansion for a one-body problem, but it is quite unsuitable for a many-body problem. If the unperturbed Hamiltonian is H 0 , with eigenstate | 0 ) and eigenvalue Wo, the true Hamiltonian H = H 0 + Hi has an eigenstate ψο) with eigenvalue E 0 , where |*o> = | ) + -, P „ffi|*o), (4.1) &0 — £10 and E 0 = Wo+ < |# |* >. (4.2) The operator P in Eq. (4.1) is the projection operator which projects off the state | ), given by P = 1 - | )< |. (4.3) The state | 0 ) is normalized to unity, but | Ψ 0 ) is normalized so that its scalar product with | 0 ) is unity. It can easily be seen that the equation (Ho -ΤΓο) | > = 0 (4.4) 33 34 IV. Perturbation Theory can be combined with Eqs. (4.1) and (4.2) to give (H - Eo) |*o> = (#o + Ηχ- Eo) |*„> = 0. (4.5) The expansion of Eq. (4.2) which is obtained by iterating Eq. (4.1) is Eo = Wo + < |# | > + < |# -, P „ ΗιΦ») + · · · , (4.6) JOJQ — /zo and this is the Brillouin-Wigner expansion.
eBook - PDF- John H. Lowenstein(Author)
- 2012(Publication Date)
- Cambridge University Press(Publisher)
4 Canonical Perturbation Theory Canonical Perturbation Theory provides a systematic pathway for going beyond the highly constrained world of complete integrability while retaining the benefits of the canonical formalism. The use of such techniques in celestial mechanics has led to an impressive level of predictability in the motion of massive bodies in the Solar System, and an ability, aided by enormously powerful computers, to simu- late the history of that system billions of years into the past. Such techniques also have important applications in atomic and molecular physics, notably in the semi- classical regime, where the non-zero size of Planck’s quantum of action can safely be ignored. 1 4.1 General approach The perturbative approach begins with approximating a given system by an integrable one described by action-angle coordinates (θ 1 , . . . , θ n , J 1 , . . . , J n ) on n-dimensional tori, with Hamiltonian H 0 ( J ). We think of the original system as inhabiting the same phase space, with a Hamiltonian of the form H (θ, J ) = H 0 ( J ) + H (θ, J ), θ = (θ 1 , . . . , θ n ), J = ( J 1 , . . . , J n ), In the perturbative approach, we solve the equations of motion for θ(t ) and J (t ) as formal power series in the parameter up to some desired order. To do this, make a succession of canonical transformations, (θ, J ) = (θ (0) , J (0) ) → (θ (1) , J (1) ) → (θ (2) , J (2) ) → · · · , 1 One could argue that the most notable contribution of canonical Perturbation Theory was its unexpected failure, within the framework of Bohr–Sommerfeld quantization, to account for the observed energy levels of the helium atom. According to Heisenberg [19], this was one of the motivating factors in his decision to introduce the radically different theoretical framework of quantum mechanics. Ironically, a more recent version of semi- classical quantization [20] doesn’t do such a bad job of accounting for the helium spectrum! 97
eBook - PDF- Mohsen Razavy(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
Chapter 11 Perturbation Theory In their pioneering work on matrix mechanics, Born, Heisenberg and Jordan among other original contributions also developed a systematic method of ap-proximate calculation of the eigenvalues based on Perturbation Theory [1]. In “three men’s paper” this Perturbation Theory was formulated in the following way: Let us write the Hamiltonian of the system which is not explicitly time-dependent as H = H 0 ( p, q ) + λH 1 ( p, q ) + λ 2 H 2 ( p, q ) + · · · , (11.1) where λ is a small dimensionless parameter. We assume that the solution for the unperturbed Hamiltonian H 0 is known and we use a representation in which H 0 ( p, q ) is diagonal, i.e. h n | H 0 ( p 0 , q 0 ) | j i = E (0) j δ nj (11.2) where p 0 and q 0 are matrices which make H 0 diagonal. Here p 0 = lim p and q 0 = lim p as λ → 0. In order to diagonalize H ( p, q ) we choose a unitary transformation U such that p = Up 0 U -1 , and q = Uq 0 U -1 , (11.3) and then the Hamiltonian H ( p, q ) = UH ( p 0 , q 0 ) U -1 , (11.4) becomes a diagonal matrix h n | H | j i = E j δ nj . (11.5) 309 310 Heisenberg’s Quantum Mechanics To determine the form of the transformation, U , we write it as a power series in λ U = 1 + λU 1 + λ 2 U 2 + · · · , (11.6) with its inverse given by U -1 = 1 -λU 1 + λ 2 ( U 2 1 -U 2 ) + · · · . (11.7) Next we substitute (11.1), (11.6) and (11.7) in (11.4) and equate different powers of λ and we obtain the following set of equations: h n | H 0 ( p 0 , q 0 ) | j i = E (0) j δ nj , (11.8) h n | U 1 H 0 -H 0 U 1 + H 1 | j i = E (1) j δ nj , (11.9) h n | U 2 H 0 -H 0 U 2 + H 0 U 2 1 -U 1 H 0 U 1 + U 1 H 1 -H 1 U 1 + H 2 | j i = E (2) j δ nj , (11.10) · · · · · · · · · · · · · · · · · · · · · · · · · · · h n | U r H 0 -H 0 U r + F r ( H 0 · · · H r , U 0 · · · U r -1 ) | j i = E ( n ) j δ nj . (11.11) In these relations all of the operators are functions of p 0 and q 0 .
eBook - PDFMany-Body Methods in Chemistry and Physics
MBPT and Coupled-Cluster Theory
- Isaiah Shavitt, Rodney J. Bartlett(Authors)
- 2009(Publication Date)
- Cambridge University Press(Publisher)
2 Formal Perturbation Theory 2.1 Background There are two stages in the study of Perturbation Theory and related tech- niques (although they are mixed intimately in most derivations in the lit- erature). The first is the formal development, carried out in terms of the total Hamiltonian and total wave function (and total zero-order wave func- tion), without attempt to express anything in terms of one- and two-body quantities (components of ˆ H , orbitals, integrals over orbitals etc.). We can make a considerable amount of progress in this way before considering the detailed form of ˆ H . The second is the many-body development, where all expressions are obtained in terms of orbitals (one-electron states) and one- and 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. 2.2 Classical derivation of Rayleigh–Schr¨odinger Perturbation Theory 2.2.1 The perturbation Ansatz We begin with a classical textbook derivation of formal Rayleigh–Schr¨ odinger Perturbation Theory (RSPT). We separate the Hamiltonian into a zero-order 18 2.2 Classical derivation of RSPT 19 part and a perturbation, ˆ H = ˆ H 0 + ˆ V .
eBook - PDF- Nivaldo A. Lemos(Author)
- 2018(Publication Date)
- Cambridge University Press(Publisher)
10 Hamiltonian Perturbation Theory C’est par la logique qu’on d´ emontre, c’est par l’intuition qu’on invente. Henri Poincar´ e, Science et M´ ethode Given that non-integrability is the rule, rare are the mechanical systems whose equations of motion can be completely solved in closed form. Nevertheless, many mechanical systems of great importance, especially in celestial mechanics, differ little from systems we know how to solve exactly. Among these, the quasi-integrable systems stand out, which are characterised by a Hamiltonian that consists in an integrable Hamiltonian plus a “small” perturbation. In such cases, from the known solution of the integrable problem, one seeks the solution of the quasi-integrable problem by a scheme of successive approximations. This chapter is a brief introduction to what Poincar´ e considered to be the “probl` eme g´ en´ eral de la dynamique”: the Perturbation Theory of quasi-integrable Hamiltonian systems. 10.1 Statement of the Problem Perturbative methods have been largely employed in celestial mechanics ever since the investigations of pioneers such as Lagrange and Laplace. A spectacular triumph of Perturbation Theory was the discovery of the planet Neptune, in 1846, from deviations of the expected orbit of Uranus calculated independently by Urbain Le Verrier and John Couch Adams. Let us start with two simple examples of mechanical systems that differ little from a corresponding integrable system. Example 10.1 Suppose one wishes to study approximately the influence exerted by Jupiter on the Earth’s motion around the Sun. In a restricted version of the three-body problem, let us make use of an XYZ reference frame with origin at the centre of the Sun. We assume that Jupiter describes a circular orbit about the Sun and that the Earth (of mass m) moves in the same XY-plane of Jupiter’s orbit (see Fig. 10.1).
