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 {$>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. This problem proved to be too complicated to produce useful results in the avail able time. Finally, it was discovered that a large part of the difficul ties with the radiation field (divergencies and runaway) can be elimi nated by using a more general theory of generalized functions. The mathematical apparatus needed to deal with these problems is developed in chapter XI. In addition, it was found that the new generalized functions eliminated the need for most of the usual approximations. There is no transient time and no restriction to weak damping, in con trast to previous work. Moreover, a new approach to the divergence difficulties, associated with quantum electrodynamics or any continuous system, seems possible. It would be useful to develop quantum electrodynamics, using these methods, as a simple model for more general continuous systems. The investigations reported in this dissertation supply the tools needed to investigate the problems that originally motivated this research. With the results that are now available, a new attack on these problems could be made.
For practical reasons natural units have been used throughout.
3 CHAPTER II
GENERALIZED FUNCTIONS DEFINED ON DISCONTINUOUS TEST FUNCTIONS AND THEIR FOURIER, LAPLACE, AND HILBERT TRANSFORMS
I* Introduction
Generalized functions were introduced by DIRAC (1930) in order to be able to deal with the continuous spectrum in quantum mechanics. Parts of the theory of generalized functions were anticipated, for instance by HADAMARD (1932), in his theory of the "partie finie" of divergent integrals. The mathematical respectability of generalized functions was proved by SCHWARTZ (1950, 1961) by considering them not as functions but as functionals on a suitable space of test functions. This theory was presented in a simple form by LIGHTHILL (1958). Gener alized functions are presented as equivalence classes of sequences of "good" functions. The disadvantage of this approach is that the generalized func tions have to be adapted to the range of the integrals we want to evaluate. If we have integrals from -<» to +» we use good functions on (-.», +») , while for integrals from 0 to » we must use good functions on (0,»). In particular, both the Laplace and the Fourier transform of the 4 It will be shown that a consistent theory of generalized func tions based on this extended class of test functions is possible. It includes the old theory of generalized functions. In addition, new generalized functions appear which were equivalent to zero in the old framework. Integration between arbitrary limits becomes possible and the Hilbert transform presents no difficulties. The usefulness for physical applications appears naturally. The presentation has no pre tense of mathematical rigour, but we feel that it will be possible to construct a rigorous foundation for the results. The exposition is based on the theory of the "partie finie* of Hadamai3. This is the most convenient way to obtain results. The first two sections give an introduction to the new generalized func tions. In the last section an application to the theory of dispersion relations is given in order to illustrate the use of the new formalism. 2. Discontinuous test functions In the following we restrict ourselves for simplicity to com plex valued functions of one real variable. We want to extend the usual class of c" test functions in such a way that discontinuities in the test functions are possible, while keeping differentiability and symmetry under Fourier transform. Ac a consequence, slow disappearance of the functions at infinity {for instance as 1/x) is unavoidable. Moreover, in order to ensure conver gence of the Fourier transform at infinity, the number of jumps and their magnitude must be restricted. These considerations lead one to impose the following require ments on the class of test functions A. f e A - (i) I) f € L2 (-co, +»). II) f of bounded variation. III) The derivative and the Fourier transform, which exist almost everywhere as a consequence of I and II, again satisfy I and II. Strictly speaking the elements of A are equivalence classes of functions. Functions differing on a set of measure zero are equiva lent. Some consequences of these requirements are: f € A •*• 1) f is infinitely differentiate almost everywhere. All derivatives are elements of A. 5 2) f has at most a countable number of jumps. The sun of the Magni tudes of the jumps is finite. 3) f behaves at infinity "at worst" as a negative power of x, multi plied by an alaost periodic function. 4) All good functions are A-functions, A function is called "regular at x - 0" if lim f *n*(x) « In) XrO lira f * ' (x), for all n. The derivatives and the limits exist as a con- xfo » sequence of I, II, and III. In the same way, using ~ » y, we define "regular at x a »*. All f € A are regular at x « •, but not necessar ily for finite x. lie introduce the notation A sequence f € A is regular if lim(f .g) exists for all g € A. Two sequences f and g are equivalent if lim ((^"S,.) »ft) * ° for a11 h € A. Generalized functions or distributions are defined in the usual way as regular sequences of A-functions or as linear functionals on A-functions. Examples: 1) The generalized function generated by g € A is the sequence g * g. 2) The generalized function 5, is defined by the sequence n n , 0 It has the property Q «yn +« 1/n |dx5,(x)f(x) - lim n ƒ f(x)dx • lim f(x) = f<0+) . (4) -« n-»-» o x*o The "delta down" is a one-sided S-function. In the same way we define 6 (x) = %{6+(x) + 5f(x)) , o (x) * %{6+(x) - 5f(x)) - 6(x).sign(x) . (5) The a-function measures the discontinuity at x » 0 (a(x),f(x>) - %(f(0+) - f(0-)) , (6) 6 which is half the jump of f at x • 0. Other sequences equivalent to (3) are x/a 6±(x) - lis» Xi Ii [#uy| e" H(x) , (7) V a-K> nT a (aj and 6,(x) - 11» i, • I-M' ~'l H(x) , (8) * a-K> a where H(x) is the Heaviside function defined by H(x) • |°' * * ° . The sequence (8) is a sequence of good functions in the sense of Lighthill. In this sense 6., 6,, and S are equivalent and o is equiva lent to zero. 3. Differentiation Derivatives of generalized functions cannot be defined by term- by-term differentiation of a sequence of test functions as is possible for good functions. For instance, the first sequence (3) defining the 6,-function is differentiable almost everywhere with derivative zero. Moreover, the term-by-term derivative of a regular sequence does not have to be a regular sequence. An example is the sequence 2 2 f (x) = n e~n x , which is regular and equivalent to /? 5(x). Its term- -n2xz by-term derivative £i(x) * -2n3x e is not regular as it gives » when tested with a test function 6 A having a jump at x - 0. Consequent ly, derivatives must be defined as limits of generalized functions. First we define generalized derivatives for A-functions. For f 6 A we define the sequences - n(f(x+2/n) - f(x+l/n)) , *•' n and - n(f d as/ 3xf + ^J-U.H,. (x-x.) . (10) gen ord i r x * 1 df~\ df Here |^ is the generalized function generated by the A-f unction g£. Jf(x.) is the jump of f in the point x.. 7 An example is given in the figure below. f(x).«",)0 sign.U) 'I ws mm 0 IU-1/2) © IU-1Ï ® Cs|t^.2(M«-ViMU.1)) Por f and g 6 A we have the partial integration formulas (11) This follows at once from the invariance of the integrals under the substitution x -»• x + a. In addition, we use the linear combinations 1 fd _ d + and ) (12) dT. 1 [3x"f 3x^ dx. I ldx+ 3xJ ' They satisfy the partial integration formulas 0 and (II ,g (13) Üe") • («* v o <* * O' A A If we do not introduce distributions, gj is the derivative and gj gives zero, except on a set of measure zero. For generalized functions we use the same procedure to define the derivative as the limit of the distribution d f f «I = nm ff 6 g Um + (die + ' l " 7 (*|(x 2e)-«i t+ = Um [g (a|)rJ(«-2c>-j(«-e?| - lim ^WT(- » E e C-»0 * J p-».e-»o - -g'(O-) = +(6{,g) . (15) 8 In this way we obtain d as/* + dV* • «; (16) By adding and subtracting we find a table of results for derivatives of 6-functions 6 = 6 #-« = *: dx+ t dx+ + dxe d*o d d - d « d rt (17) 0 " aie° Note that a* is not the derivative of o but %(6|-6|). The rules are simple to memorize: the arrow on the result is the arrow on the differentiator, even derivations give 6-functions, odd derivations give o-tunctions, differentiating a o gives zero. At first sight it is surprising that a non-zero generalized function can have all derivatives equal to zero. It is a consequence of the possibility of having non-zero test functions with derivative zero almost everywhere. This possibility does not arise for good functions. For good functions A6,(x) + (1-X)6.(x) is equivalent to 6tx). The simple 6-function of the usual generalized functions has been split into a one-parameter family of generalized functions. At first sight, one would expect the same to be true for functions behaving like H(x) or sign(x) near the origin. However, the generalized func tions 0 1 x<0 Hjtx) - lim ^ x/e 1 0 0 1 x<-e H. H2(X) * lim • 1 + x/e 1 -e is equivalent to zero. To distinguish it from zero we need a test function having a 6-function singularity at x = 0 and we do not allow this. We conclude that in this framework there is only one H(x) and one sign(x) and we have £ BID = 6+(x) £ H(x) » «+ 4. Powers of x. The partie finie The next step is to introduce powers of x as generalized func tions, we do this by means of the partie finie of Hadamard. We inves tigate «•> |dx xXf(x) with f{x) 6 A . (19) o The treatment here is a generalization of the method used by GELFAND and SCHILOW (I960) for good functions. The integral is defined as an ordinary integral for - 1 < Re A < 0 and it is in this strip an analytic function of A. The distribution x H(x) is defined for other values of A by analytic continuation. We have (- 1 < Re A < 0) CD ^ CO A A fi0 dx x f(x) = dx x (f(x)-f(0+)) + dx x*f(x) + *\+* • (20) o o a The right-hand side is defined and analytic for - 2 < Re A < 0, A * -1. In the strip - 1 < Re A < 0 it is identical to the left side, so it is the analytic continuation. For - 2 < Re A < -1 we define it to be the "partie finie" of the integral. We write it as Pf 7dx x*f(x). It is independent of the choice of a, as for -2 10 4» «O» Pf fdx xXf(x) • fdx xX(f(x)-f(0+)) . In this way we find the continuation in the half plane Re A < 0; A #-1, ~2, -3, .... In the strip -n -1 < Re X < -n we have Pf fdx xXf(x) - fdx xXff(x)- l fïk)(0+) JU • (21) o o Jtas° The residue in the poles is easily found. For example GO lim (A+l)Pf fdx xAf(x) - A—J L «I w lim lirn (A+l)x H(x) = <5 (x) (22) A--1 + Here the limit is a limit in the distributional sense. In general we find lim (A+n+l)xXH(x) = tt- Z{.n) (23) A->-n-l n* * The numerical factors are chosen in the conventional way, such that dx ójn)(x)f(x) - (-)nf(n)(0+) (24) The Pf x can be defined for negative integral values of A by subtracting the pole part. Pf x~ H(x) is defined as Ln-1 n Pf x" H(x) » lim ***M - lxl~nUn-l)\ «{"""W • <25) A+n Example: Pf f X A a f lim fdx x (f (x)-f (0+)) + fdx x f(x) + f(0+) rTT- - jffi lO a |dx f(x)-f(0+) + |dx flsl + fi0+)log a . (26) 11 Obviously, Pf * is undetermined up to an arbitrary multiple of 5+{x). N» could fix this by fixing a, but then In cases where the value of a can be fixed, for instance when the upper limit is finite, t.e indeterminacy can be avoided. In other cases the only way is to admit the indeterminacy, we write Pf i H{x) = Pf i H(x) + u6+(x) (28) for arbitrary \i. An indeterminate equation like (28) means that we can H/v) replace one side by the other. In the same way Pf „ is undetermined (n-l) \ x up to yfi* '. The same method applies to (-x) H(~x). It has poles for integer A. Up to a sign the residue.is again a 6-function, but the arrow is reversed. The residues are collected in Table 1. The principal value is defined as a linear combination. P AH + n A, t 2 <29) (^l ' Ji»n(" "" <-» <->" "- > - iuHun-ot'] • Other definitions are possible, but then the principal value is no longer even or odd but a mixture. In particular, P^ = P| + yo(x) . (31) p7*T = pTxT + »tM • (32) For good functions o gives zero and P- is defined uniquely. In our treatment the relationship between the two is more symmetrical. For some applications we need z with complex z. It is defined as zA = , .A.iA arg(z) _ (33) We de fitte P(x + io)* » Pf xAH(x) + e±ilTAPf (-x)AH(-x) . (34) This expression has poles for X = -1,-2, .... with residue n n l (-) -V - > 12 Por these values of X we define 2 P(x±io) = limjcx+io)* - jx:nM;.lH j • OS» If we expand the exponential in (34) we find 6(n-l) P(x+io)~n - Px~n + iir +_^ , . (36) As Px~n is undetermined by o ~ we can disregard the arrow on the ó-function. This gives t\~l)(x) « -lim XPf xXH(x) . (38) T A to The / through a symbol means it is a function at infinity. Our test functions f 6 A behave at infinity as a negative power of x, multi plied by an oscillating function. lim xfdx xXf(x) Ato I is different from zero only if f behaves as —- near x = +». Then we have 00 lim Aldx xXf(x) = c. . (39) Ho o| Now that the f is defined, and c. found, we can write oe. a « . fdx xXf (x) = fdx x*f(x) + fdx xAff (x) --jij - x ci * (40) o o a This gives the analytic continuation in the strip 0 < Re A < 1. Then the f'1-2) ' can be defined as the residue at A * 1 and the continuation to 1 < Re A < 2 is possible. In this way continuation to all non-inte gral values of A is possible. The residue at A«n is proportional to ,(-n-l)# 13 By subtracting the pole part we can define ft and f at infin ity. Example: ft H{x) = Pf x°H(x) • lim |xXH(x) + i#j_1)*x)] (41) Xfo *• ' It is undetermined up to an arbitrary multiple of the ^-function at infinity. The f at infinity is defined by adding or subtracting. We write ?I(x) = fl for the generalized function fx°. In this way fI is undetermined up to r" at infinity. All our test functions are reg ular at infinity. This means that all the rf-functions are zero, and the f is defined uniquely. They have been kept in the table for pos sible generalizations and for symmetry. All residues can be found in Table 1 : (42) X A X X X - x H(x) (-x) H(-x) W |x| sign(x) negative odd .(-X-l) .(-X-l) 26(-x-n 20*"*-1» (-X-1)! (-X-l>! (-x-i)i (-X-D! negative even -(-X-1) *(-X-l) _25(-A-l> -2,—4,—6,.... <-<-i>i (-X-D: (-X-D: (-X-DÏ positive even 1 .(-X-l) i ,(-x-u -2 rf(-X-l) , n ,, 0,2,4,6, -X! 't +X! '• IT * if « - - positive odd 1 .(-X-l) i ,(-x-n -2 .(-n-1) x I) 1,3,5,7, IT * if *«- - The defining relations are lim (X+n)xXH(x) = 'r'.ju «*"n_1) (43) X-*— n A lim (X-n)x H(x) = -*—•,n * , (44) A -*-n (ó(n),f) « (-)nf(n,(0) , n « 0,1,2,3, (45) C -(-nl n) «t _ n _ f d 1 f(«) (l " ',f) (ÏÏ=ÏTT \Tm\ nl(n-l)* ' n - 1,2,3,4, (46) The <5- and ^-functions act as a filter which measures coeffi cients in a power series expansion. They can be defined with an arbi trary extra factor. For 6-functions the normalization is fixed in the usual way, such that (5(p,,f(q)) - (-)r(ó(p*r),f 14 The normalization of the f-functions was chosen in such a «my that the sane formula applies. He also want to give a value to Pf fdx x , which is not yet defined. For non-integer values of A we define it by «o a •» Pf fdx xX - Pf fdx x* + ft fdx xX+1 i . (43) o o a Depending on the value of Re A the first or the second tern is defined as an ordinary integral. For all non-integer values of A the result is f . A-U A+l Pf Jdx x* = a__ _ a__ . 0 (49) A+l A+l o By analytic continuation it is zero also for integer A. In particular, f fdx - Pf te - 0 . (50) o o The same is true for integrals from -°° to +°°. In particular we have +«° P?j|^=0 , (51) —00 for all values of y. Remark I It is also possible to use test functions with a different be haviour near 0 and *, for instance x° near " and x1 near x=0, if the powers are chosen in such a way that a strip of at least unit width remains as a basis for analytic coninuation. All residues remain the same 6-functions, but they may be at infinity or zero in a different way. In our example 6 is shifted to infinity, because a strip of width one remains, a given power of x can occur either near 0 or near », but not both. Remark II Coordinate transformations which do not have an infinite range for the new coordinates, for instance polar coordinates, are simpli fied by the use of 6's with an arrow. An example occurs in the treat ment of the H-atora in the following chapter. Remark III In the usual generalized function theory we can define the support of a generalized function. For instance, the support of the 15 old ó-function is the point x*0. This is no longer possible with 6^. Here the "support" is the positive infinitesimal neighbourhood of the point 0 or the "point* 0+. The theory of infinitesimals as invented by ROBINSON (1970) would probably simplify the preceding exposition. 5. Fourier transforms For all f 6 A the Fourier transform is defined for almost all k by +« Ff(k) - fdx e~ikxf(x) , (52) +<*> _1 l + F f(k) •*ƒ= ^r '|d x e^ftx) F~ - ^ F . (53) —ikx It is undefined for those values of k for which e f(x) behaves as 1/x near «». Then it may be defined by means of the principal value at infinity. The 2TT is placed in the same way as in BATEMAN. As all A-functions have a Fourier transform in A, we define the Fourier transform of a generalized function by taking the term-by- term Fourier transform of a regular sequence. Parseval's theorem for L2-functions, (f 'g> = 'h (Ff'F9> ' <54> guarantees that the term-by-term Fourier transform of a regular sequence is again a regular sequence, and equivalent sequences remain equivalent. In particular, we want to find the Fourier transforms of powers and 6-functions. The Fourier transform of x is easily found for - 1 < Re A < 0 by means of a change of variable and using the definition of the r-function (see e.g. Gelfand Schilow I or Bateman I, 1.3.1., 2.3.1). We find F(xXH(x)) » r(-u)(e11Tv/2(k)*,H(k) + e-±iru/2(-k)wH(-k)) , (55) F((-x)XH(-x)) = r(-M)(e"ilTy/:2(k)wH(k) + eiwy/2<-k)wH<-k>) , (56) with M » -A-l. By analytic continuation we extend the result to all non-inte ger A. For integer A the expression on the right has poles. The linear 16 combination ei*u/2xuH(x) + e^'Vx^lM-x) , (57) considered as a generalized function, has poles for integer u. For p -* -n the residue is _..