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Perturbation Theory and Exact Solutions PERTURBATION THEORY AND EXACT SOLUTIONS by J J. LODDER R|nhtdnn Report 76~96 DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS J J. LODOER ASSOCIATIE EURATOM-FOM Jun»»76 FOM-INST1TUUT VOOR PLASMAFYSICA RUNHUIZEN - JUTPHAAS - NEDERLAND DISSIPATIVE MOTION PERTURBATION THEORY AND EXACT SOLUTIONS by JJ LODDER R^nhuizen Report 76-95 Thisworkwat performed at part of th«r«Mvchprogmmncof thcHMCiattofiafrccmentof EnratoniOTd th« Stichting voor FundtmenteelOiutereoek der Matctk" (FOM) wtihnnmcWMppoft from the Nederhmdie Organiutic voor Zuiver Wetemchap- pcigk Onderzoek (ZWO) and Evntom It it abo pabHtfMd w a the* of Ac Univenrty of Utrecht CONTENTS page SUMMARY iii I. INTRODUCTION 1 II. GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNC­ TIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS 1. Introduction 4 2. Discontinuous test functions 5 3. Differentiation 7 4. Powers of x. The partie finie 10 5. Fourier transforms 16 6. Laplace transforms 20 7. Hubert transforms 20 8. Dispersion relations 21 III. PERTURBATION THEORY 1. Introduction 24 2. Arbitrary potential, momentum coupling 24 3. Dissipative equation of motion 31 4. Expectation values 32 5. Matrix elements, transition probabilities 33 6. Harmonic oscillator 36 7. Classical mechanics and quantum corrections 36 8. Discussion of the Pu strength function 38 IV. EXACTLY SOLVABLE MODELS FOR DISSIPATIVE MOTION 1. Introduction 40 2. General quadratic Kami1tonians 41 3. Differential equations 46 4. Classical mechanics and quantum corrections 49 5. Equation of motion for observables 51 V. SPECIAL QUADRATIC HAMILTONIANS 1. Introduction 53 2. Hamiltcnians with coordinate coupling 53 3. Double coordinate coupled Hamiltonians 62 4. Symmetric Hamiltonians 63 i page VI. DISCUSSION 1. Introduction 66 ?. Table of results and discussion 66 3. Discussion of the Pauli equation 68 4. Discussion of the Randan Phase Approximation 69 5. Dissipative equation of notion for arbitrary systems 70 VII. LANGEVIN FORCES AND NYQUIST'S THEOREM 1. Langevin forces 71 2. Transient times 74 3. Nyquist's theorem 74 VIII. THE H-ATOM 76 IX. LITERATURE 81 REFERENCES 83 APPENDICES A. Commutator and trace properties 86 B. Harmonic oscillator, boson operators 87 C. Translation operators 88 D. Coherent states 90 E. Normal ordering, special operators 91 F. Canonical transformations and diffusion 93 G. Phase functions 98 H. Characteristic functions 103 J. Strength functions of the radiation field 105 K. Continuum limit with coordinate coupling 106 L. Continuum limit with symmetrical coupling 111 ACKNOWLEDGEMENTS 117 il SUMMARY The subject of this report is the description of dissipative notion of classical and quantum systems. In particular, attention is paid to systems coupled to the radiation field. A dissipative equation of motion for a particle in an arbitrary potential coupled to the radiation field is derived by means of perturbation theory. The usual divergencies associated with the radiation field are eliminated by the application of a new theory of generalized 1 unctions. This theory is developed as a subject in its own right and is presented independently. The introduction of classical zero-point energy makes the classical equation of motion for the phase density formally the same as its quantum counterpart. In particular, it is shown that the classical zero-point energy prevents the collapse of a classical H-atom and gives rise to a classical ground state. For systems with a quadratic Hamiltonian the equation of motion can be solved exactly, even in the continuum limit for the radiation field, by means of the new generalized functions. Classically the Pokker-Planck equation is found without any approximations, and quan­ tum mechanically the only approximation is the neglect of the change in the ground state caused by the interaction. The derivation is valid even for strong damping and arbitrarily short times. There is no tran­ sient time. For harmonic oscillators complete equivalence is shown to exist between quantum mechanics and classical mechanics with zero- point energy. A discussion of the derivation of the Pauli equation is given and perturbation theory is compared with the exact derivation. The exactly solvable models are used to calculate the Langevin force of the radiation field. The result is that the classical Lange­ vin force is exactly 6-correlated, while the quantum Langevin force is not 6-correlated at all. The fluctuation-dissipation theorem is shown to be an exact consequence of the solution to the equations of motion. iii CHAPTER X INTRODUCTION The subject of this dissertation is the description of the motion of dissipative systems, both classically and quantua-mechani- cally. in view of the difficulty of this subject, the best-tint could be done was to obtain dissipative equations of notion for a number of simple systems. These examples provide model equations that can be used to introduce damping in more general systems. We use the term "dissipative equation of motion* in a very general sense