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A New Sum Rule for the Two-Body Distribution Function of an Infinite Saturated Fermion System

R. Guardiola

Published Jan 1, 2007• 0 citations
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Abstract

where e is the dens i ty of the system, xi represents the set of space, spin and isospin co-ordinates , xi = {ri , ai , ~i} and the in tegrals include t h e summat ion ove r discrete var iables as wel l as the in tegra l on the space co-ordinates. Fu r the rmore , eq. (1) mus t be unders tood in the A + ~ l imit . The funct ion ~(Xl, x2) is an opera tor in the spin and isospin space, and is t rans la t iona l ly invar ian t , i.e. i t depends only upon the r x r 2 var iable . W i t h t he a id of the d is t r ibut ion func t ion one m a y compute the expect a t ion va lue of any local, two-body opera tor by means of the fo rmula

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The origin of the wave-part ic le dual ism of quantum theory can be t raced back to Einstein 's analysis of the fluctuations of b lack-body radia t ion in 1909 (1-3). Based on this invest igat ion EINSTEIlq proposed the in te rpre ta t ion tha t the electromagnetic field in thermal equil ibrium has as well par t ic le as wave propert ies . In his analysis concerning the ideal quantum gas of 1925 he came to the conclusion tha t this dualist ic in te rpre ta t ion applies to the par t ic les of the gas as well (a). Recent ly we have shown tha t the fluctuations of b lack-body radia t ion can be explained from the assumption of s ta t i s t ica l ly dependent indist inguishable part icles (5). Moreover, i t tu rned out tha t the so-called wave contr ibut ions to the var iance are ent i re ly due to the s ta t is t ical dependence of these part icles. F rom this viewpoint Einstein 's dualist ic in te rpre ta t ion is incomprehensible. I t is the aim of this note to explain a weak point in Einstein 's analysis in detai l and to show tha t wave proper t ies emerge in the classical l imi t only. Planck 's formula gives the mean value of the energy densi ty g* (with respect to dco) of b lack-body rad ia t ion :

•Jan 1, 1985•0 citations
where e is the dens i ty of the system, xi represents the set of space, spin and isospin co-ordinates , xi = {ri , ai , ~i} and the in tegrals include t h e summat ion ove r discrete var iables as wel l as the in tegra l on the space co-ordinates. Fu r the rmore , eq. (1) mus t be unders tood in the A + ~ l imit . The funct ion ~(Xl, x2) is an opera tor in the spin and isospin space, and is t rans la t iona l ly invar ian t , i.e. i t depends only upon the r x r 2 var iable . W i t h t he a id of the d is t r ibut ion func t ion one m a y compute the expect a t ion va lue of any local, two-body opera tor by means of the fo rmula
In recent years, some interes t has been raised by the problem of the quan tum harmonic-osci l la tor wi th a coupling cons tant K(t), which is a funct ion of t ime (1.2). The impor tance of this p roblem has been stressed by the analysis of the mot ion of a charged part ic le in the presence of a magnet ic field H~(t), which has a fixed direct ion bu t changes i ts in tens i ty wi th t ime (1). The reason of this note is tha t , among these recent studies, the most na tura l approach is surprisingly absent. We will describe i t here, and we will analyse some easily solvable potent ials . W e say t h a t i t is a ve ry na tura l approach, because i t consists in a ve ry na ive general ization of Dirae 's method. I n order to in t roduce it, we remark tha t the Dirae creat ion opera tor (in the t imeindependen t case) could be considered as the opera tor t ha t gives a solution once applied to another solution (for ins tance the vacuum). So we in t roduce an opera tor A ( x , ~ /~x , t) with the p rope r ty tha t , once appl ied to a solut ion of the SehrSdinger equat ion , i t gives another solution. The expl ici t form of this opera tor A is obta ined by means of a simple differential equat ion wi th one and only one solution. W e then find an explici t solution of the t ime-dependen t SchrSdinger equat ion , as a simple general izat ion of the vacuum wave function. App ly ing i tc ra t ive ly the A opera to r we obta in an explici t form for a set of solutions. Noth ing can be said, however , abou t the general case.
The quan tu m m echanical correlation problem, involv in g th e ev a l uation of expectation values of many particle op erators f or a given eigens tate, is formulated, by m eans of t h e matrix sum ru le, in term s of unobservable or virtual on e particle trans it ion s t o other eigenstates of the system.This formulation is set forth for t he two-fold purpose of both demonstrating t h e natur e of t h e many particle correlations from t he po in t of view of matrix m echani cs, and mak ing poss ible the d er ivation of sum ru les which may be used in t heir analysis.Th e analysis shows tha t t he trans it ion integrals < "'IAil "" > and < ",IAil",' > , where Ai and Ai are arbitrary one particle operators involving t he ith and jth particles, resp ectively, are not n ecessari ly equal, sin ce such trans it ions can in vol ve q uantum states of differen t coordinate sy mmetry, i. e., the antisymmetric and sy mmetric states \"h en N = 2, an d the states o f mi xed sy mmetry, in add it ion, wh en N? 3. Th e indistingu isha bili ty pr inciple is d e monstrated to r equ ire on ly t hat ~1< "'I A i/"" >1 2 = ~/< ", / A i /"" >12 .a' ex' B y detai led analys is of one particle tra ns itions to states of "al'ious sym m etri es, t he relationsh ip between the va ri ous one part icle trans itions is d eterm ined.Th e expectat ion valu e fOI' a n ant isymmetri c state of t h e two part icle operator :wh ere t he sum "'~ is over a ll a n t isy mm etri e states, and the s um a ;' " is over a ll the nonantisymm etri c states.This equ ation is used to obta in an approx imate relation between t hese ex p ectation values and obser vable quantities.From t he H eise nb erg co mmu tation r elat ions, th e sum rule N .z=~ (E.,-E.)/<", lri/",' >1 2=3h 2 N/2m i = l a 'is obtained an d used to d emonstrate some ge nera l properties of systems in t he nonrelativ istic approx im ation.
The origin of the wave-part ic le dual ism of quantum theory can be t raced back to Einstein 's analysis of the fluctuations of b lack-body radia t ion in 1909 (1-3). Based on this invest igat ion EINSTEIlq proposed the in te rpre ta t ion tha t the electromagnetic field in thermal equil ibrium has as well par t ic le as wave propert ies . In his analysis concerning the ideal quantum gas of 1925 he came to the conclusion tha t this dualist ic in te rpre ta t ion applies to the par t ic les of the gas as well (a). Recent ly we have shown tha t the fluctuations of b lack-body radia t ion can be explained from the assumption of s ta t i s t ica l ly dependent indist inguishable part icles (5). Moreover, i t tu rned out tha t the so-called wave contr ibut ions to the var iance are ent i re ly due to the s ta t is t ical dependence of these part icles. F rom this viewpoint Einstein 's dualist ic in te rpre ta t ion is incomprehensible. I t is the aim of this note to explain a weak point in Einstein 's analysis in detai l and to show tha t wave proper t ies emerge in the classical l imi t only. Planck 's formula gives the mean value of the energy densi ty g* (with respect to dco) of b lack-body rad ia t ion :