«n (n—1) , . 2(i) cr (x) „ • i -> •» iz&\ - (n-l) i * i,2,i,.... (sol For y = 0,1,2,.... the poles cancel as all ^-functions are zero. For these values of p the r-function has poles. This means that there is a first order pole for all integer values of y. In the following we need the expansion of r(-y) around its poles. This is found from 11 ' " r(l+z) sin(ïïz) " r(l+n) + (z-n)r*(l+n)-K>(z-n)z (z-n)+0(z-n)3 " J 2 = n\ ( z_n) (1- (z-n)iji(n+l) +0(z-n) ) . * = T'/T (59) The Fourier transforms of the 6-functions are found by calcu lating the residue of both sides at X = -n. n) n+1 i,Tll/ i7ry/2 w Fö] = lim (-) n!(y-n)r(-M)(e VH(k)+e- (-k) H(-k)) - v.*n = lim ( - F6<"> - r(ik)n - ïiï^i ^""-^ . (60) In the same way we find F&\n) - * By adding and subtracting we find F6(n) - (i)n?kn , (62) Fff(») .=H4^#(-n-l) . (63) n # As expected, the Fourier transform of a o-function at the origin is a /-function at infinity. This must be so, for jumps at the origin in a test function transform into negative powers at infinity. 17 Inverting the previous results gives F|»xn - 2*(i)n5(n) , (64) F*("n) = -2i(i)n"1(n-l}:o(n"1) . (65) We also need xnsign(x), so we calculate F(pf xnH(x)) = F lim A-*n = lim rtm-l) . [(-iJ^V^^'^tk^Htt) + u+- n-i *• By expanding this we find F[ff xnH(x)) « n!(-i)n+1P(lt)"n"1 + ir{i)n6(n) . (66) In the same way we find F{n (-x)nH(-x)) = nMD^Pfk)'""1 + Tr(-i)n«(n) . (67) Adding and subtracting these gives F(fxn) * 27r(i)n6(n) , (68) which we had already, and r(?xnsign(x)) - 2n!(-i)n+1P(k)~n_1 . (69) Notice that the first one of these is completely determined, while the second one is indeterminate up to Aa r which is just the Fourier transform of the indeterminacy in /(xnsign(x)). These are the Fourier transforms we need in the following. For completeness we also calculate Wpf x~nH(x)) « "^j„lij {— (log|k| + Af sign(k) + c) , (70) where log|x| has been found as lim - (|x|e-l) * log|x|. eto e 18 c is an arbitrary constant which appears as a consequence of the inde- terminacy of Pf x H(x). In the same way we find n ( i c ) f(Pf{-x) H(-x)) = ~^( n J I ), (log|k| - ^ sign(k) • c) (71) Adding and subtracting these gives f(px-n) = -i^sign(k) ?(_ik)n-l § (72) F(px"nsign(x)) = "**{»*£{»— (log|k| + c) . (73) In particular we get PP- = -in sign(x) (74) This result will be needed for the introduction of the Hilbert trans form. Table 2 summarizes all Fourier transforms derived in this section. f(x) f(k) * Ff(X) Pf x*H(x) r<-u) (eifflj/2(k)vH(k) • e~i,ril/2(-k)uH(.k)) X lffw/2 U i,rii/2 W Pf(-x) H(-x) r(-M) (e- (k) H(k) + e (-k) H(-k)) ?f xnH(x) n!P(ik)"n-r +Ti(i)n6(n) pf(-x)nH(-x) n:P(-ik)"n-1 + 1r(-i)n6(n, n Pf x~ H(x) "*{iffi)" (lo9|k| + V sign(k) + c) -f(ik)n- 1 Pf(-x)nH(-x) Vn1.^""; (log|k[ - Af sign(k) + c) ?xn 2n(i)n5(n) fxxgign(x) 2n!P(ik)~n~l -n i (k) 1 Px " (n!l?? F(-ik)"- Px~nsign(x) , n+1 , ,. (n) (n) r(lk)« + l£± |("«-!> n: (n) Kik)1 . n (nfl) (n) iri_ j.(-n-1 ) n! n = 0,1,2,.. m = 1,2,3,.... -A-l 19 6. Laplace transforms For completeness we make some remarks on the subject of Laplace transforms. The Laplace transform as introduced for distributions on (o.°°) by Schwartz gives results like £6*1. The fiher e corresponds to our 5,. We have L«+ =1, £5+ - 0 . (75) In analogy with sine and cosine transforms we define y x{ £ef(y) = | fdx e" l f y x L0f(y) = ^ jdx e" ' lf(x)sign(x) , (77) —<*> or, equivalently, CO f(x) f( x) Lef - lfe = jdx .-** ^ - . (78) _xy f(x> f( x) L f - 1Lf* - fddx e ** *w :,M- M . (79) o) oO JJ 2 o All the usual results for Laplace transforms follow at once. We remark once more that the <5 in £5 = 1 is not the same as the 6 in P& = 1. The difference does not appear in the theory using good functions for lack of test functions which can distinguish between them. 7. Hilbert transforms The Hilbert transform for f 6 A is defined by +« Hf(y) = i P {dx M —oo so . I fdx f 20 For generalized functions we use the same definition. Hubert transforms can be found by using (81) and the table of Fourier trans forms. . . , .n+1 ._ —n—1 BSM = iz> £•£* , (82) HaM = 0 , (83) as a consequence of I sign = 0= 0. n Hfx = lz£l j(-n-l) m (84) We see that polynomials in x, which have no Hubert transform in terms of good functions, do have a Hilbert transform in this framework. He also see why attempts to define them in terms of good functions yield zero. On the other hand, we have not succeeded in providing a Hilbert transform for every generalized function. A generalization in which this is the case seems possible, but as we do not need it for applica tions to perturbation theory we do not attempt it here. All "old" generalized functions do not involve o-functions. They now all have a Hilbert transform. 8. Dispersion relations We illustrate the use of Hilbert transforms by means of a simple example. For this we choose the theory of dispersion relations. For a linear system satisfying uhe causality requirement we have a relation between input X(t) and output Y(t) of the form Y(t) = jdT G(T)X(t-T) , (85) o where G(T) is the impulse response function or Green's function. If we Fourier-transform this, we obtain y(u>) - x(ü))g(w) . (86) As a consequence of the causality requirement, g(u>) is an ana lytic function in the lower half plane (lm w<0). If, in addition, g(ü)) goes to zero for w + <*>, we have a Kramers-Kronig relation or dis persion relation between the real and imaginary parts of g(») on the 21 real axis +» lm g(tó) = - P j ^ 1 +» D« „/,,\ - "* B fdw'Im g(u>') . ffl7v Re g(u>) = — P j jjrrg • <87> —CO The real and imaginary parts are Hilbert transforms of each other. Difficulties arise in this framework, for instance if X(t) re presents an applied voltage and Y(t) the resulting current, in the case of an ideal resistor or an ideal capacitor. Then G(T) contains a 6- or 6'-function and g(w) does not disappear at infinity. There is no dispersion relation. The usual way out is to introduce subtracted dispersion relations (see e.g. BYRON and FULLER II.6.6). Take gtwj-gddj) g, (o>) * with a. arbitrary. I to—bi. 1 Then, if g(cj) is bounded for w-»•<», we can write a dispersion relation for gj and from this a once subtracted dispersion relation for g. +» Re ,(U, = Re g(Ml) - (U^JP jff^t^^ , +» im g(M) - I» g(Wl) + (u-u^P (|^y . (88) —CO In general, n +• 1 subtractions are necessary if g(w) « ton near «°. In the new framework all this is true but unnecessary. If we have a resistor, then G(x) - Ó+(T) and g(u) - FKw) + iTr|(-l) (w) , (89) and we have Im g(«) = H Re g(iu) For a capacitor G(T) » 5|(x) and g(co) - iPw - TT/("2)(W) . (90) 22 fie see that in this framework we always have a Hubert transform rela tionship between real and imaginary parts of g(u), without having to subtract. Notice that only the 6,(1) satisfies the causality requirement: An instantaneous response must be understood as a response to the in finitesimal past. The splitting of the 6-function into 6. and 6, cor responds here with the splitting of the present instant in time into an infinitesimal past and an infinitesimal future. 23 CHAPTER III PERTURBATION THEORY 1. Introduction In this chapter we derive dissipative equations of motion for a small system coupled to a heat bath of harmonic oscillators. Similar derivations have been given by many authors for the special cases of a harmonic oscillator, a spin, or a two-level atom. For references see chapter IX. Here we present the derivation for a more general case, where the small system consists of a particle in an arbitrary, possibly time-dependent, potential. The derivation is representation-free and both classical and quantum- mechanical. For convenience quantum-mechanical notation has been used, but, at all stages in the calculation, conversion to classical mechan ics is possible. The method employed is to solve an initial value problem, with a suitable initial state of the whole system, to second order. The time derivative of the reduced density operator is found by summing over the bath variables. The validity of this result is extended beyond the range of validity of the second-order perturbation theory by means of the Random Phase Approximation (R.P.A.). In chapter VII we show that in the special case of a harmonic oscillator the result of perturbation theory and R.P.A. agrees with the exact solution described in chapters IV and V. This agreement sup ports the approximations used in this chapter. 2. Arbitrary potential, momentum coupling We investigate systems described by the Hamiltonian H ' 5 + V(ro't} * \l^bffy + ^o^j + W • {1) The form of the Hamiltonian is suggested by the theory of radiation from an atom in the dipole approximation, but the bath may also 24 consist of the normal modes of a lattice or the sound waves in a fluid. For the time being we have a finite number of oscillators. Later on in the derivation we will take the continuum limit, which means that the number of modes goes to infinity, while the coupling of each mode goes to zero. The only special case we consider in detail is the radiation field where the spectrum covers the positive real axis. The initial state at t = 0 is supposed to be given by the uncor rected density operator P(r0,po,rj.p;.,0) = Po{rQ,po,0) 0 Hp.. (lyp^n^,2^ = PQ»Pb» <2a) where p (0) may be an arbitrary density operator of the small system. The bath oscillators are in displaced thermal states (see (E.8)). The displacement is given by 2., while n. is the number of thermal quanta. The temperatures of the bath oscillators are not necessarily equal. The first and second moments are given by - Zj As we go only to second order in the perturbation theory, we do not need the higher moments. It is possible to show (LOU1SELL (1964)) that the displaced thermal state is the state of maximum entropy if the energy and displacement are given by Eq. (2b). In the following we suppress the arguments of the density operators, as they are clear from the context. The equation of motion for the whole system is £ ? = -MH,p] (3] having the formal solution p(t1 **\) =_ e _™*"t^#p(0)rtVe_iH t (4) In the interaction representation the density operator is given by iHft -iHft Pitt) e x p(t)e r (5) 25 with Hf * Ho + Hb " Po/2a * V(ro't> * I MibIbj (6) The equation of notion for p. (t) is ^ Pi = -ifHi(t).pi(t)l , (7) with H.(t) given by iH.tf . > -iH-t (t)«e t [%p JCAjbI J + Xjbj)]. f . (8) Hi Q j As H~ is a sun, H,(t) can be written as Hi(t) - po(t)»B(t) , (9) with iH t -iH t po(t) = e ° Poe ° , and r iw.t -iüi.t •« B(t) a *JlAje bj + Aje bj * (10) We solve the equation of motion (7) to second order t P (t) = p (0) ± dt i i ' / 'lHi(f)tpi(0)} + o - jr Jdt' Jdt'MHjCtMrlHi(t"),pi(0)]] , (11) " o o or for simplicity ^ Pi = -i[Hi(t),pi(0)J - jr /df[Hi(t),[Hi(f),Pi(0)]] . (12) o The first order term in the perturbation expansion [Hi(t),pi(0)] = [po(t)B(t),po(0)pb] can be written in the form [Hi(t)fpi(0)J - %[Po(t),Po(0)l(B{t)rpb) + + %(Po(t),Po(0)J[B(t),pb] (13) by means of the formula (A.2) for the commutator of a product. The reduced density operator is obtained by tracing over the bath variables. Traces of commutators are zero, compare (A.12), since all 26 expectation values are finite. The first order contribution to the equation of motion iw.t -iw.t-j p (t) 3 3 é i red= UPoCt>,Po<0n {[yj*e* + X*z e = iF(t)[po(t),Po{0)] (14) for the reduced density operator represents a time-dependent driving force. In the continuum limit for the radiation field an arbitrary incident field can be specified in this way. In order to find the dissipation we have to go to second order. Again, using (A.2), we find [Hi(t),[Hi(f),pi(C)]] - - *H [Po [B(t),[B(f),pb]J 2B(t)[B(t'),pb] -+ red "'red 2[B(t),B(f)]pb -2i I A. A* sin u>, (t-t')pb red red = -2i I A.A* sin u.(t-t') , (16) and the term with u = - and v=-, which contributes 2[B(t),B(f)]p + red - I A.. A* cos ^(t-f )(b;jbj + bjbj)pb red - 2 I X .A*(2n.+l)cos u.(t-t') . (17) The second-order contributions from the z, are omitted since they are merely an iteration of the first-order contribution. In the following we take z. = 0 for all j. We can always restore these terms if we need 3 * them. 27 Combining the partial results (14 through 17), we find p (t) = i t, (t> Now that we have eliminated the bath variables we drop the indices red and o. We can write Eq. (18) formally as •^ Pi(t) = i /df[P(t),[p(t'),Pi(0)]] Jdü) Y(«)sin w(t-f) + o +— o (19a) O» ) becomes a smooth function of w. For the radiation field this means letting the quantization box become infinitely large. The strength function of the radiation field is in the continuum limit given by 2e2 as argued in App. J. In the following we consider only the radiation field or other fields that have a strength function proportional to u>, for instance an ideal ized sound field. Using generalized functions, the frequency integrals in Eq. (19) are easilv evaluated. The first integral (19a) is (t-t» = T) GO +0D i /düi u'Sin UT - H Jdw we~±WT = iit6'(T) . (20a) O —on The second integral (19b) can be split into a zero-point part, which is present even when the bath is in its ground state, and a thermal part which contains the thermal occupation numbers. The zero-point part is proportional to /dtü u'cos UT • k /dw|u|'e~ wT • 77 • (20b.1) o -*> 28 If we take the bath in thermal equilibria» with all bath oscillators at the sane temperature, the number of thermal quanta is given by n(w) « (e^-l)"* , where B is the inverse temperature. Then the thermal part can be evaluated as an ordinary integral. It is (BATEMAN Vol. I 1.4.8) proportional to 00 2 2 2 ldu 3&T «• « - IT) * lesinhVr/e)) • <20b-2> o e Combining the zero-point part and the thermal part, we find for the integral in Eq. (19b) • 2 Jd» ü>(2n +00 7A -i«(t-t') Jdw w»e ..AO as a principal value at «• , we obtain 29 f /du u>-e"XWT - infi" = %i*(«;(t) +«;(TÏ) . —•» Only the 6' contributes to the time integral t /dT p(t-T)(ó|(T) +6+(T)) , O which is now well defined, so the result is /dT p(t-r)«; For low temperatures the situation is more complicated. In particular for T=0 (at absolute zero), we obtain t P |dT R!£IL . (24b.l) o We approximate this by +» P |dT Rl£li = H |f(t) = ff jï(t) . (24b.2> 'CD It is not possible to justify the approximation (24b) at this level of perturbation theory. For the special case of a harmonic oscillator it can be justified by comparison with the exact solution. For the gener al case we show in the next sections that the consequences are reason able from a physical point of view. The problem is caused by the procedure of solving an initial value problem. As we show in the next chapter for the harmonic oscil lator, the level shifts are of the same order as the damping and we have no way of separating the two. Neglecting the time-dependent dif fusion-like terms corresponds to neglecting the change in the states 30 brought about by the interaction. A fuller discussion will be given after we have solved the special case of the harmonic oscillator. Some idea of the physics involved can be obtained by investi gating a free particle (V=0) coupled to the radiation field. In this case Eq. (24b.1) takes the form t è pi(t) = Ü? IP'tP'PllP {£7 , (25) o which is a diffusion equation with a tine-dependent diffusion coeffi cient. The width of the Green's function o at time t is given by t' 2e2 _ f .. . _ a P jdf P jf£ = ||4 log t/t . (26) 'x 3m- o t is an arbitrary time scale which arises from the indeterminacy of the principal value. As the only time scale in the problem is the mass of the particle, we take t = l/m. Then, even for macroscopic times, the width is of the order of l/m, which is the Compton wavelength of the particle. The approximation (24b.2) replaces Eq. (25) by _d_ P (t) = dt i Ü£ iP'lP'Pll* ffi = ° ' <27> which is acceptable from a physical point of view, as effects involv ing frequencies equal to the mass are not included anyway in the pre sent (non-relativistic) framework. 3. Dissipative equation of motion We introduce the notation A = i P V eiHTA e"1Ht (28) for the Hilbert transform of A(t). The operator A depends both on A and on H. Now we collect all partial results (19 through 24) of the previous section into the dissipative equation of motion for a system coupled to the radiation field A pi + I irfP^Mf^UfPtO)!] + ^r[p(t),[p(t),p(0)n • (29) 31 The validity of a derivation using perturbation theory is limited by two time scales. The times considered have to be short enough for second-order perturbation theory to be applicable and long enough to allow transient effects» caused by the high-frequency cutoff, to die out. The use of the ?u strength function avoids this complication. There is no high-frequency cutoff and no transient time, so that there is always a time scale on which the derivation is valid. The only essential approximation is the neglect of the shifts in the levels. If this is unimportant, as it is in the classical domain, the deriva tion is valid also for strong damping. The next step is to make the random phase approximation. This is the assumption that the bath is again uncorrelated at later times. This allows replacing p±(0) by p,(t) in Eq. (26). A better formulation is that from the infinite order perturbation expansion only the terms corresponding to repeated second-order terms are retained. The final dissipative equation of motion (omitting the uninteresting driving force), obtained by means of perturbation theory and the R.P.A., is (t) = ar <>i i£ [p + 3ÏT Ip + 3mTflP(t)4P(t),Pi(t)]] . (30) This equation can be transformed back from the interaction representa tion, which gives ^ p(t) = "i[H,p(t)l + ^[p,[|^,p(t)]] + + WÏP'tH'^n - 3fflTg[P,IP,P This is the operator equation for the (reduced) density operator. 