which in­ cludes all differential equations describing the behaviour in time of a small system coupled to a heat bath after the bath variables have been eliminated from the description. Dissipation is r. subject that deserves study by its own right. A great many problems in plasma physics (e.g. Landau damping, quasi- linear theory, and radiation from a turbulent plasma) involve dissi­ pation. The aim of the present work is to contribute to understanding dissipation in general. The examples that will be solved are chosen to be both manageable and representative of practical problems. The starting point of the present work was the point of view expressed by Harris and others {see chapter IX for references). These authors demonstrated that the usual formulation of quasi-linear theory is mathematically very similar to the time-dependent perturba­ tion theory as used in quantum mechanics. There are a number of dif­ ficulties with this approach. The most obvious difficulty is that it is necessary to derive the classical dispersion relation of the sys­ tem by taking the classical limit of a quantum-mechanical result. Then, the dispersion relation is quantized again and perturbation theory is used to find the damping of the quasi-partides. This procedure of deriving the quasi-linear theory is not straightforward. The objection can be raised that the classical dis­ persion relation should contain the damping of the waves. In the quantization process the damping has to be neglected. It is reintro­ duced by means of perturbation theory. If attempts are made to extend the theory to higher order in the perturbation, the usual difficul­ ties, which are familiar from their appearance in quantum electro- l dynaai.es, occur. All this raises the question how a quantum•mechanical system with damping should be described or how a yjawpuwl classical system should be quantised. in classical mechanics it is usually not difficult to add damping to the description of a conservative system. Friction forces can be added» either directly to the equation of motion» or indirect­ ly» in the form of Rayleigh's dissipation function to the Lagrangian. This procedure cannot be used in quantum mechanics. The reason is that» regardless of the initial state» the damping violates the uncertainty relations if we wait long enough. There axe a number of incorrect ways to circumvent this» using time dependent Hamiltonians» non-Hermitian Hamiltonians» or non-linear Schxödinger equations» which still recur in the literature. A step in the right direction is made by realizing that a dis- sipative equation of motion cannot be factorizable. If it were f ac­ tor izable, then x -• 0 and p + 0 implies fx,p) * 0 and AxAp •* 0» which is a violation of the uncertainty relation. This shows that in quan­ tum mechanics damping and a factorizable evolution are mutually ex­ clusive. This means that a dissipative equation of motion is neces­ sarily an equation for the density operator of the system. Even if the density operator is a product of the form {$><?[ at one instant in time» it will not be of this form at other times. A SchrOdinger equation or any other equation for the wave function of a dissipative system is impossible. When this point has been understood, it is simple to write dis­ sipative equations of motion for the density operator of a spin or a harmonic oscillator with given phenomenological damping coefficients. For instance the Bloch equations for a spin can be produced in this way. It is simple to write these equations in a representation-free way. This leads to the question of the validity of these phenomeno- logical equations of motion» which amounts to the question: Is there a microscopic model which will produce a given macroscopic damping? At this point we come into contact with the large body of theory developed in the past twenty years to describe spin-resonance experiments or lasers. In this field many authors (see chapter IX) derived dissipative equations of motion, using perturbation theory. On the other hand, exactly solvable models for Brownian motion were developed by Ullersma and others, fie will combine these developments. The present exposition is a generalization with respect to Ullersma. More general Hamiltonians are considered and the off-diago­ nal elements of the density operator are investigated. The exact solv­ ability is an improvement on the perturbation theory. On the other 2 hand* the perturbation theory is applicable to more general systems. The different derivations of the same final results sake it possible to compare perturbation theory with exact solutions, this can be used to demonstrate the validity of the approximations that have to be made in perturbation theory. The perturbation theory can be used for systems consisting of a particle in an arbitrary potential coupled to the radiation field. As a special case, the H-atom was investigated. This has some experi­ mental interest, as it would be useful to give a calculation of the radiation from an H-atom in a turbulent electric field.
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