eBook - PDFOrdinary Differential Equations
Introduction and Qualitative Theory, Third Edition
- Jane Cronin(Author)
- 2007(Publication Date)
- CRC Press(Publisher)
Chapter 7 Perturbation Theory: The Poincar´ e Method Introduction Chapters 7, 8, and 9 are all concerned with the same problem, which can be stated roughly as follows: Given a differential equation dx dt = F ( t , x ,ε ) (7.1) where x is an n -vector, F has period T ( ε ) in t , where T ( ε ) is a differentiable function of ε , and ε is a real parameter such that | ε | is small; suppose that for ε = 0, system (7.1) has a soluton ¯ x ( t ) of period T (0). Then does (7.1) have a solution of period T ( ε ) for all | ε | and is this periodic soluton “near” the given periodic solution ¯ x ( t )? Equation (7.1) is said to be unperturbed if ε = 0 and perturbed if ε = 0. Thus our objective is to use knowledge of the unperturbed equation to study the perturbed equation. We shall be concerned primarily with establishing the existence of periodic solu-tions and investigating their stability. Students who have already encountered compu-tational perturbation methods may have doubts about all this emphasis on existence. Why not just use such a method and just go ahead and compute? Unfortunately if one “just computes,” complications arise even with simple equations. For an enlightening discussion with examples, see Greenberg [1978, Chapter 25]. The problem we will study is very old and very important; old because it has been regarded as a serious problem for centuries in celestial mechanics; very important because it has arisen in disparate subjects, especially since the beginning of the twentieth century. The earlier work, inspired by celestial mechanics, was taken over for use in electrical engineering, in particular radio curcuits (see Andronov and Chaikin [1949]). Later it was used in control theory and most recently in biology (see Murray [2003], Keener and Sneyd [1998]). We will use a method of Poincar´ e for dealing with this problem.
- Stanley I. Sandler(Author)
- 2015(Publication Date)
- Wiley(Publisher)
Chapter 14 Perturbation Theory As should be evident from the previous chapters, considerable effort is involved in obtaining the values of the thermodynamic properties and the radial distribution function for a chosen interaction potential at a single temperature and density, regard- less of whether an integral equation method or computer simulation is used. Also, either of these methods only results in numerical values of these properties at the chosen temperature and density; they do not provide an analytical equation for use in calculations with other interaction potentials or at other state points. One method of extending the usefulness of the thermodynamic properties and radial distributions functions that have been obtained for one interaction potential (which we call the reference potential) for use with a different potential is by using Perturbation Theory , wherein one does a Taylor series expansion of a thermodynamic property or the radial distribution function in the difference between the new potential of interest and the reference potential. An introduction to this method is the subject of this chapter. INSTRUCTIONAL OBJECTIVES FOR CHAPTER 14 The goals for this chapter are for the student to: • Understand Perturbation Theory using the hard-sphere as the reference potential • Understand Perturbation Theory for other reference fluids • See how Perturbation Theory is used to develop thermodynamic models of use in chemistry and chemical engineering 14.1 Perturbation Theory FOR THE SQUARE-WELL POTENTIAL As one can see from the previous sections, the calculation of the thermodynamic properties and radial distribution function for a liquid is a difficult task. One case where we do know the thermodynamic properties with reasonable accuracy is the hard-sphere fluid.
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