4. Expectation values The dissipative equation of motion for the state can be con verted into an equation for (expectation value of) the observables. We multiply Eq. (31) by an arbitrary observable A(r,p) and take the trace. Using cyclic change under the trace, we find g - MH,A1 - gJ[|J,[p,All • + 3¥?[!lF'lp'A1J " 3¥73IP'tP'A]1 ' (32) 32 Strictly speaking, this is an equation for the expectation value of A, but we can interpret it as an operator equation. Putting the time dependence in the observables instead of in the states corresponds to changing from the Schrödinger to the Heisenberg picture. For observables depending on the momentum only» there is no contribu tion from the dissipative terns. For the first moments we find 2 d E 2e 3V ,33. 3t r = m W 3r" ' <33> d 3V -,.. dt P " ' 3? * <34) Equations (33) and (34) are valid both classically and quantum-mechan ical ly. The radiation reaction occurs in Eq. (33). It cannot be inter preted as a force. A force should occur in Eq. (34). If we accelerate a particle by means of a potential, the radiated energy comes from the potential energy and not from the kinetic energy of the particle. In the usual treatment of the radiation reaction (see e.g. LANDAU and LIFSHITZ (1971)) a force is found which gives on the aver age (for periodic motion) the correct energy loss. Equations (33) and (34) give a correct description of the motion without averaging over the period. This conclusion has already been found (ROHRLICH (1965)), using other means. For the energy we find .2 d - _ 2e2 dt E " ~ 3m? [Ir-J + 3Ï?[^'3?J * <35) The first term is the classical energy loss of an accelerated charge, the second term is the quantum radiation reaction which can be as cribed to the zero-point energy of the radiation field. In the next section we show that for a quantum system in its ground state the two terms cancel each other. The fall of an electron into the nucleus is not modified, but the zero-point fluctuations of the radiation field maintain the ground state. This is worked out in more detail in chap ter VIII for the H-atom. 5. Matrix elements, transition probabilities For the interpretation of the QH equation of motion we can take matrix elements in the representation generated by the eigenvectors of V H|k> - uk|k> , H - ?^ + V(r) . (36) 33 All matrix elements can be expressed in those of r <3lfjl*> * •(»j-wk>* The last line follows directly from the definition of the Hubert transform +00 i(tu.-u>. )t < k k " 5i|?l >? jf* •" - i sign(u>ru,k) We introduce 3r ~ 3r " x 3r t 3V~ 3V . 3V (40) 3r " 3r 3r - wr • They have the matrix elements 2 <31|^ |k> = 2H(u)j-a)k)(u)j-ü>k) 2 <3l|J |k> = 2H(ü)k-o)j)(ü3j-ü)k) The equation of motion can be rewritten as 2 2 dp ,,„ , . ie . dV~ . 3V , e - r ,, ,.,, £ = i|HfPJ + -y- [Pf ^ p+P?? ] - -^jjy ïPi[p,p]l . (42) We see that the operator p • |0><0| is the stationary state if T=0. This is in accordance with the neglect of the change in the state caused by the interaction. 34 In terms of matrix elements the equation of motion is ^ 2 + T II (o) -(ü.) (w -u ) H(u,ri-u)m) 2 + (a)_-o)v) (to -co ) H((i> -d))< j | p|m> + (UJ -to ) (w -ai. )< j |p|mXm|r|nXn|r|k> (43) m n n K The classical term in Eq. (31) produces both upward and downward tran sitions. From Eqs. (33), (34), and (36) we see that the term with the Hilbert transform of -53V— cancels the upward transitions, leaving only transitions to lower levels. If the density operator is diagonal in the number representation at some time ([p(t),H] = o), then and the diagonal elements at later times can be found from 3 ^ This has the form of a gafn-loss equation (or Pauli equation) A with wkn = I e7('VCun)3H(wk"un)| 35 stimulated emission as a result of the temperature of the bath. The result is valid only because we assume p(t) to be diagonal. However, a diagonal p will not remain diagonal. The off-diagonal terms influence the motion. This will be demonstrated in the next chapters for a harmonic oscillator. A discussion of the derivation of the Pauli equation is given in chapter VI. 6. Harmonic oscillator The simplest special case is the harmonic oscillator. The solu tion to the undamped initial value problem is r(t) cos ID t sin u) t o o (46) P(t) -sin <») t cos ui t o o The Fourier transform of p(t) is p(ü)) = >sp(ó(u)-u>o) + ó(d>+ü)o)) + H r (6((i)+uo) - 6(u-uo)) +» -j3V- = /do))u)| 'p(u)) = ui p . (47) 3r This, together with -53V— = u r, enormously simplifies the operator equa- ÓT O tion. We obtain 2 j|£ = -i[H,pJ + § ie wo[p,[r,p]] + J 2 2e + y e wQ[p,[p,p]] - -jg- [p,[p,p]] (48) The solution to this equation is discussed in section V.l, and a com parison between perturbation theory and the exact solution is given 1n chapter VI. 7. Classical mechanics and quantum corrections The quantum-mechanical calculation of the preceding sections can be repeated classically, but it is possible to convert the quantum results directly into the corresponding classical results. Straightforward conversion of Eq. (35) yields 2e: 3V 3V JLP . -{H,P) +gr (P,£P} + ^ {p,{p,p» -fctp.tjfp», (49) 36 where the bracket is a Poisson bracket, and gr has been written as rr, in accordance with the classical convention. As a result of using nat ural units, some care is needed to distinguish classical and quantum- mechanical terms. Replacing a [ , ] by { , } introduces an ft, and B contains an ft, so the first three terms on the right-hand side of Eq. (49) are classical, while the isest term is quantum mechanical. If we keep only the classical terms, we have obtained the classical Kramers equation (which is the Fokker-Planck equation with an arbitrary poten tial) . The derivation is valid for strong damping and does not involve a transient time. Only the R.P.A. is essential. It is possible to give a classical interpretation to the quan tum-mechanical term in Eq. (49) by introducing the concept of a clas sical zero-point energy. If we take the classical temperatures of the radiation oscillators to be given by kT. - %hü). or 0.w. = 2 , (50) 3 3 ] ] the classical thermal energy equals the quantum-mechanical zero-point energy. This classical zero-point energy gives rise to the quantum- mechanical term in the equation of motion (49). In chapter VIII we show for the special case of the H-atom that the introduction of clas sical zero-point energy is sufficient to prevent the collapse of the classical H-atom. It remains true that r goes to zero, but the phase density function reaches a steady state with a finite energy. In chapter IV the same correspondence is found for the harmonic oscillator (compare section IV.4). For temperatures different from zero we obtain (49), if we replace the classical temperature kT by kT» = hhoi coth h jgj - (51) For high temperatures or low frequencies the new temperature equals the old one, while for low temperatures or high frequencies the oscil lators have the zero-point energy. We calculate the stationary state at zero temperature of Eq. (49) for a harmonic oscillator. Sufficient conditions for stationarity are {H,p} = 0 , 2wrp + {p,o) = 0 , (52) which means that p has to satisfy p = p(r2+p2) , 2rp + |£ = 0 (53) The solution is (unnormalized) .2_,_ n2, ^z-5!7w * ^ p = e r p = (54) 37 This, we recogni2e as the Wigner distribution of the QM ground state (G.15). We have ^ We also see that 8. Discussion of the piü strength function In order to understand the meaning of the pu strength function, we compare it with 7(01) = Ü)C(ÜO), where C(OJ) is a cutoff function, 2 ' ft 2 which we take even in u, 1 + °(^) f°r w < 9, and 0(—5-) for w > 6 . In Eq. (19a) we have to evaluate t 00 t T°o /dr p(t-T)'/dü> Y(w)sin <»n = Jdx p(t-t)'/dw iwCMwJe T 00 o -°° A simple and often used choice for C(CD) is 6' 1 • C(w (57) > =137+u ? H i+iu/e * i--iu /e) We obt ain t p(t- •i )e sign( T)e •e|t| (58) o For «T > 1 this is Gp(t) + p'(t) + 0(6_1) . (59) This diverges linearly with e. The same happens with other choices for the cutoff function. The reason is obvious. We are attempting to inte grate half a 6'-function. 38 We now use the theory of generalized functions (as in chapter II), and ask for a cutoff that will make the Fourier transform of the Y-func- tion resemble *<6;+Ó;) . A simple choice is m * tf 1 ^ 1 1 Qk -92u>2 ,,.. C(U)) = '|(1 + W8)^ + (l-io)/6)2j " (82+0,2)2 • (60) With this cutoff the Fourier transform of the strength function is 2 9 T 8 (l-6|T|)e" I lsign|T| , which is (compare Eq. (II.7)) a regular sequence equivalent tc 6'(T). For 9t > 1 the time integral equals p'(t) + OO-1) , (61) which is (59) without the divergent term. Now the limit 9 -* » can be taken. The only difficulty is that Y(I*>)C(UI) is not positive definite. This is so, independent of the particular cutoff function, as in the limit QO P Jdu) y(w)C(to) = 0 . O A negative strength function cannot be obtained as the continuum limit of a finite system, since the strength function is given by y(u) = lim AjX^g(w ) . A negative strength function can be obtained only by assuming a nega tive density of modes or a non-Hermitian (non-real) Hamiltonian, both of which are unphysical. However, in the limit the negative strength occurs only for in finite frequencies which are not observable. The continuum theory with fdi is not the limit of a discrete theory or of a continuum theory with an ordinary cutoff. It is a different postulate which should be judged by its consequences. Even in the incomplete theory we have developed so far, it recommends itself by its conceptual simplicity and the automatic disappearance of disagreeable divergencies. The theory should be developed further to check its agreement with experiment in more detail. 39 CHAPTER IV EXACTLY SOLVABLE MODELS POR DISSIPATIVE MOTION 1. Introduction In chapters IV and V we investigate quadratic Harailtonians which can be diagonalized explicitly. This makes it possible to solve the equations of motion and to calculate the density operator for the whole system or the reduced density operator for parts of the system for all times. The outline of chapter IV is as follows. First, the time-depen dent operators are expressed in the corresponding time-independent operators and supposedly known functions of time, which is possible as a consequence of the linearity of the equations of motion. Then, the reduced density operator of the small system is expressed in terms of the reduced density operator at t= 0 and the same functions of time. For the purpose of comparison with perturbation theory we de rive, from the known density operator, a (differential) operator equa tion which it satisfies. It is convenient to use a characteristic function instead of the density operator itself. In special cases the method of ULLERSMA (1966) can be used to calculate the time functions. This is done for coordinate coupled HamiLtonians in App. K and for symmetric Hamiltonians in App. L for the radiation field, which is the most interesting special case. The theory of generalized functions, as developed in chapter II, makes it possible to avoid the usual divergencies without introducing a cut-off function. Consequently, there is no transient time and the derivation is valid for strong damping or even overdamping. This is an improvement over previous results. In chapter V the time functions found in App. K and L are substituted into the general framework of chapter IV. In chapter VI the differential equation for a damped harmonic oscillator, following from perturbation theory and the Random Phase Approximation (R.P.A.), is compared with the exact solution of the same problem, and the results of chapter IV and V are summarized. The different derivations of the Paul! equation are compared. 40 The presentation in these chapters is more general than the work of Ullersma, since it is not restricted to coordinate coupled Hamiltonians and to diagonal elements of the density operator. It is an improvement over the derivations of the Fokker-Planck equation in the theory of the laser (see chapter IX), since the R.P.A. or an equivalent stochastic assumption is not needed. Chapters IV and V are a combination of these two approaches to the description of a damped harmonic oscillator. The language and notation are quantum mechanical, but at all stages in the calculation it is possible to change over to a classical description. The final results are used to obtain quantum corrections to classical mechanics and to prove an equivalence theorem between quantum mechanics and classical mechanics with zero-point energy. 2. General quadratic Hamiltonians We want to investigate the behaviour of one harmonic oscillator in an assembly of coupled oscillators having a quadratic Hamiltionian. Without loss of generality we can assume the other oscillators to be uncoupled, since this can be brought about by a suitable canonical transformation. Then the Hamiltonian Can be written in the form H - KbObO + *£Mjbjbj + Vy*jbObj + XjbObj + ^jbObl + ^jbObj)' (1) J 1 J X where N is the as yet unspecified number of oscillators and the b's are boson operators, apart from a factor of two, satisfying the commu tation relations [bjrbjl = 26jR , (b^b.] - [bt,b£] - 0 . (1) The extra factor 2 is introduced to give a closer correspondence with complex numbers (see App. B). The resulting (operator) equations of motion are linear ax bo = ifH'boJ = "i(Jobo " *\ A*bj " ^ wjbj ' ^ bj - MH.bj) = -iUjbj - iA.jb0 - iUjb* . (3) The formal solution to this set of equations iHt iHt b0(t) = e boe" " I -4^ lH,bQJ , (4) n=o n 41 can be written in the form bo(t) . gl(t)bo • gJ(t)bo + Hh^CWbj + hJjCUbJ) , (5) since the commutator of H with a linear function of the b's is again a linear function of the b's. The functions g and h can be obtained by solving a system of linear equations. This is done for special cases.in App. K and L. The commutation relations do not change in time, [b0(t),bo(t>] = [b0.bol = 2 , so that the functions g and h must satisfy »1«* -*2*2 + nhljhïj-h2jh2j) = X ' (6) where we have dropped the time arguments to simplify the notation. The density operator at t = 0 is chosen in the usual way (see III.2, E.3, and E.8). p(bo,bJ;b.fbj,0) = Poü>o,b*,0) en Pjü ^ynyzj , (7) where p (0) is an arbitrary density operator for the small system, and the p.'s are displaced thermal states. We want to know the reduced density operator of the central oscillator at later times p (t) = Tr (e"iHtp(0)eiHt) . (8) all j In order to make manipulation of the operators under the partial trace possible, we multiply (8) by a suitable function of b and b and trace over the Hilbert space of the central oscillator to get a trace over the full Hilbert space of the system. As we do not want to lose in formation all expectation values must be represented in this function. In classical mechanics a suitable function is exp(-ik x-ik p). The x p resulting function of k and k is the Fourier transform of the phase x p distribution function. In quantum mechanics we choose an analogous function, but x and p are operators and we have to order them, we choose the following characteristic function, ik,b lkbt C(k,k\t) =iTr(e^ e^ po(t)) , (9) o but this choice is a matter of convenience and personal taste only. The motivation for this choice is given in App. G and H. 42 Given the characteristic function C we can find a phase distribution as its Fourier transform i k2#+k#Z f (Z.Z*) = JL |d2k e^ < >C(k,k*> The density operator in normal ordered form is found by substituting z -> b and z* + b p(b,bf) = N f(b,b+> . The necessary properties of coherent states, phase-distribution func tions, and characteristic functions are summarized in App. E through H. A combination of (8) and (9) gives a trace over the whole Hilbert space. A cyclic change under the trace gives iH ik b %ikb iHt C(k,k\t) - JLTr(e V^ * e- V p(0)) . (10) The time evolution is factorizable (see (A.6)), which means that A B r an two (AB). = t t f° Y operators A and B, which can be used to bring the time evolution under the exponentials , , -%ik*b (t) -^ikL^(t) ï C(k,k»,t) = j^ Tr e ° e ° p(0)j . (11) Now we express C(k,k*,t) in terms of C(k,k*,0) and the known functions g and h of time. Using the product property (7) of p{0) and the commutativity of the b and b. operators we can factor the trace into a product of partial traces l C(k,k*,t) = ^ Tr e ° * ° e -«oio po(0)J x f -^^(hijb-i+h'.bt} -%ik(h b.+hj b+) in J 3 21 } x n Tr e e -«D D *3 D p.(0) # j j I 3 J (12) The traces can be evaluated by reordering the exponentials. This can be done by means of the Baker-Hausdorf lemma (A.7), since all commu tators of b-operators are constant. The first trace can be expressed in terms of the C-function at t = 0, 2 *(k*2g g*+2kk*g g*+k g*g2) , -%ib (k*g +kg ) -%ib0(kg*+k*g*) l 4 w x 01 l = e * * Tre ° * e pQ(0) 2 \(k2g*g +2kk«g2g*+k* g g*} = 2ÏÏ e * z l * l * C(k',k,#,0) , (13) 43 with k' given by k' k k = f Re(g1+g2) Imtg^) j [kl (U) = or [ il = k«* g -lm(gi+g2) Re(gi-g2)j(kJ V J 1*2 J l*pJ In the same way we reorder the b.-operators in Eq. (12) a (...) me*(k hïjh2j + 2kk-h2jhJj + k«hljh5j) K Tr 3 ~%ibj (***>! j+khj .) -liibjfkh^+k'h^) x Tr P5(0) . (15) 3 The remaining trace is simply evaluated» using operator techniques (H.9, H.10), as it is the characteristic function of a displaced ther mal state. (k'X jh2 j + 2kk#h2 jh2 j + k2hI jh2 j) Tr( ) = e k j # (-%iZj(k h1:J+kh2:.) + -isiz^kh^+k'h^)) -hln^+D (k*h1;.+kh2j) (Wi'j+k'hjj) (16) Finally, combining all partial results, we find a simple expression for the C-function at all times 2 # 2 -U(kw*+k*w) -k(k b11+2kk b1_+k* b„) C(k,k»,t) = C(k',k'*,0)e e l w = I w, , w. = hjjZj+h5.z5 , bn = I (2nj+l)hljh*. - gig- = b*2 , b b 2l - i2 - I ("j+Dhj.h». • I lyh^hjj - g2g* . (18) The notation can be simplified considerably by introducing vector notation in the k and z planes. The vector k" can be represented by k and k or by k and k*. The scalar product is diagonal in terms of k and k and given by # # (k,z) = xkx + yky - %(kz +k z) . In a scalar product we omit the arrows on the vectors. 44 In this notation (17) takes the form C(k.t) = C(Ak\0)e-i(k'w)-*(k'Bk) . (19) In the k »k representation A and B are real, B is symmetric and given by hlj+h2.|2 2lm(hljh*.> i B = *sl - >5A A + l(n.+h) (20) J 3 2I«(hljhSj) |hirh2j|2 The matrix A is given by Eq. (14). We will see later that the matrix B is not necessarily positive definite. This does not matter, as the only requirement is that C(k*,t) represents a physical state. As it is obtained by reducing a physical state of a larger system, this is obviously true. From the C-function we can find the f-function by Fourier transforming Eq. (18). For two-dimensional Fourier integrals we do not use the same convention as was used in chapter II for one-dimensional integrals. i(k z f<1,t> -£ Jd*k e ' >.C i(k z) i(k w) %(k Bk) ,£ |d2ke ' .C(A^,0)e- ' - ' = = _^ |d2k Jd2z. ei(k,Z)e-ilAk,z')f(.,0)e-i(k,w)-Js(k,Bk)> (21) This can be written in the form f(z,t) = d2z' G(z,z\t)f(z\0) , (22) with G given by 2 T G This is a simple Gaussian integral which can be evaluated at once T G(z\z',t) = i ^((Z-AV-W) , B-l(z-A --w)) (24) 2irdet(Bp It is a Gaussian distribution centered at -z+ = A.T-* z'. + •w* . The matrix B"1 gives the shape and orientation of the Gaussian. Now that we have found f(z,t) and hence p(b,b*,t), all expectation values and all matrix elements can be evaluated in terms of expecta tion values or matrix elements at t = 0 and known functions of time. 45 3. Differential equations In this section a differential equation for the density opera tor. p(t) will be derived for the purpose of comparing perturbation theory with the exact solution, which is done in chapter VI. The most convenient procedure is to derive a differential equa tion for C, which can be converted into a differential equation for f, which can be transformed into an operator equation for p. Starting from Eq. (19), differentiation with respect to t gives (k Bk) i(k W> |f = (gi , Vk,)C(AÊ,0)e-* ' - ' + C(k\t) , (25) - U(k,si k) + i(kfg) and differentiation with respect to k gives V.C(k\t) = ATV C(Ak\0)e~*(k'Bk)"i(k'W) - (Bk+iw)C(k\t) . (26) K Ak The initial conditions can be eliminated from Egs. (25) and (26), which gives 1T 1T £ C(k\t) - (§& k,A- Vk)C + (k,f£ A" Bk)C + lT - %(kf§5 k)c - i(k,fj[ - A- w)C , (27) where A is assumed to be non-singular. The times for which det A = 0 must be investigated separately. Equation (29) can be written in the form ~ C = (k,aVk)C - (k,Dk)C + i(F,k)C , (28) where a - (r' £)T . * - | - A"1T5 . (29, D = %(§f " "B ~ B« ) . (30) We transform Eq. (27) for C into an equation for the phase distribu tion function f (see H.l) f<*,t) -^ d2k ei(k'z)C(k,t) . (31) The Fourier transform interchanges multiplication and differentiation. The transformation rules are Vk - -iz , jc - -iVz . (32) 46 Applying these to Eq. (27) gives D7 )f + (33) || - ~(Vz,az)f + zf «--»• pb z*f +•*• bTp , |§ <--> % [ p, bt ] = MpbVp) , 8f jpt •*-* %[b,p] = fc(bp-pb) (34) In order to avoid vectors with (super)operator components we write Eq. (33) out in the z,z* representation. If we write out the matrices in the x,p representation. -Yi Dll D12 tt = D = D12 = D21 ' 'w2 "y2 D21 D22 we find in the z,z* representation i w +a3 - + +i ü) w T1-Y2~ < 1 2^ Y1 Y2 < i~ 2^ El = kf_L 3 9t ^{dz Tz* Y + Y 1 a) tü +i (j) +u l 2~ ^ l~ 2^ -Y1-Y2 ( 1 >2^ 3 Dn-D22+ i(D12+D21) Dll+D22 _L 3 1 d 9z 3Z Dll+D22 D11-D22-1(D12+°21) (35) Fè-F*^|9z< f Applying the transformation rules (34) and working out the commutators gives + + f + + |f = -fciUi+Wj) (b bp-pb b) + i(Y1+Y2) (bb p+pbb -2b pb) + w b2 c + i(Y2-Y1+i((i>2" l^) (hpb-P ) + h« ' 47 + + + + • |(D11+D22)(bpb +b pb-bb p-pb b) + 2 2 • (Du-D22+i(Dl2+D21)) (b p-2bpb+pb ) + h.c. , (36) where h.c. stands for Hermitian conjugate. For special choices of the Hamiltonian Eq. (36) simplifies con siderably, as we show in the following chapter. A special case of Eq. (36) is what we call a diagonal equation for p. By this we mean that a diagonal p in the number representation remains diagonal in the course of time. From a diagonal equation for p the Pauli equation can be obtained at once by taking matrix elements. The necessary and sufficient conditions for the uncoupling of diagonal and off-diagonal elements are = Y T1 = >2 Wj = di2 = u Du - D22 = D Dl2 = D2J » 0 , (37) which gives the equation of motion || = -Uu>[b+b,p] - %Y(bb+p+pbbf-2bfpb) + + »jD In this equation D and y cannot be chosen arbitrarily. In order to represent a physical state p must be positive definite. The coeffi cient of b pb is %D - y. If we calculate the -rr of the ground state we see that this coefficient must be positive, otherwise p will not remain positive definite, so that a necessary condition for the acceptability of Eq. (38) is D > 2y . (39) In App. G we see that the phase function f, associated with a positive definite p, has a given minimum width. This is a result of the quan tum-mechanical uncertainty. By inspecting the Fokker-Planck equation (33) we see that the damping terms tend to decrease the width. If D - 2y the diffusion is just strong enough to maintain the ground state, which has a minimum uncertainty. The condition (29) is also sufficient to ensure positive definiteness of p. Writing D = 2y + Dtn , (40) Eq. (34) takes the form + + + t + |g = -%iü>[b b,pj - %Yib bp+pb b-2bpt' ) - »sDtn[b,[b ,p]l . (41) 48 The first term gives the undamped motion of the oscillator, the second term gives the damping, and the last term represents the thermal dif fusion. In the next chapters the coefficients in Eq. (36) will be cal culated for special cases. In chapter VT a discussion of the deriva tion of the Pauli equation is given. This amounts to showing in which cases the conditions (37) hold. 4' Classical mechanics and quantum corrections If we are content with a classical description we can stop at the moment the Fokker-Planck equation (33) is established. In this section we derive quantum corrections to the classical Fokker-Planck equation. In classical mechanics the ordering of the operators plays no role so the classical equivalent C , of the characteristic func tion (9) is just the Fourier transform of the phase distribution func tion. The n.'s are given by fftü,il kTi fftwll kT n. + \ = hcoth The result of the classical calculation can be found from the quantum kT 1 calculation by replacing n. + % by =•— = s—r# and omittinga the term T 1 * fwj Bwj ^1 -^A A which results from the non-commutativity of the b-operators. Of course, this can also be found directly by repeating the whole deri/ation in the classical notation. The occurrence of ft in a classi cal formula is a consequence of the normalization as explained in App. B. From (19) and (20) we see that the classical characteristic function is given by Ccl(k\t) - Ccl(AÏt,0)exp(-i(ie,w) - *(Jc,Bclïc)) (43) with B , given by |h h |2 2Im(h .h* )- kT. lj+ 2j 1 j B i = l ÏT-1 (44) Cl L. fid) .^IrMh^h*. ) |hirh2j|' fc/jjotions (19) and (44) differ by the term \1 - ^ATA. The appearance of this tern is a consequence of the choice of the characteristic function. We define a new QM characteristic function by C(k,k*) - e*kk*C(k,k*) . (45) w 49 In App. F, G, and H it is shown that this C is the Fourier transform of the Wigner distribution function. In terms of C Eq. (19) becomes Uk k,k, Cw(ïc,t) = e *e~* *Cw(AÊ,0)exp(-i(iï,w)-ïs(£,BÏc)) . The extra exponentials just cancel the extra teriüs in B. The result is C (Ê,t) = Cw(A£.0)exp(-i(Ê,w}-»i<Ê,Bwk)) (46) with B given by 2 |h.,+h, . I 2Im(hljhJj) B = y(n.H) [47) 1 ^imth^h*.) |hirh2j|2 By comparing (43) and (44) with (46) and (47) we see that the corres pondence between quantum mechanics and classical mechanics is closest when we use the Wigner distribution. The correspondence can be made even closer by introducing a classical zero-point energy. By this we mean that the temperature of the classical oscillators is raised in such a way that they have the same energy as the quantum-mechanical oscillators at a given quantum- mechanical temperature. Comparing (44), (47), and (38) we see that for the classical temperature we have to take ha). (48) ]qm In particular for T. =0 the classical temperature satisfies r jqm kT. . = Ijtiüj . . For low frequencies the energy is independent of the frequency (classi cal equipartition), while for high frequencies the energy is ^hw which is the QM zero-point energy. With this assumption (42) the Fokker- Planck equation for the Wigner distribution is identical to the Fokker- Planck equation for the classical phase distribution function. As shown in App. G, expectation values may be evaluated as phase integrals. There it was shown that to every QM observable A and every QM density operator p phase functions A(p,q) and p(p,q) can be assigned in such a way that :A> = Tr(Ap) = /dpdq A(p,q)p(p,q) . 50 In particular, it is possible to take p(p,q) to be the Wigner distri bution function. This allows us to formulate the following equivalence theorem: For systems with a quadratic Hamiltonian the quantum-mechanical dissi- pative motion is indistinguishable from the classical dissipative motion with zero-point energy. Indistinguishable means that from the solution of the classical prob lem with zero-point energy it is possible to compute all quantum-me chanical expectation values. It does not matter that the Wigner distribution is not positive definite. It is not a probability distri bution in the classical sense. Correspondingly, the 6-function is not observable. The Wigner distribution can be converted into a probabil ity distribution by a suitable diffusion and a corresponding change in the observables. This has been demonstrated in more detail i-i App. F and G. 5. Equation of motion for observables The equation of motion for the state which we derived in the preceding sections, can be converted into an equation of motion for the observables. This is analogous to the change from the Schrödinger to the Heisenberg picture in ordinary quantum mechanics. As an example we take Eq. (41), multiply it by an arbitrary observable A, take the trace, and use cyclic change under the trace, which gives t + + + + ^ A = ssiu)[b b,A] - W (bb A + Abb - 2b Ab) - ^yn[b , [b,AJ ] . (49) This is an equation for the expectation value of A as a function of time, but it is true for any density operator, so we can interpret it as an equation for the observable A as a function of time. In particular, we find ^ b = (-iu-Y)b , oT b+ = + + ~ b b = -2Y(b b-2n) , (51) g^ bb+ = -2Y(bbf-2n-2) . (52) 51 For products we use the notation A.A2(t) for the evolution of A.A-f while A,(t)A2(t) denotes the product of the evolution of A, and A-. This distinction is necessary, since , dA, dA_ , £ (Al(t)A2(t)) = -g^(t)A2(t) +Ax{t) -^ (t) *± (AlA2(t)) (53) The chain rule is valid for the products of the evolved operators, but not for the evolution of a product. When the coefficients in Eq. (41) are constant in time, we find the solutions b(t) = b(0)e-iwt-^ , b+(t) = bf(0)eiu>t-Yt , (54) b+b(t) = e"2Ytb+b(0) + (l-e"2yt)-2n , (55) bb+(t) = e"2Ytbb+(0) + (l-e"2yt)(2n+2) , (56) which shows that b(t)b+(t) - b+(t)b(t) = 2e"2yt , (57) and that bb+(t) - b+b(t) = 2 (58) for all times. It is instructive to compare Eqs. (57) and (58) with the derivation of Eq. (6). There, t>(t) is an operator in the Hilbert space of the small system and the bath, while in Eqs. (57) and (58) we have operators in the Hilbert space of the small system alone. 52 CHAPTER V SPECIAL QUADRATIC HAMILTONIANS 1. Introduction The general quadratic Hamiltonian (IV.1) can be diagonalized explicitly only for special choices of the coefficients A. and y .. Three special cases are investigated in the following sections. The actual calculation of the time functions is done in App. K and L. 2. Hamiltonians with coordinate coupling In this section the special case of the Hamiltonian (IV.l), which has A. and \i. equal in magnitude for all j'S, is investigated. In order to make the notation conform to ULLERSMA (1966) we put with the commutation relations [x ,p 1 = ij , [x ,x 1 = [p ,p 1 = 0 . (3) 1 n Mir nm ' l n nr ltn'MnJ With this notation the standard Hamiltonian (IV.l) takes the form H = *( 2 + 2x2) + *Hp2 + <^x2) + x x (4) vtPo W o _ n n n; IL cn o n o' n n We call Hamiltonians of this form coordinate coupled as only one coor dinate (in phase) of the first oscillator couples to the bath. It does not matter if we take xx, pp, or xp, as x •* p, p -* -x is a canonical transformation. 53 Hamiltonians of this type have been discussed by Ullersma to describe an electron coupled to the radiation field. Notice that the x and p variables are not dimensionless, in contrast to the ones used in chapter IV and App. B. This is necessary for the diagonalization of the Hamiltonian. The solution to the equation of motion can be written in the form • x (t) A A X A A 1 X. o 3 f ° ] *• • + I » • (5) p0(t) A A p P ° j 3 The A's are in principle known functions of time. They were calculated by Ullersma for some special choices of the e.'s. This calculation is repeated in App. K and, in addition, we demonstrate that the general ized functions of chapter II make it possible to calculate the A-func- tions when the bath is the radiation field without further approxima tions. The A-functions and the g- and h-functions of chapter IV are easily converted into each other * A gx = A - %i di A O u) O (6) o' r /u h, . = ^A. 2 y^ - ^i IA . /ui w . - J— 13 3 + V h- . = ^A . '".Ï (7) 23 3 J /(JU Ü! . ' o 3 The commutator property {IV.5) is now given by A2 - AA + l{k2. -A.A.) = 1 . (8) 3 3 3' The A-matrix (IV. 14) I*. given by a: A O [9) k' (ill A 0)_A I PJ 54 In order to express the elements of the B-matrix in the A-functions we need 2 ü) In. . + h,. I = A2 — + wu.At ? ü). A. 2 |h.1 . - h,.| = A -i + -J- 1J] 2Jn ' Jl uoi uoi o)j. 2 2 l9i + 9: = A + tij^A 2X2 I9i ~ 92 I = A- + w-*A imfh^h*.) *KIAA + WJAJAJ3 ' lm(gig*) = %(u-iA& + u>0AA) (10) The B-matrix (IV.20) reads T B * *jl - )sA A + J(n, + ^)ATA. , (11) j J J where the matrix A. is given by u)0 A3 - A. = (12) 3 /V j [%"jAj wjAjJ The fi-matrix (IV. 27) is now easily calculated d T -IT (13) Q - ^(A )A t-u(t) -2y(t)J with -1 A2 -AA AA-AA -in (t) = -2y(t) = (14) % A2 - AA A2 - AA 2 -2Y(t) = A log|A -AX| . A useful identity follows at once A(t) + 2y(t)A(t) + w w(t)A(t) E 0 . (15) In general, the B-matrix cannot be evaluated in a simple way without taking the continuum limit. However, for sufficiently high temperatures (n. > 1 for all j's) the sums can be evaluated directly without taking the continuum limit. 55 The n.'s are given by n. + = *scoth %6^w^ = h j"j " e^ + 0(Bi-j> • Retaining only the first term, and taking 6- = B independent of j, we obtain wo (Aj + ujAj) ">oA j (Aj + w jA j) B » ^1 - *ATA +i[-l (16) 6(i>0 3? U34 u A. (A. + w2A.) A? + w?A? We see that B B 8 (17) 12 21 2 u o dt 11 which is an exact consequence of choosing a coordinate coupled Hamil- tonian. The sums (16) can be evaluated directly, even for a finite system. This was done by Ullersma. The result is T 0)T2 • (A? + U)?A?) = (i>72 - A2 - w2A2 , j D v 3 3 J' 1 1 ,2*2 I tuT2 • (A? + d)?A2) = 1 - A2 - w2i ? 2 I uJ Aj(Xj*«2Aj) = -A(A+(D A) (18) where A(t) is defined by . t E? A(t) = -A- + /dT A(T) , u)2 = u)2 _ I _| (19) "'•1I' o' 10. (1) . Introducing this in (16) we find 2 2 2 2 'W0(*D7 " A - wjA ) -(JOA(A+(O A) B « ,1 - *ATA • JL (20) -UA(A+UJA) 1-A2-W2A2 The diffusion matrix defined by (IV.27) D = * dT B " *nB " ^BQT can be written in terms of the A-functions. The hi part contributes 0 u>(t) - u), " k(n+n ) = k ui(t) - ui 4Y 56 T The term with HA A does not contribute since gr (A A) - A A AA-AAA'A=0 The contribution of the thermal part to the diffusion matrix elements is D1X = 0 (21) for all t, which can be seen from (IS) and (17), or directly from the form of the Hamiltonian, 2D12 = 2D21 = dT B12 " woB22 + »(t)Bll » • /• D22 = *B22 + w(t)Bi2 + 2^^>B22 = = 1 |2y(t) - (A)2A(A+2Y(t)A + u) w(t)A)| . (23) This is as far as we can go without taking the continuum limit. The result in the continuum limit depends on the strength function. The simplest case is the radiation field with strength function yU) = (2e2/3nm2)-?w2 . (J.6) Then we find e~TIt| 2 2 2 2 A(t) - H Sin ut Y = i. E_ M ui = ai - Y • (K.14) w J m* o o Notice that we use three different Y'S. y(u>) is the strength function, y(t) is the time-dependent damping, and y is the constant value which Y(t) assumes in the continuum limit. The result is valid for all times. There is no transient time. It is also valid for strong damping as long as w2 > 0. Substituting (K.14) into (14), we find Y(t) = Y'Sign(t) , w(t) = u (24) for all t. Now Eq. (15) can be integrated 2 A + 2Y'A + wtoA A + 2YA + ü) A * 0 . (25) t=o 57 Substituting this in Eqe. (22) and (23), we find as final results w. 0 Q = D = (26) -w 2y o 0 •('•£ which is valid in the continuum limit for high temperatures (8 < w ). The zero in the (1,1) place is an exact consequence of the special coordinate coupling in the Hamiltonian. The results D.~ = D21 = 0 and ^12 - ~^2l are a conse(3uence °f choosing fu2 as the strength function. For low temperatures the sums can be evaluated in the continuum limit. We do not bother to do this since we are interested in the dis sipation and not in the details of the adaptation of the uncorrelated initial state to a correlated final state. In order to see how impor tant these effects are, we calculate the reduced interacting ground state. This is done in App. K, where we find the characteristic func tion (K.21) under the assumption y < u> -JL-(k2+k2) C(k,k*|4>) = C (k,k*) e (K.21) This shows that the decrease in the width of the ground state is of order Y/^« We keep only the lowest r.on-vanishing order. The undisturbed width of the ground state is equal to one, so we can neglect the change. In this approximation we have X2 ov y s x± = w„ and (27) "v"'ov I s w. V V These considerations lead us to replace for instance 2 2 2 I uTMAjUu)^) by UQ J UK (A + w?A ) , (28) which is one of the sums (18) which were calculated by Ullersma. If we carry this through in Eq. (16) we find the same B-matrix with 1/6 replaced by %u> . We interpolate by putting u> A? + a)?w2A2 A.A. .2A o j "J o 3 + jAj A 2 B = hi - *iA A + ^coth(^6wo)^woü)~ 2 2 2 A .A. + u^A.A. u£(A +u> A ) (29) 58 In the same way as before we find 0 0 pw D = Y(1 + coth IjBu) ) = 2Y + 2Y(e -l) - Do * Dth" 0 1 (30) Finally; if we substitute D and ft into (IV.32) we find the damped equation of motion for the harmonic oscillator which is coordinate coupled to the radiation field. + || = -Uw0[b b,p] + + ^y (bb+p + pbbf - 2bfpb + b+2 p - b+pb+ + pb2 - bpb) + + *SY (l + coth %3t»j ) (bpbf + bfpb - bb+p - pbfb + + 2bpb-pb2-b2p +2b+pbf- pb+2 -b+2p) . (31) This can be simplified by combining the zero-point diffusion terms with the damping terms dP I • r , t, , •£ = -*jiw[b b,p] - JJY (bfbp + pb+b - 2bpb+ + pbf2 + b2p - b+pb+ - bpb) + (32) + + + + Yn0([b,[b ,p]] + [b,[b,p]l + [b ,[b ,p]]) -l n = feBU I o *• -1) Here too (compare IV.41) the zero-point diffusion compensates the terms that lift the oscillator out of its ground state. This equation differs from the simplest possible one (IV.37) by the presence of terms coupling diagonal and off-diagonal elements. Even when p is diagonal at t=0, it will not be diagonal at later times. It is possible to express all matrix elements at later times into the matrix elements at t=0 by using (IV.19), (IV. 20), and (H.6). This is straightforward in principle, but a lot of work in practice. Instead, we calculate some matrix elements and some of the moments to illus trate the general principles. We have a quantum-mechanical version of the discussion of Brownian motion by Wang and Uhlenbeck. For the first moments we find 'bit -2Y 59 The phase orbit is an exponentially contracting ellipse. Of the second moments we consider only the energy. We find 2 2 A<£> = -2Y( For a diagonal density operator 2 2 0 < h The energy loss is oscillatory and on the average equal to -2y <0|p(t)|2> = <2|p(t)|0>* and <0|p(t) |0> - 1 - They satisfy the equations of motion (for simplicity this calculation is done with 8 = °°) d d <0|p|2> = 2(iüj -y)<0|p|2> - 2y/2" 60 Solving these we find Re<0|p|2> = J e"2ïtsin wt , There is a first-order correlation between x and p, but the diagonal elements are unaffected to this order. The coupling between diagonal and off-diagonal terms can be eliminated by averaging over the period, which means replacing all sine and cosine factors by zero and all sin2 and cos2 terms by h. In order to average the A-matrix, it is necessary to go to a system rotating with the mean angular velocity. Taking the average, the result in the non- rotating system is 1 + I cos (jt sin ut (JL) (JL) A = %e'-Y t (38) (i) .1± _. ±±iy 1 + -sin tot cos ut U) Ü) This represents uniform rotation and damping. The average fi-matrix is IT --IT 1 0 0 1 ft = A A = -Y + Ü) (39) 0 1 o -1 0 Averaging the B-matrix gives 2Yt B = (no+^j) l-^e- ll (40) Co2 For the average diffusion matrix we find D = Y(no-t-*s)I (41) From Eqs. (39), (40), and (41) we see that the conditions (IV.37) are satisfied so that we find Eq. (IV.41) + + + + t §£ = -%u>0[b b,P] + *5Y(b bp+ pb b-2bpb ) + yno[b,[b ,p]j , (42) which is diagonal. The same procedure may be followed in the case that the strength function is not fm7 . Then, with the more stringent re quirements t > T and Y ** u» we do not find the diagonal -rir . We also have to neglect the additional frequency shifts (ft-i ** ^i o^» *n or<3er to obtain a diagonal equation of motion. 61 A frequency shift should not be confused with a level shift which is a change in the ground state (energy). 3. Double coordinate coupled Hamiltonians In the previous section we have seen that the off-diagonal elements arise from the damping of only one coordinate of the oscil lator. In order to investigate a system with damping of both coordi nates, we investigate the Hamiltonian H b b + u b b + W b b + = ^o o o « j Ij lj « j 2j 2j + *Xo pjbIj+Xjblj) + ÏPo Pjbïj + *jb2j) ' (43) which we call double coordinate coupled. Here, b,. and b_. describe different oscillators. It is not possible to combine the couplings into the symmetrical coupling, to be described in the next section, because the damping results from the second-order terms in the equa tion of motion. If the bath and the coupling are identical, the exact solution carries through in the same way as before. In order to see what happens to the dissipative equation of motion, we consider the canonical transformation x -* -p b -» ib t t p -+ x b -* -ib The diagonal terms in Eq. (32) are invariant, while the off-diagonal terms change sign. The dissipative equation of motion for the two-bath Hamiltonian is found by adding the dissipative equation with p-damping to the one with x-damping, which gives again + + + f t §£ = -i*iwo[b b,p] - *SY (b bp + Pb t> - 2bpb } + Yno[b,[b ,p]] . (44) This is the equation of motion with diagonal terms only. Again, the change in the ground state has to be neglected as it does not change sign under the canonical transformation. This is to be expected since non-interacting ground state is not an eigenstate of the interacting Hamiltonian. A simple model for this Hamiltonian is an L-C circuit. The current and voltage are the coordinates of a harmonic oscillator. R1 •r^o 92 The resistance Rj damps the current, while R» damps the voltage. 62 By using suitable idealized components, R. and R2 may be radiation resistances. If we arrange matters such that the damping of the current is equal to the damping of the voltage, the circuit is described by the diagonal g£ . An arbitrary mixture of the two kinds of damping can be obtained by choosing Hc = *nxo HXjblj + Xjblj) + ^-^Po ï(Xjb2j + Xjb2j) ' (45) with 0 < n < l , as the coupling term. 4. Symmetric Hamlltonians The symmetric Hamiltonians are obtained from (IV.1) if we take U- to be zero. We call them symmetric on account of the symmetrical role of the b- and b -operators or the x- and p-operators. The symmetric Hamiltonian can be seen as an approximation to the general Hamiltonian (IV.l) or the coordinate coupled Hamiltonian (V.l) by noticing that the X.-terms vary slowly in time for ID , * u , while the (j .-terms have a rapid oscillation. Keeping only resonant terms and leaving out the non-resonant ones seems to be a reasonable approxima tion. By analogy with the corresponding spin problem, this i3 often called the rotating wave approximation. The symmetric Hamiltonian is given by H = feub +b + hi w.btb. + hlU-b bt + Atb^b.) . (46) ^ o o o 5j 3 3 3 jj^}03 3 o j> It has the advantage that the ground state is not changed by the inter action. The equations of motion are simple. The b- and b -operators fc = do not mix, so that in Eq. {IV.4) we have g2( ) h-.(t) = 0 for all t. In the following we write gj(t) = g(t) , nij(t) = hj(t) ' The commutator property (IV.5) is gg* + I h,h* = 1 . (47) j J J The A-matrix is Re g lm g A = (48) -lm g Re g 63 The B-matrix is proportional to the identity matrix 1 0 B-^(Vl>h.hJ = (1 -gg*+ X) * I , 0 1 with A given by X = 7 n.h.h* (49) j J 3 3 The ft-matrix is -y u a = (50) [-Ü) -Y with log|gi and w = --rr-argfg) (51) Y = ~ dt dt The diffusion matrix is given by D = h[A ^nj+l,hjhj+ 2Y i(nj+i)hjh*j x r (52) The diffusion can be split into a thermal part D , and a zero-point th part D . The zero-point part is Do = * £r I h.h* + 2y I h.h* x I (53) By using (51) and (47) we find DQ = hi ^ gg* + 2Y(l-gg*) J = yl . (54) The thermal part is -£: 7 n.h.h* + 2y Y n.h.h* (55) Dth " *T Combining all this we find an equation of motion for the density oper ator of the form (IV.41) |H = -fciutt) [b+b, J - %y(t) (b+bp + pb+b- 2b b+) + dt P P - D (t)[b,[bT,pj] . b = b (56) th o This equation of motion for p leads to an equation for the diagonal elements only. We obtain a loss-gain type equation for the probabil ity of finding the oscillator in a given state 64 with the time-dependent transition probability given by wj>j+1(t) - 2(j+l)Dth(t) (58) wjfj.l(t) = 2jDth(t) H- 2jY(t) fc and all other w's are zero. The terms with y{t.) and ELh( ) represent spontaneous and stimulated emission, respectively. However, for a finite system y{t) and D(t) are not necessarily positive, so we might sometimes speak of spontaneous or stimulated absorption. This will certainly occur on time scales of the order of the Poincaré recurrence time of the system. For the radiation field in the continuum limit we find only the emission. The calculation of the q~ and h-functions in the continuum limit has been done in App. L. The calculation is more difficult and more restrictive conditions are necessary than in the corresponding calculation in App. K. Moreover, it turns out that the interacting thermal state is not the product of tha non-interacting thermal states, except when T= 0. The main results ai-o •iui t g(t) - e^l o 0o> -l Mt) = d-gg#) e -1 y(t) = ysiqn(t) -1 ( 6aiO D(t) = Y(t) e -1 = Y(t)x, (59) The restrictions are v < w and t > 1/ia . Changes in the states caused by the interaction are neglectedo . There oi s no quantum transient time longer than 1/y. We have obtained a diagonal equation for -«£• with constant coef ficients, but under more stringent conditions than in the preceding chapter. + J t + + f g£ = -*aw0[b bfpl - iY(b' 'bp+pb b-2bpb ) - Yn0Ib,fb ,pl] . (60) The corresponding Pauli equation has constant transition probabilities pwo . wjfj+1 - 2,(j+l, e -1 Wj,j-1 = 2^ + YJ '>-/•' (61) The presence of off-diagonal elements of the density operator has no influence on the diagonal elements. 65 CHAPTER VI DISCUSSION 1. Introduction In this chapter we collect the results of chapters IV and V, give a discussion of the derivation of the Pauli equation, and compare the perturbation theory of chapter III with the exact solutions. 2. Table of results and discussion Table 1 summarizes the results of the exactly solvable models for different choices of the coupling. In this Table "Fokker-Planck equation" refers to a partial differential equation for the classical phase density which is first order in time and at most second order in x, p, 3/9x, and D/9p, "operator equation" refers to an equation for -£ which is at most second order in b, b , [b, ], and [b , J, and Pauli equation refers to an equation for the diagonal matrix elements of the density operator in the number representation. The following notation is used CM = classical mechanics, QM = quantum mechanics, Y = damping, ui = undisturbed frequency, B = inverse temperature, T = transient time. 66 Table 1 "-\^_^ H = coordinate double coordi symmetrical coupled nate coupling coupling results -—-^^ V.4 V.43 V.46 CM a Fokker-Planck equation with IV. 33 time-dependent coefficients finite systems an operator equation with QM time-dependent coefficients IV.36 V.32 V.44 V.56 continuum constant coefficients limit App. K App. K App. L CM t > T t > T 0)T • 1 and Y -4 ui y < w y < u) approximations QM needed with an ordinary u>B > 1 or strength wB < 1 or QM Bw 1 and t 6 or neglect (JJB * 1 and function < > only change in ground state neglect change in thermal state CM none. The derivation ut > 1 is exact Y "^ OJ approximations needed with oiB > 1 or a principal wM 1 or value strength neglect change in ground 6 * 1 and QM function state or Bw < 1 and t > 6 neglect change in thermal state averaging over the Pauli V.37 the period or equation can V.42 p(0) diagonal take matrix elements be obtained V.57 and restric by means of V.61 tion to first order in y/u The equations of motion with time-dependent coefficients are an exact consequence of the quadratic Hamiltonian and a suitable initial state. The time functions were calculated by Ullersma in the contin uum limit with an ordinary strength function. In the QM calculation these results were extended to include off-diagonal matrix elements. The continuum limit with a fw strength function yields new results. The restriction to weak damping and to times longer than a transient time is not necessary. Classicallly the Fokker-Planck 67 equation is an exact consequence. It is found without making approxi mations. In the QM calculation the only approximation is the neglect of the change in the ground state. 3. Discussion of the Pauli equation The Pauli equation can be found in four different ways: 1) By taking a diagonal initial state ([p(0),H 1 = 0). and assuming weak damping. Then it is correct for the diagonal elements. The density operator at later times is not diagonal to first order, but the difference does not appear if we restrict ourselves to diagonal observables {[A,H J =O) . 2) By averaging over the period. Then it is true also for non-diagonal initial states. 3) By modifying the Hamiltonian such that the damping of r and p is equal. This is the double coordinate coupled Hamiltonian. The Pauli equation follows at once. 4) The rotating wave approximation is not very useful. Its only advan tage is that the ground state is not changed. It has the disadvan tage that the calculation of time functions is complicated by the absence of negative frequency oscillators. The Pauli equation can be derived exactly if we take the symmetrical Hamiltonian and add the negative frequency oscillator?. The physical interpretation of this procedure is unclear. It is a model equation which does not correspond to a physical model. We now have four ways of deriving the Pauli equation. The approximations are collected in Table 2 for the case of a principal value strength function. Table 2 - Derivations of the Pauli equation ~~~ ---^approximations neglect high temperature "^--^ needed y •< hi dit > 1 change of wB 1 and t 6 method ~"~^-^_ ground state < > coordinate coupling yes no yes no + |p(0),Ho) = 0 no yes coordinate coupling yes no + average over no yes the period no yes double coordinate no no yes no coupling no yes symmetrical coupling = coordinate coupling yes no no + rotating wave yes approximation 68 There seems to be no way of obtaining the Pauli equation without approximations. Which way is to be preferred depends on practical considerations. 4. Discussion of the Random Phase Approximation In chapter III the time-dependent perturbation theory was used to obtain a dissipative equation of motion for the special case of the radiation field with a fm strength function. For a harmonic oscillator the final result is the same as the result of the exact solution, but the nature of the approximations is less clear and the whole deriva tion is less transparent. The essential difference is the need for the Random Phase Approximation. The term Random Phase Approximation (R.P.A.) is used in two different meanings. The first meaning is the assumption that the small system and the bath are uncorrelated at all times or, equivalently, that the density operator factorizes into a product of a density oper ator for the small system and density operators for the bath oscilla tors. This is a quantum-mechanical equivalent of the Stosszahlansatz, which makes it possible to derive irreversible behaviour from revers ible equations of motion. The R.P.A. allows one to use the time deriv ative found at t = 0 for all times. The second meaning of the term R.P.A. is the same as the first one, with the additional assumption that the (reduced) density operator of the small system is diagonal in the number representation at all times. This allows a simple deriva tion of the Pauli equation for the occupation numbers and the transi tion probabilities. Whenever it is necessary to make a distinction between these two, we use the terms weak R.P.A. and strong R.P.A., while R.P.A. without further specification always means weak R.P.A. Originally, Pauli assumed all phases to be random, without making a distinction between parts of the system. This is equivalent to the strong R.P.A. In chapter V it was shown, for the special case of a harmonic oscillator coordinate coupled to the radiation field, that the strong P..P.A. is both incorrect and unnecessary for the derivation of the Pauli equation, which can be obtained by assuming weak damping (taking the weak coupling limit), or by averaging over the period, or by introducing double coupling. The (weak) R.P.A. is (at least for harmonic oscillators) cor rect, since the results of perturbation theory and the R.P.A. agree with the exact solutions. It should be possible to derive the dissipa tive equation of motion from infinite order perturbation theory with out using the R.P.A. 69 5. Dissipative equation of motion for arbitrary systems For an arbitrary potential the equation of motion is more cora- plicated as a result of the presence of the operator xl. it is not possible to obtain the Pauli equation by any of the approaches of section 3. The trick that can be used for the harmonic oscillator, using the double coordinate coupled Hamiltonian, does not work here. Even the procedure of adding to -£ the same equation with p replaced by ^~ -*--* ^ ut j dr and |^ replaced by p, which produces the diagonal -3^ in the special case of a harmonic oscillator, does not work in general. Sometimes it is possible to produce the Pauli equation by averaging over the period to eliminate off-diagonal elements. However, in the case of an atom with a more complicated spectrum, it is neces sary to average over times long compared to difference frequencies, which may be very long. An example is provided by the Stark effect of the second level of the H-atom. If the density operator is not diago nal in the eigenstates of the Hamiltonian including the electric field, the emission probability of a photon oscillates with the Stark splitting frequency. The other possibility is to have p diagonal at t = 0 and neglect second-order effects. This may work sometimes, but it fails again for the second level of the H-atom when a time-varying electric or magnetic field is present. There seems to be no generally valid procedure to obtain the Pauli equation. The restrictions needed exclude some simply realiz able experimental situations. More work on the contributions of the off-diagonal elements, for instance for an H-atom is time-varying fields, is needed. 70 CHAPTER VII LANGEVIN FORCES AND NYQUIST'S THEOREM 1. Lanqevin forces For the classical Brownian motion UHLENBECK & ORNSTEIN (1930) demonstrated that a small system, close to equilibrium, in contact with a heat bath can be described as an isolated system under the influence of a Langevin force. The Fokker-Planck equation can be de rived if suitable assumptions about the force are made. For the ex actly solvable models of chapters IV and V the force acting on the small system as a result of the coupling to the bath and its auto correlation function can be calculated exactly. We do the calculation only for the case of coordinate coupling with a fu2 strength function, which describes the radiation field. We follow the procedure of ULLERSMA (1966). The equation of motion follows at once from the Hamiltonian (V.4) X 0 ' 0 1 o - dt 2 c x -w 0 P I n n I ° J o, n The dissipative equation of motion for the observables x and p is in this case (V.33) X 0 1 X o = o p o ' o Po Comparing these, we see that the force can be split in a systematic part and a fluctuating part. The fluctuating part is found by sub tracting the systematic part from the total force F(t) = 2Ypo(t) - I cnxn(t) . (3) n We can express F(t) as a function of the initial conditions by sub stituting the known solutions of the equation of motion. 71 We find F(t) = 7 — (w x (O)cos ui t + p (O)sin ID t) . (4) n n In order to evaluate the autocorrelation function *(x) of F(t), we note that the expectation values in the initial state are given by 2n(uj )+l *{T) = E2 7 = LT — lf2n(u» ) + l)cos u) T = dm ^-^- v(2n For a quantum system with all bath oscillators at the same temperature 2n(iii) + 1 = coth ^gu) , and we find, recollecting Eq. (III.20bJ, 6 sinhU c/B) f du) ID coth(^BID)cos WT = — . (8) o 1 By comparing 3 with the time scale on which PQ(t) varies (for instance l/wQ)/ we can distinguish a high and a low temperature regime. For high temperatures and t > B Eq. (8) can be approximated by a 6-func- tion. 72 We find the classical result * * This corresponds to the term with the Hilbert transform -5— in Eq. (III.26). It should be possible to start with a Langevin force with autocorrela- p tion function proportional to "fnK2~" an<^ to derive the dissipative equation of motion (III.26). Many authors (see e.g. HAKEN (1970) or LOUISELL (1973) and references cited there) describe a dissipative system with a factoriz- able equation of motion by adding to equations of motion like (2) or (IV.50) a quantum-mechanical Langevin force with autocorrelation func tion *(T) = 2yw -coth(ljB(D )6(t) , (10) which serves to maintain the quantum-mechanical uncertainty. With this assumption the correct equation of motion for the harmonic oscillator, including the zero-point effects, can be found, but this is merely a lucky coincidence. The reason is that for the harmonic oscillator or a spin there is only one frequency, and then there is no difference between the Hilbert transform and the operations — -rr or -uj Jdt, since and sin to T = cos to T . (11) It is possible to extend this approach to an N-level system, but this entails the introduction of separate Langevin forces for each allowed transition. In reality, the autocorrelation function is given not by Eq. (10), but by Eq. (8). For all systems, except the harmonic oscil lator and the spin, the autocorrelation function (10) produces errone ous results. The more complicated Langevin force with autocorrelation func tion (8) diminishes the attractiveness of the Langevin force method. 73 2. Transient times If we describe the radiation field by means of a strength func tion with a high frequency cutoff (6), there is a transient time of order 1/e both in the Fokker-Planck equation and in the autocorrela tion function of the Langevin force. With the description of the radi ation field by a pw strength function this transient time disappears and the derivation is exact. In the corresponding quantum-mechanical problem there appears in the high temperature case an extra transient time equal to 8, both in the derivation of the dissipative equation of motion (III.23) and in the autocorrelation function of the Langevin force (8). Some au thors (e.g. MACDONALD (1962) and ULLERSMA (1966)) notice this diffi culty and conclude that there exists a universal quantum transient time equal to 6, which is a lower limit for the time needed to reach equilibrium. This conclusion is incorrect. It is necessary to separate the zero-point term from the thermal term in the autocorrelation func tion or the Fokker-Planck equation. This can be seen from the autocor relation function (8) by noting that B can be interpreted as the width of a <5-function only for high temperatures. For low temperatures the zero-point effects with autocorrelation function 1/T2 (9) dominate and the concept of a transient time loses its meaning. The same thing can be seen in the derivation of the Fokker-Planck equation which is sim ple only for high bath temperature and where a separate discus: ion of the change in the ground state is needed. As a demonstration we calculated the number of thermal quanta as a function of time for the case of symmetrical coupling. In this case the computation is simplest. We find (App. L) that for times such that yt > 1 the oscillator has reached its final state independent of the initial state or the temperature of the bath. There is no quantum transient time longer than 1/Y, which is the usual time scale for reaching equilibrium. 3. Nyquist's theorem The Hamiltonian (V.4) can be modified by the addition of a term F(t)r , which represents a time-dependent external force. The calcula tions can be repeated for this Hamiltonian as before. The final result is the appearance of an extra term iF(t)[r ,p] in the dissipative equation of motion. This result can be derived without imposing re strictions on F(t). 74 This means that the coefficient y gives the dissipation result ing from the applied force, so that it can be interpreted as a macros copic resistance. Nyquist's theorem or the fluctuation-dissipation theorem can now be found by taking the cosine transform of Eq. (6), which gives W(w) = \yw coth ^u> , (12) where W{w) = idi t('t)cos CUT o is the spectral power density of the fluctuations. NYQUIST (1928) found this result from statistical considerations. CALLEN and WELTON (1951) derived it, using perturbation theory and (implicitly) the R.P.A. Ullersma derived Eq. (12) from the exactly solvable models with the restrictions y < u , t > t, and to < to . o o These restrictions are not needed when the fw strength function is used. The present derivation is exact. The only approximation is the neglect of the change in the ground state as a result of the interac tion. We remark once more that for quantum systems the fluctuation- dissipation theorem is incompatible with a 6-correlated Langevin force. We have shown in this chapter that in the quantum domain Nyquist's theorem is correct and that Eq. (8) gives the correct auto correlation function of the Langevin force. 75 CHAPTER VIII THE H-ATOM In this chapter we calculate the ground state of the H-atom which results from adding zero-point energy to the classical radiation oscillators, as discussed in sections III.l and IV.4- There (III.49) we found the classical equation of motion, for a particle in an arbi trary potential and coupled to the radiation field, including quantum corrections |f = {H,P} + 27TY{p,|| P) - TTY{p,{||,p}} . (1) In or^er to find the ground state, the temperature is taken to be equal to zero. For the special case of an H-atom th*s classical initial vaJue problem can be solved explicitly, and Zl can be calculated as a function of -* -+ 3r r and p. The coordinate system is chosen such that the orbit lies in the x-y plane and the "perihelion" passage occurs on the positive x-axis. Then, the position as a function of time is given by x(t) = 2a I - J' (m;)cos(nft(t+t )) - ea n=l. n n "• ^p ' y 76 This leads to p (t) = -2a I J^ p (t) = 2b y - J (ne)fi cos(nft |^(t) = -2b y - J (ne)fi2sin(nl2(t+t )) . 9y L e n v p y 3V The Hi\bert transform of •=— now follows immediately. 3r J -(t) = -2a y n J'(ne)f!2sin(nfi(t+t )) |^(t) = 2b J - J (ne)fi2cos(nft(t+t )) . (5) dy L e n v p ' This expression for 4— can be used to obtain a stationary solution to the dissipative equation of motion (1). Sufficient conditions for stationarity are tH,p} = 0 #, and ., 3V . ,3V , . , . 2 TT— n - {^— , p) = 0 (6c) or or The solution of this problem can be simplified considerably by the introduction of action angle variables. We use the same action angle variables and notation as GOLDSTEIN (1950). In natural units the action variables are given by Jx = 2nLz J2 = 2TT|L| f-E J, = 2TT E < 0 (7) E o' where L is the angular momentum and E = \mt2 is the QM ionization energy of the atom (Rydberg energy) . The corresponding angles w a.n d w~ give the orientation of the orbit, and the angle w, = fi(t+t )t which is call the mean anomaly ir; astronomy, gives the position in the orbit. 77 The eccentricity of the omit is given by EL = * = V1 + 1 - (8) The action variables j,, J2, and j, must satisfy o < IJJ < j2 The first inequality follows at once fro,. "ïq. (7) , the second is a consequence of 0 < e < 1 for bound orbits (E < 0). The Hamiltonian is given by 4TT2E c (10) H = - 4 The orbital period is 8ÏÏ3E 3/ 9H 2 ft = 2T? o = 2E (ID 3JI As a simple approximation we try a density function containing only circular orbits and neglect terms that change the eccentricity. We choose a p of the form , ? P(r,p) = P(J2,J3) = P'(J3)6f(^-n/4) P (J3)<5v(c ) , (12) rJ-2 where t = arctg with 0 < $ <• IT/4 ^3 The arrow under the S-function indicates that the limit should be taken from above or below, according to the definitions of chapter II. In terms of the action angle variables the Poisson bracket is simply r : V , - *V (131 t-,— ,o = 2TT it ;>r )j, t r p ( ; u3 The :,:t -derivative equals P M - -43 n^J* ;m.) cos l(n:>(t+t ) (14) ')\:,t 'fx n p This is the same Fourier series as the one for -=-;»v— (see Eq. 4)), but with coefficients proportional to n-J instead of n. At this point in the derivation we approximate b (15) ^ (p'(J3)Vr.')) * 7j; f0'(J3))6t(,-) thai is we neglect the change in the eccentricity. 78 As the leading term in J'(nr) and - J (ne) is of order cn_1, the n r n c,-function annihilates all terms but the first one. Then, substitution of Eqs. (4) and (14) into Eq. (6) leads to P' (J3) + TT P'(J3) = o 16) Up to a normalizing constant the solution is P'(J3) = e :i7) and the full distribution function is -h -2 E -JT/TT o' P(r,p) = P(J2,J3) = *+ (*-TT/4) = 6+(G2) {18] This distribution tends to a constant value as E •+ -«, and disappears with an essential singularity as E -*• 0. A simple but tec.ious calcula tion shows that the volume in phase space enclosed by a surface of - y constant energy goes as (-E) l]. This means that the distribution (18) is normalizable. It is a simple matter to calculate the expectation value of the energy- After carrying out all trivial integrations, we obtain -Jo/* 2 -4^ Eo /dj. 2 fdV(E) EpiE) o 4 TT E o _ •2E JdV(E) p(E) Ïï •J7/ O /dj, j; e (19) This is twice the QM ionization energy cf a QM H-atom. We do not know how to interpret this factor two, as we do not know which observable represents the ionization energy. The general discussion of the rela tion between classical mechanics and quantum mechanics in App. G (see Table (G.15)) suggests that this factor two may be contained in the observables, since the phase distribution used here corresponds to the Wigner distribution. Nevertheless, it is clear that the classical col lapse of the H-atom has disappeared and that a stable classical ground state has been found. The form of the stationary state (18) suggests that it may be possible to give a general formulation of classical statistical mechanics, including the zero-point energy. If we do not make the approximation (15) the derivation is more difficult, since for elliptical orbits the diffusion and dissipation occur in different parts of the orbit, so that the stationary state does not commute with H any longer. This makes averaging over the period necessary. It is not clear what the influence of these terms »<'~uid be on the energy of the ground state 79 The preceding discussion (which shows that classical zero-point energy results in a classical ground state) is not only interesting in its own right, but it is also useful as a preparation for a quantum- mechanical treatment of the H-atom. It is possible to obtain operators which are analogous to the various quantities that occur in a classi cal treatment, but so far only preliminary results were obtained in this direction. More work on the dissipative motion of a quantum-me chanical H-atom is needed. 80 CHAPTER IX LITERATURE A list of references, which places the subject of this thesis in its proper context, is given. The emphasis is on original papers and on recent reviews, without any claim to completeness. The classical description of dissipative motion (which is in this context cabled Brownian motion) of a small system coupled to a heat bath or undei. tb'. influence of a Langevin force, is found in the papers uf UHLENBECK nnd UrtNSTEIN (1930), WANG and UHLENBECK (1945), and KRAt^KS (1940», P 1 in „nn reprint collection edited by WAX (1954). The • *sci •'-t.ion of dissipation in quantum mechanics began with the use of ti.ne-oependent perturbation theory by DIRAC (1930) to derive transition probabilities. This was combined with the (strong) Random Phase Approximation by PAULI to discuss irreversibility in quantum mechanics, which led to the equation which now bears his name. The meaning of the R.P.A. was made explicit by VAN KAMPEN (1954). Subsequently, a derivation of the Pauli equation in the week coupling limit was given by VAN HOVE (1955, 1957), without using the R.P.A. This approach is insufficient for the description of spin reso nance experiments or lasers, since information about phase angles is not obtainable. Phenomenological equations for a damped spin were introduced by BLOCH (1946). These equations were derived from a microscopic theory by WANGSNESS and BLOCH (1953), BLOCH (1952, 1557), and REDFIELD, using perturbation theory and the R.P.A. or equivalent stochastic assumptions. The same approach for a harmonic oscillator was used by SENITZKY (1960), SCHWINGER (1961), GEORGE (I960), and others. An (incorrect) attempt to describe the damped oscillator by means of a time-dependent Hamiltonian was made by KERNER (1958). An example of the use of non-linear equations to describe damping (also incorrect.) is given by CRISP and JAYIJES (1969). In the sixties a great deal of work was done to describe the laser by means of a,density operator formalism, or a Lanyevin force approach, or both. Representative papers are: LAX (1966), HAKEN and WEIDLICH (1966), LOUISELL (1964), LOUISELL and WALKER (1967), SCULLY and LAMB (1967), GORDON (1967), WEIDLICH and HAAKE (1967). 81 Reviews of this development are: VARENNA (1967), HAKEN (1970), RISKEN (1970), KAY and MAITLAND, editors (1970), AGARWAL (1973), and LOUISELL (1973). The phase space methods which are useful for developing the laser theory are based on the coherent states which were introduced in quantum mechanics by SCHRÖDINGER (1927) and developed, among others, by GLAUBER (1964). A collection of papers on quantum coher ence ':heory is MANDEL and WOLF, editors (1970). The first example of a phase distribution function was given by WIGNER (1932). The Wigner distribution was further developed by GROENEWOLD (1946) and MOYAL (1949). The connection between phase distributions and ordering of operators was treated by GLAUBER (1963), CAHILL and GLAUBER (1969), KLAUDER (1963, 1964), MEHTA and SUDARSHAN (1965). For a review see KLAUDER and SUDARSHAN (1968), AGARWAL and WOLF (1970), or ARECHI et al. (1972). Exactly solvable models for classical or quantum dissipative motion were worked out by ULLERSMA (1966), following earlier work on lattices by HEMMER (1959), RUBIN (1960, t^l), MAZUR and MONTROLL (1960). A review of his work is MARADJD-i.N, MONTROLL, and WEISS (2nd edition 1971). The approach of Ullersn.^ i as been used by others, for instance by DAVIDSON and KOZAK (1973) for a spin. The equivalence of quasi-linear theory in plasma physics and time-dependent perturbation theory was emphasized by HARRIS (1969), following the pioneering work of PINES and BOHM (1952, 1953) and (1963). Other work along the same lines is for instance from SAGDEEV and GALEEV (1969), TSYTOVICH (1972), COr I, ROSENBLUTH, and SUDAN (1969). For a review of quasi-linear theory see for instance DAVIDSON (1972), MONTGOMERY (1971), or KRALL and TRIVELPIECE (1973). 82 REFERENCES 1. AGARWAL, G.S., and E. WOLF, Phys. Rev. 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New York, Dover Publ., 1954. 69. WEIDLICH, W., and F. HAAKE, Z. Phys. 201.396,-204.223(1967). 70. WIGNER, E., Phys. Rev. 40.749(19J2) . 85 APPENDICES APPENDIX A Commutator and Trace properties This appendix summarizes some frequently used commutator and trace properties without giving proofs. The commutator and the anticommutator are defined by [A,B] = [A.B] i AB-BA , [A,B] = AB+BA + Some simple consequences of the definition are [A,BC] = [A,B]C + B[A,C] , (A.l) [AB,CD] = \[B,f]D + ^[A,C][B,D] + ^[A,C]IB,D) + C[A,D]B , (A.2) _ + + n-1 . . [A,Bn] = I B1[A,B]Bn~i" , (A.3) i=o and the Jacobi identity, [A, [B,CJJ + [b,iC,A)| + fC,[A,BJ] = 0 . (A.4) Repeated commutators are defined by A B B IA,B]n = [A, [ArB]^] . f ' l0 " Straightforward manipulation yields CO A A exp([A, |)B = I ±. [A,B\n = e Be" . (A.5) n=o For a function of B having a power series representation, we have eAF(B)e~A = F(eABe'A) , (A.6) -A A as it is allowed to insert e «e between all factors. A A A A A A e (BC)e- = (e Be" )(e Ce" ) Ail reordering of exponentials is based on the Baker-Hausdorf lemma. If [A.[ABU = [B,[A,B|] = 0 , .. A+B A B -MA,B] B AJ[A,B| then e =eee1''=eee' , A B and e e = eBeAeIA,B] (A7) 86 The trace of an operator can be calculated by means of a com plete orthonormal system of vectors. Tr(A) = I Tr(AB) = Tr(BA) = J APPENDIX B Harmonic oscillator, boson operators 2 The harmonic oscillator Hamiltoniaa is H = |- + ^kx2 (B.l) with the commutation relation (x,p) = iti. (B.2) We take the unit of length to be „£.. and the unit of momentum to be /hmu , where m, 2 =- -* . (B.3) m The normalized Hamiltonian is H = ^w(x2+pz) (B.4) with the commutation relation fx,p| = i. (B.5) Instead of the usual boson operators we introduce the operators b = x+ip, b - x-ip. (B.6) They satisfy the commutation relation |b,br| = 2. (B.7) The reason for introducing the b-operators without the usual L//2 is that this gives a closer correspondence to complex variables, which is useful to avoid errors with factors of two in the calculations. The relation to the usual boson operators is b = a/2, b! = a+/2, \a,a'\ = 1 . (B.8) 87 The inverse transformation is b +b b-b ,_. ft> x = —^— , p = ~2J— . (B.9) The Hamiltonian is H = ^u(b+b+l) = o){a+a+^) (B.10) with the commutation relations + + [b,b | = 2 and [a#a l = 1 . (B.ll) The orthonormal eigenvectors of H are given by +n jn> = — |Q> , b|0> = 0|0> . (B.12) /2V The action of the operators or the eigenvectors is given by b|n> = /2n|n-l>, b+ | n > = /2(n+l) ln+l > , btbjn> = 2n|n> . (B.13) APPENDIX C Translation operators The commutation relation (B.3) of x and p,[x,p] = i, can be represented by mean.s of multiplication and differential operators on a Hilbert space. The spectrum of these operators is continuous. In this appendix we will write x for the operator to avoid confusion with the coordinate x which is used both as a number and as a label for an eigenvector when written in a bracket. The coordinate and momentum eigenvectors |x > and p ' o | p > have the properties »|x>=x |x> ftlp r > = -i ~-\u > ' o o' o ' o 3p '*o *o plp>=p|p> P ! * > = i ~-1 x > o ipx I = Jdx |x> 88 The phase translation operators are defined by U(x,p) = exp(i(px-x£)) - exp(Hpx) *exp(ipx) «exp(-ixp) = exp(-bipx)-exp(-ixp)«exp(ipx) . (C.3) The last two steps follow using the Baker-Hausdorf formula (A.7). The action of u on the coordinate and momentum eigenvectors is found by using (C.3) and (C.l). L • ipx U(x,p)|x >= e~*ipx • e °|x+x > , o U(x,p)|p>= e^ipx • e P° |p+p > (C.4) o |r ro U is unitary and U+(x,p) = U_1(x,p) = U(-x,-p) . (C.5) The action of U on the operators is found by means of (A.5) U+(x,p) x U(x,p) = x + x , Ur(x,p) p U(x,p) = p + p . (C.6) The addition formula of the translation operators, *ii(x p -x p2) U (x1(Pj) U(x2,p2) = e • U(x1+x2,p1+p2) , (C.7) can be found by repeated use of (A.7). The U's ar a projective repre sentation of the translation group of the phase plane. In terms cf the boson operators U is given by U(z) = exp ^(zb1 - z*b) = U(z) = exp(-^zz*)»exp(+^izb )«exp(-^iz*b) = Uiz) = exp( + ^zz*) -exp(-ijiz#b) 'exp( + Hzb' ) , (C.9) where U(z) = U(Re(z),im(z)j In this and the following b is always an operator and z is always a complex number. With this convention the " is no longer needed. We rev/rite (C.6) and (C.7) in terms of z and b. U+(z) b U(z) = b + z, U+(z) bfU(z> = bf + z* . (CIO) U(Zj) U(z2) = expfU'Im(z*z2))U(z1 + z2) . (C.ll) 89 APPENDIX D Coherent states In this and in the following the word "state* refers both to a state vector in the Hubert space of the systen and to the density operator which is the projection operator on a given state vector. Also a superposition or a statistical mixture of states is again called a state. A state that can be described by a state vector is sometimes called a pure state to distinguish it from a statistical mixture. The coherent states are defined as vectors in the Hilbert space by jz> * u(z)|0> , where |0> is the harmonic oscillator ground state. The coherent states can be expressed in the number states by means of (C.8) and (B.?2). |t> - e"*zz* I -£= |*> , <0|z> = e"izz* . (D.l) They are eigenvectors of the b-operator. bU(z)fO> * U b|z> = z|z> . The scalar product is given by hllmlz.zV 1£ 2 Two coherent state vectors cannot be orthogonal, but they are approx imately orthogonal if |z.-z2| » 1. The \i> are an overcomplete set of vectors giving rise to a resolu tion of the identity 1*^ Jd2z|z> Consequently, there exists a linear dependence among the jz> |z> ** ± }C2z' The trace can be calculated by means of the coherent states. 90 TrCAï - T Vectors can be represented by entire analytic functions: |*> -+ for a normalizable j*> f(z*) is an analytic function of z*. Operators can be represented by This means that This operator has the right value of APPENDIX E Normal ordering, special operators Any operator having a power series representation can be written in normal ordered form by means of the commutation relations. This is simple if we know the diagonal elements in the |z> representation. For example: zz# ( z#z) b bfibj -%bfb |iXj| = -2 |0><0| . (E.2) ii! J?ï\ Wi^hiy. e 91 By neans of (E.l) any operator can be put in noraal form if its matrix eleaents in the number representation are known. Sane examples are given for later use: 1} The rotation operator in phase is the evolution operator of the harmonic oscillator A = e =i© |n>vn| , n ian ian + ia + A = e*i«*> *> » H exp(%(l-e )b b) . (E.3) This represents a rotation in the phase plane of magnitude a in the negative direction. ft special case which we will need later is the parity operator which has a = +iw. It has the properties b b ± p = jve" P = P+ = p-i p2 = i . (E.4) It represents a reflection in the phase plane. 2) Thermal density operators p(6) = (l-e"Btla') I e*Btlti,k|k> p(6) = (l-e-6^) N exp(-%b+b(l-e~Phw)) . (E.S) This can be written as p(n) « -j^j N exp n+lj ' (E.6) where n is the expectation value of the number operator in the ther mal state. 3) Displaced states t If A = Nf(b b) is an operator in normal form, + # then 92 f + and u (s )a0(so) - «(b +mJ,b+*0) . In particular, |z><*|« K exp(-%(bt-a*) (b-z)) . (B.7) lie also use displaced thermal states p(n,*) = U+U)p(n)UU> m jjlj s expf-% (b "n*}***"**] • These states are a generalization of the coherent state and the thermal state. They are useful in calculations (chapter III and IV), since they are states of maximum entropy if the displacement and the energy are given. One of the advantages of normal ordering is that commutators with b's are simply to evaluate as normal ordered operators. From (B.2) we have lb,b+lt] = 2k- b"^1 [bf,bk] - -2k«bk"1 . k > 1 (B.9Ï For any normal ordered operator p we have {b,pj » 2 -i^ , 3bT [b+,p] - -2 || , (E.10) as it is true for all terms in its power series expansion. APPENDIX F Canonical transformations and diffusion In this section we introduce some operator transformations which are useful for comparing phase distribution functions and for the discus sion of the relation between quantum mechanics and classical mechanics. A systematic method for introducing groups of operator transfor mations is to investigate operator equations, and their formal solu tions, of the form g£ - £p , P(t) - eLtp(0) , (F.l) where l is a linear superoperator. The term superoperator is used for operators that act on operators instead of vectors in a Hilbert space. Generally, the collection of all operators is again a Hilberv. space, so there is no difference in principle between the two types of oper ators. Italics are used for superoperators. 93 The normal ordering introduced in Appendix S can be used to convert operator equations into partial differential equations. The notation can be simplified by working with the diagonal matrix elements in the coherent state representation. We write The transformation rules are pb •«--»• zf b p •*••*• 2*f [b,p] ~ 2 ^ [b+,p] «-> -2 || . (P.2) These rules transform operator equations into partial differential equations. The simplest evolution equations are the factorizable equations which can be written in the form |f = i[A,p] = Ap, A * i[A, ] , (F.3) having the formal solution nn-\ - ^Atn _ „iAt -iAt ,v Ax p(t) • e p = e pe . (F.4) There are two reasons for calling these equations factorizable. The first one is that the product commutes with the evolution ^At, . „iAt „ -iAt e e (PJJPJ) • e P102 " = e p^e e P2e » Pjitjpjit) . (F.5) The second reason is that the class of projection operators of rank one is an invariant subclass. If p = |*><*| then p(t) * |*(t)><4>(t) | with |f (t) > - eiAt|*> . Superoperators acting on operators factorize into operators acting on vectors. Transformations of the phase space generated by factorizable equations with a Hermitian generator are canonical transformations. Some exam ples of canonical transformations are given below. 1) The quantum-mechanical equation^ of motion for a conservative system are of the form (F.3) both in the Heisenberg and in the Schrödinger picture. Factorization of the equation of motion for the density operator gives the Schrödinger equation. 94 2) The translation operators defined earlier can be seen as a two parameter group. Any linear combination of x and p or b and b gener ates a translation in the phase plane. The infinitesimal generator is P4^. ]orc£-C^ The corresponding finite transformation is written as r 3) The rotation operators in phase space are a special case of example 1. The generator is the Hamiltonian of the harmonic oscilla tor, ijinb b. This generates a rotation in phase in the negative direc tion. The corresponding superoperator is written as iab+b iab+b R{*)p - e* pe-* . (F.7) 4) Scale transformations (SU)) are generated by the operator -%(xp+px). The equations of motion are |*- = -%i[xp+px,xj - x , |E = -U[ (xp+px),p] • -P • The finite transformations are S(A)x - eXx , S(X)p - e"Xp . (P.8) The dissipative quantum systems for which we derive equations of motion in this thesis have non-factorizable evolution equations. This is true for all dissipative motions. A simple example of a non-factorizable equation, which will be used later on for comparing phase distribution functions, is the dif fusion equation |f - -%[b,[b+pJJ = -*[bf,[b,pj] . (P.9) The equality of the two forms follows at once from the Jacobi identity (A.4). This equation can be solved by converting it into a partial differential equation by means of the rules (F.2). We write p(z,ztt) * 95 The solution of this two-dimensional diffusion equation is p(x,p,t) - ^ jjdx'dp* e 2t pfx'.p'.o) or in terms of z and z* 2 zz pU,*\t> - •%— jd z' e p A special family of solutions is given by pCz,z«,t> » ^ e 2t . t > 0 (P.12J The corresponding operator family is h+b f "2F p(b fbft) - ï¥t ffe ** . (P.13J Special cases are 1) t • 0 leads to the 6-function, which can be interpreted as a generalized function. It will not be used in the calculations 2) t = % -> p » i S e"b b = \ P. (F.U) TT W This is the parity operator introduced in (E.6). 3) t = 1 - p = ^ ff e**b b = ^|0><0| . (F.15) This is the harmonic oscillator ground state. b+b 2(n+l) 4) t - (n+l) * p « IFflnT e - £ p(n) . (F.16J This is the thermal state with expected number of quanta n. We introduce the diffusion (super)operator defined for functions by 2 2 A f 3 . 3 ex P(\) - exPl \$p + jp) * P 2A jij^l , (F.17) and for operators by 0(A) = exp(-%A[b,[b+, 1]) . (F.18) The diffusion operators are defined for all positive values of A and have the semigroup property o(A)0(y) - Z>{A+p) • A,M > 0 (F.19) For negative values of A the operators 0(A) is not defined on the whole domain, but only on those functions which can be written as P(-A) acting on another function. 96 For negative values of A and it equation (P. 19) r MI ins valid provided we restrict the domain of application sufficiently, line diffusion operator can be used as a superoperator acting on operators, or as an ordinary operator acting on functions defined on the phase plane. A useful relationship is |0><0| « hDlh)P P • 2IM-%>|0><0| . (P.20) This follows at once from (F.14) and (P.IS). Under the trace, the (super)operators can be Moved according to Tr((TU)A)B) * Tr(*(3M-*)B)) , Tr((ff(a)A)B) « Tr(A(i?(-o)B)) , Tr((s(X)A}B) = Tr(A(S(-X)B}) , Tr(fo(A)A)B) * Tr(A(0(A)B)) . (F.21) For the diffusion operator this property can be proved by proving it for the generator. Tr((b,[b+A]JB) - Tr((bbtA-bAbt-btAb+Abtb)B) * Tr(A(Bbb+-b+Bb-bBbt+b+bB)) * Tr(Afbf,[b,B]]) = Tr(A[b,lb+,B)]) . The relationship between the superoperators is seen most simply from their generators in the x,p representation. a2 a2 D ""•" aï?+ apT 3 & R " x T? "p »* ' The diffusion commutes with the rotation and the translation. All other commutators do not vanish. 97 APPEHDIX G Phase function» Ne can define a family of phase functions corresponding to a given density operator by t fx(x,p) - fx<*.**) « 7; Tr(UUJ(p{X-l)|OXO|)U <*)p). \ >9 (G.l) That is, f is the expectation of a certain operator as a function of its displacement from the origin. Since diffusion commutes with translations, we can write + fx+1(2,2*) = ^jr Tr(ö(A)(ü(2)|0><0|O (s))p) « jj Tr(ü<2) |0><0|U+U) (D(X)p)) »^rTr(|2><2|(P(X)p)) fx+1(2,2*) = i As shown in App. F we can take out the diffusion operator, so that we obtain fx(z,2*) * j^ 0(A-1) 1) For X * 1 we obtain flU'z#) " 1Ï 0 < f < 1/2K , and normalized to 1 if p is normalized to 1, since Jd2z f(z,z#) - ^ Jd2z The sharpest possible distribution is obtained by taking for p a coher ent state p * |z > -%|Z-Z0|2 fj(z,z*) » e ° (G.Ó) 98 2) X «• n + 1 gives a similar result as example 1 with coherent state replaced by displaced thermal state. 3) X - \ leads to V2'2** = éi Tr(O(z)(D(-%)|0><0|)üt(z)) - = ^ Tr(ü(z)Pü+(z)p) - jL Tr(PU+(z)pÜ(zi) , (G.7) where P is the parity operator as defined in (E.6). It is obvious that f, is not positive definite. For a state with negative parity we have ft(0,0) = T- . In general we have " TF « Vz'z#) < Tn as a consequence of -1 < < 1. if we calculate the trace in the coordinate representation, we obtain f^(XrP) = J$ fax.' 2ipx - —- Jdx' In the special case that p is a pure state, we obtain Vx'p) " TS 'dx' * W(x,p> - j^ U(x,p)PU+(x,p) , W(x,p) = f^(x,p) * The Wigner distribution is the expectation value of the Wigner opera tor, which is a displaced parity operator. The phase distribution functions can be used to find expecta tion values. Expectation values of antinormal ordered operators are simply found from the f,(z,z#). The expectation value of an operator A is calculated by means of a phase integral 99 This is seen readily for a product Jd2zA(z,z*)f D(X)b+b = b+b + 2X « (l-X)b+b + Xbbf , D(X)bbT - bb+ + 2X * <1+X)bb+ - Xbfb , and Dl\)zz* = zz# + 2X . (G.13) By induction this can be extended to all polynomials and to all con vergent power series. If we take as an example the harmonic oscillator energy above the ground state f b b bb f H - %b b = + - k - %bb - 1 , then the phase functions given by substitution are f(z,z*) • ^zz* , \zz* - k , Szz* - 1 , (G.14) which corresponds to A » 0 , X - *s , X - 1 , respectively. The sharpest possible 3tates and observables are the coherent state projection operators. They are represented by Gaussian distributions. 100 Table 1 gives soae properties «f phase functions. CG-15) Table 1 X - 0 X - h X » 1 ordering of normal symmetric * antinonsal observables * Weyl ordering of states antinornal Weyl normal sharpest possible -zz» state |0><0| ó(z)d(z») e e"*"* sharpest possible -%zz* 5(z)6(z*> « observable 10 ><01 e - S(xH(p) energy above ground *zz* - %zz* - ^ - %ZZ* - 1 » f state %b b = %(X2rp2) - *U2+p2-l) • % There are no physical reasons to prefer one of these distribu tions to the others. They can all be used to calculate expectation values. There are some practical reasons for preferring the Wigner distribution (A = *) . 1) It is the only one which transforms in the classical way under seals transformations. S(A)W(x,p) = w(eAx,e"Ap) as a consequence of S(A)P = P . The parity operator is invariant under scale transformations as it commutes with the generator. 2) As a consequence of this the equations of motion for the damped oscillators (chapter IV) are simplest with A = ^. Also when doing per turbation theory (chapter III) the correspondence with classical mechanics is closest. 3) There is symmetry in the treatment of states and observables. The states as well as the observables occupy an area of at least in « \ in phase space. 4) The connection with the coordinate operators is simple. This can be seen by noticing that a coordinate eigenstate is the limiting case of a coherent state (under scale transformations). or more 101 directly by integrating /dx W(x,p) = ^L /dx U(x,p)PU+(x,p) « j^ /dx U(x,p)/dp'|p'><-p'|u+(x,p) 2ixp, - i /dp' /dx e- |p+p'Xp-p'| = /dp' 6(p')|p+p'> /dp ft(x,p) = |x> By tracing this with p we see that integrals of the Wigner distribu tion function are the coordinate probability distributions, /dx W(x,p) = On the other hand there are reasons to prefer A « 1 when the scales of x and p are given. 1) The state is represented by a positive definite phase function, which can be interpreted as a probability density, and 6-functions are allowed observables. This situation is familiar from classical mechanics. States occupy an area of at least *jh in phase space, while observables can be infinitely sharp. 2) The quantum-mechanical derivations are often shorter and without superfluous factors (see chapter IV), and there is a direct connection with the coherent states. In classical mechanics states are represented by positive defi nite phase functions and observables by arbitrary phase functions. In the classical domain only areas in phase space much greater than hh are considered. Then, the phase distributions corresponding to different values of X are equivalent, and the harmonic oscillator ground state is indistinguishable from a <5-function. Classical mechanics can be modified by introducing zero-point energy to Include quantum-mechanical effects. This was done for special cases in chapter III, IV, and VIII. In general, there are two approaches corresponding to the choices X « *f and X * 1, as in the previous dis cussion. The same considerations apply and a choice can only be made on practical grounds. Finally, we remark that even though the number of observables is formally much greater in quantum mechanics than in classical mechanics, this is only apparently so as the preceding discussion shows that we 102 can use the commutation relations to reorder then. There is a one to one correspondence between equivalent quantum observables and phase functions. The correspondence between classical mechanics with zero- point energy and quantum mechanics has been shown in detail in the chapters on perturbation theory, the H-atom, and for the exactly solv able models. APPENDIX H Characteristic functions He can take the Fourier transform of the phase distributions defined in (G.l). (Note: in chapter II a different convention for F was used.) C (k = dxdp ex Jc x+x {x p) A x'V 1Ï ' P(-i< x pP))fx ' ' i(k2 +k#z) CA(k,k*) «£ /d*. e-* * fxU,Z*) , 2 i(kz#+k#z) fx(z,z*) - ^_ Jd k e* Cx(k,k*) . (H.l) Using the definition of f we have i(kZ +k#Z) CA + 1(k,k«) = ^ /dxdp e-* * ^ <2|D(A)pi2> ikbf k b = ii7/d^ bt tb = lTr(e^ (P(>)p)e^ ) , k k %ik iikb A^+l< ' *> = -h Tr(z?(A)(e" *V )p) . (H.2) a2 Using D(l) « exp 2A- gives 3b3bt j k ikbt Cm CUl For convenience in the derivations we will use A = 0. All other pos sibilities are easily found from this C = C. C(k,k*) = ± Tr(e"*lk#V*ikb+p) , (H.4) £<«,«•> -£ /d'k .W(te#+k#ï,C For this f there exists a simple connection with the matrix elements in the number representation. 103 If p * I then and (H.6) There is a simple connection between the characteristic function and expectation values of operators. We have i(k2 zk ) C(k,k*) - -^ fa*z e^ ^ * i+j 1 <2i) IM (•HE * C(k,k*) - -^ ;d2z B-hu*z*+**z)z.izi (2i) i+j ™*Y c(k,k*> ^ Tr(bjb+ip) (H.7) \M k=o The derivatives of C at k = 0 correspond to expectation values of antinormal ordered operators. A special case to be used later is a displaced thermal state + p - U(zo)p(n)U (zo'i . Its distribution function is given by (z-z_)(z*-z*) ^ Its characteristic function is given by the Fourier transform of this C - C Conversely, any p that has a characteristic function of this form is a displaced thermal state. Conjparingj this with the definition (H.3) yields Tr(.-*ik*V*lkb+p<.,,s>) = exp(-fc(kk*(n+l) - ikz* - ik*z)) . (H.9) 104 APPENDIX J Strength functions of the radiation field The interaction Hamiltonian describing the coupling of a charged par ticle to the radiation field is given by e -+ •+ .-* Hi = mP-A(r't) • The radiation field is quantized in a cubical box of volume V. The dipole approximation means expanding to order zero in the wavelength. Then X(r,t) is given by 3 a /w.V J J. J where j is the mode number, o the polarization index, V the quantiza tion volume, e. is a unit vector perpendicular to the direction of propagation, and the b's are modified boson operators. This has the form of the Hamiltonian in chapter IV with A . given by (J.2) The density of modes per unit frequency per unit solid angle is given by g(w)dü)dn = jJf duidfi . {J.3) The sum over the polarizations and the average over angles can be carried out z dn V,-p| - ^151' • "•« Finally, multiplying everything together, we find In.some cases we use a different normalization corresponding to P = p'/ü , /tö With this normalization the strength function is given by Y(w) - T&»2 ' (J-6) In an unrationalized system of natural units the fine structure constant ,2 , „ * _L 105 APPENDIX K Continuum limit with coordinate coupling In the notation used by ULLERSMA (1966) the Hamiltonian is given by (K.l) We have to diagonalize the matrix wo £1 en (K.2) The eigenvectors i corresponding to the eigenvalues s2 satisfy u2X +7eX = s2 X , (K.3) o ov ni- n nv v ov e X + u)2 X = s2 X , n ov n nv v nv ' z (K.4) nv S -0)2 OV v n The characteristic equation is s2 - u>2 - , _, n (K.5) L Sz-üi/ n v n We need the function A(t) = V X2 cos s t J ov v en (K.6) If we introduce G(Z)=Z-Ü)O 2-Y* • -—5Z-üH- , O n** Z—liln* the sum can be converted into a contour integral kit) = JL ft dz cos t/z where the contour encircles all poles. in the continuum limit the contour can be contracted to the real axis to give ds Y(s)cos(st) A(t) « -, , (K.7) o ("'""J "*/$&) + ^<*> where the strength function y(s) is the product of the coupling strength and density of modes. 106 At this point we can choose between two approaches. The first is to consider Y(S) as describing a real bath, for instance the phonons in a lattice, or as the radiation field with a high frequency cut off. Then, under the conditions that Y(S)/S2 is almost constant over a frequency interval around u and that Y * « (weak damping), A(t) is given by A(t) - e"Y't'(cos uc - >/u> sin w|tj) , (K.8) valid for |t| > x, where the transient time T is roughly the inverse of the available frequency range. All this has been worked out in more detail by Ullersma. The second approach is to consider Y(S) as describing the radiation field without a cut off and use the generalized functions of chapter II to avoid the divergencies. Then the strength function is proportional to fu2. we put In order to calculate A we have to calculate first oo oo o o From the definition of f this is n \A .." i n [dutfdt» sS 2 P d(0 «- « - 1 = P —n—-w oI U?-S2 oJu^-S' * The integrand is even, so it is possible to extend the integration to -». Splitting in partial fractions yields | fdw-~ - -f fdu-4= • 0 , (K.9) 2 ' t»5-s 2 ; u+s tfvOD — OD as demonstrated in Eq. (11.51). We now have (K.10) 2S2 The denominator ensures convergence at infinity, so the f is not needed. The integrand is again even, which makes extension to -°° pos sible. The integral is now easily evaluated, using contour integra tion. We find, without further approximations, A(t, - e'^'fcos mt - J sin w|t|) . w2 - IÜ|-Y2 > 0 (K.ll) 107 This is the same result as before, but it is obtained here without assuming y/u < 1 and there is no transient time. It is even possible to consider an overdamped oscillator. Then we find in the same way Y e - Y2 e 'l ' o A(t) - 2y —- 2 5 . (K.12) Y y l " 2 7 Y2 = Y ~ /Y -^ In the limit Y ^ <»>0 this is simply A This shows that even the strongly damped oscillator can be handled in this way. In the following we consider only the underdamped case. In the same way the other A-functions are found. The results are -Yt A(t) = - —T- {cos wt + ^ sin ut) , o e~yt A(t) = - • sin cot A(t) = e~y (cos tot - ^ sin ut) , A(t) = -we"Tt (^1 cos tot + sin wt) . t > 0 (K.14) For negative times the signs must be changed in such a way that A and A are odd and A and A are even in t. The results of the second approach make it possible to reinterpret the first approach. Instead of taking a realistic y{ia) it is simpler to approximate y(w) by fts>2 first and evaluate the integrals afterwards. In this interpretation the derivation of strong damping for the radiation field can also be considered as an approximate derivation of strong damping for other systems having approximately the same strength func tion, for instance the sound field. For Hamiltonians with coordinate coupling it is not true that the interacting ground state is the same as the non-interacting ground state, as b + b > vpj ? »; j)iv°j * ° 108 To see what happens to the ground state we calculate the reduced inter acting ground state. Its characteristic function is given by -%ik»b -%ikb; C(k,k*;*) - i Tr(< . -^i^*^ -%ikb' b = x c L/5 + o h J*• ov 4 I *™c* v w_ OV V . i /ui /s•> i /to /s "\ We substitute (K.16) into (K.15) and use the Baker-Hausdorf formula (A.7) to bring *-he c's to normal order. Using Ac e v|$> = |»> , we find 2 C (k,k*;*) = J- exp -^ I X* f (k +k»2) (-2 - -1) + 2kk»b°- + _S» + 2)| exp (K.17) •sr v *• v r o ' j From this we split off the characteristic function of the non-inter acting ground state C0(k,k*) = ~ exp -* C(k,k*;*>) - C0(krk*>expj-U£ I X^v(^2 . i)J -4kZ I X* fc - 1)1 . (K.18) p L 0VlU_ 'J 139 Vie have to evaluate X2 I s X* and I *ov V V He evaluate the sums for the particular case of the radiation field with strength function yiu) =* fu>2. In this case X£ 4Y<*> r j „ «•»„ 2r- -s-OV1 * —v' Q— I f,»_...»dS öS ^ j„g,a s2 = y "o * s * ÏÏ oJ(s 2-w2)2 + 4y2s2 2yw «__2 ÉY. . (K.19) 2 0(y-^) +4Y y This is a simple integral (see e.y. GRÖBNER * HOFREITER II, 131, 3a). The result is Ü1 ( 01 — arccot To first order in ^ this is 1 - - * + 0|-C] . CK.20) TT Ü) [ü)2j The other sum is J_ J X2 , . ±L ? f ê**! . (K.21) Again we put s2 = y ,Wo i(y-«2)%4 2y o — o Y This equals 2 d((y~w£) + 4v2y _1_ f 2_ + _JL_ (2Ü)2+4Y2) - -**, r + 4 2 +4 2 *° ' o To first order this equals the previous result. Our final result is now _L_ [k2 + k2] x p C(k,k*;*) * C0(k,k*) «*™ I ' . (K.22) 110 From this we see that to first order in y/u the distribution in phase space has been narrowed as a consequence of the damping. It is how somewhat sharper than the uncertainty relations permit for a free oscillator. The ground state energy is also lowered to first order. This has nothing to do with a frequency shift. All levels will be shifted in the same way. The frequency shift is second order in -y/u. In the equations of motion we must expect first-order effects result ing from the adaptation of the non-interacting state to the interact ing state. The violation of the uncertainty relation should not worry us as the system coupled to the radiation field with strength function fu> is not a QM-system in the usual sense. Moreover, y_ _ 2 e2i» 10 J m Fcr electromagnetic systems u * e^m. The narrowing is of order e6 = a3 * 10~7. This is not likely to be observable. APPENDIX L Continuum limit with symmetrical coupling In order to calculate the functions g(t) and h.(t), we have to find the eigenvectors of the matrix _ y ) o 1 n \* Ü). -e- i n 1-> A = v. (L.l) i -s 1 >. -G- %. •% V X( * w„ n n This has been done by Ullersma (1966) for a similar problem. The eigen vectors X and the eigenvalues z satisfy AX = z -X (L.2) V V V o ov fr i IV vi 3 ov A*X + to X » z X (L.3) n ov n nv v nv _ Ax*n*ox v £ (L.4) nv " Vcon 111 The characteristic equation is A A* A . A . (L.5) u - z + y * i - o O V k Z -tl) . 3 v 3 (L.6) " G^ - % " Z+ A.A I ^* = ° Normalization : A A* X X* + 3 = 1 ov ov 1 I (2 ^ )2J 3 v 3' ' z is real, choose X real. V ' OV -1 Then X* = dG (L.7) ov dz In terms of the X and z , g(t) and h.(t) are given by V -iz t g(t) = XI*2 e. I ov "izvt h.(t) = Y X^ X, e 3 ^ ov jv -iz t v X' e = X ov (L.8) 3 I z -w. Using (L.7) and Cauchy's theorem, these functions can be expressed as a contour integral. -izt dz e g(t) ITMT G(z) a.9) izt h ,,. 1 I dz e" (L.10) V ~ 2rri J The contour encircles all the poles on the positive real axis. We can now take the continuum limit «"•> - • - "o - P52^ • (L.ll) Y(uj)dü) = g(oi) A (ui) A#(u)du) , where g(w) is the density of modes. 112 The contour can be contracted to the positive real axis, ««*> = 2ÏI \d*[ G+(x) = lim G(z) - z+x+ie ^S-lM + iir Y(x) , = x - UQ + P X-Li which gives -ixt d« ï(x)e ^ g(t) = 1 , ' - "'T . 1 '»>• — U.13) x-„ + p [doLlïil + TT2Y2(X) J X-Ü) t If the bath is the radiation field, Y(w) = f -f ?u> (L.14) O If we extend the lower limit of the integrals to -», we can use +OD = 0 (L.l-5) P? da) tü-X We obtain dw uie -iwt • Here too we extend the lower limit to -», Now, the integral can be evaluated by contour integration. The poles of the integrand are located at U) - (L.17) To first order in ^ we find ID -iü)0t-y|t| g We estimate the errors introduced in this wny. In the first place the principle value integral in the denominator is not zero. However, if the damping is weak y < w , we can approximate it by a constant frequency shift. 113 The cut off error for the negative frequencies is -io>t dw e j_ fdu_« The graph of CO (ü)+'i) ) ^ looks like For t = 0 the integral is of order — . To get the integral small the exponential must oscillate fast under the hump, so we must have t > 1/w (L.19) To summarize, we have obtained ,it> - .-TI*!."""' . (L.20) under the restrictions y/w < 1 (L.21) and t > 1/w . These restrictions are caused by the absence of negative frequency oscillators. In the corresponding problem in Appendix K these restric tions are absent. The reason is that there we had a second-order sys tem and only squares of the frequencies occurred. We also need the h.(t), which we calculate from the g(t) found earlier, using (L.4) gj h^t) = -iWjh^t) - i^gft) (L.22) hj(0) = 0 The solution is -iw.(t-t') h.(t) - -iA. dt' e J g(t»j J J . o -Uü^-ojjjf -Y|t' -iu) .t - -iA.e -i(u> -ID.) -Y sign t' 114 This gives -ieu t-if|t} -iw.t hj(t) = Xj ai -u. - iy sign t and (I..23Ï o 3 A.X»[l-2e"Ytt'cos(« -ti.)t + ê"2l,|tlj 3 L t) = -1_2J ,••,,.> °- * (I..24) v*< (it) — U . ) ^ l vz What «re need is X = I h.h^-n, - I ^J- 3 3 J $ 6.w. 3 e 3 3 _ j In the continuum limit this equals no Y t 2 du vca 1 - 2e- ' tco-3U-u,0)t + e' ^} = 1 [e6«"<« - lJ[(u,-* )* + r*J (L.25) o o| o We now take the bath in equilibrium 6(w) = 6 . Then, the function looks like J«-l For high temperatures, &u < 1, we can take the exponential out of the integral. Then we find A = (l-gg*) EL.2C) 8wrt e6"°-, e °-l The last step follows directly from equation IV.4. The diffusion coefficient is then given by »'« •» ft •>» - -^- • e °-l For low temperatures, ï«uo kT « ito the contribution of the low-frequency oscillators for t * » is given roughly by i l Y l l l l y f 1 I2 If It Ü1 B I Üi TT 115 On the other hand, the contribution of the resonant oscillators is of order , ••!, „2T"° , • .Ml- » i e'6"° . res it Uu l 8w^ ] 3w_ ir °{7^ 2 e °-l For sufficiently low temperatures the contribution of the low fre quency oscillators is dominant. This means that the oscillator does not reach the bath temperature for t -* °o. This begins at temperatures given.by 8w = 2«log 3<»> - log ^ . For ^ < 1 we have Bw > log Btu , so approximately Bw * -log Y/W. This results from the fact that the interacting thermal state is not the product of non-interaction thermal states except when T = 0. It is possible to show that the oscillator reaches, for t = », the reduced thermal state corresponding to the bath temperature. The effect is an uninteresting shift in the state. As we do not want to find the equi librium state to this precision, we approximate X(t) by foi all temperatures. We also see from (L.25) that X(t) is independent of time for yt > 1. There is no QM-transient time longer than 1/y. 116 ACKNOWLEDGEMENTS In the first place, I want to thank Prof.Dr. N.G. van Kampen for his interest in this work and for many stimulating discussions. The theoretical group TN I has been a stimulating environment for the development of this work. In particular, I want to thank Prof. Dr. M.P.H. Weenink for his interest and for his part in providing the atmosphere in which this work was possible, Dr. T.J. Schep who read the entire manuscript and suggested many improvements and clarifica tions, Dr. R.W.B. Best for some stimulating discussions on generalized functions, and Dr. D.A. D'Ippolito who read the drafts and corrected many mistakes in my English. The theoretical physics group at the T.H. Eindhoven has been very hospitable during my stay in Eindhoven, and the discussions I participated in considerably deepened my understanding of quantum mechanics. I also want to express my thanks to all those who contributed to the realization of this report: the secretariat of the FOM-Instituut which prepared the final draft in a very short time, Ms. E. Postma who carefully typed the final manuscript, Ms. H. Toft-Betke and Ms. H.J.C. Thoden van Velzen who corrected the final version and made me aware of some of the errors in my use of the English language, and the repro duction department of the FOM-Instituut which prepared the illustra tions and the cover design. This work was performed as part of the research programme of the association agreement of Euratom and the "Stichting voor Fundamen teel Onderzoek der Materie" (FOM) with financial support from the "Nederlandse Organisatie voor Zuiver-Wetenschappelijk Onderzoek" (ZWO) and Euratom. It is also published as a doctor's thesis of the Univer sity of Utrecht. 117