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 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.
In Chap. 2 we solved for the probability amplitudes Ca and C.b explicitly. The results of our discussion, such as the probability of a transition or the value of the induced dipole moment, were invariably expressed in terms of bilinear combinations of the amplitudes, such as CaCa* and CaCi>*. In fact, the expectation value of any observable involves bilinear combinations. Hence an alternative formulation of quantum mechanics consists of dealing with the bilinear quantities directly. The method involves organizing the quantities in a matrix form called the density matrix. There are two principal advantages in doing so. First, the resulting mathematics is often simpler. Sec ond, when the wave function for an ensemble of systems is not known but the probabilities for having various different wave functions are known, the en semble can be described by a weighted sum of individual density matrices. In particular, the amplitudes cannot themselves describe simply certain com mon statistical phenomena, such as the effects of elastic collisions on the in duced dipole moments of atoms.
Corrélations quantiques : une approche de physique statistique La notion de cohérence, intimement liée à celle de dualité onde-corpuscule, joue un rôle central en mécanique quantique. Lorsque la cohérence quantique s'étend sur plusieurs particules au sein d'un système, la description en termes d'objets individuels devient impossible en raison des corrélations quantiques (ou intrication) qui se développent. Dans ce manuscrit, nous étudions les systèmes à l'équilibre, pour lesquels nous montrons que les fluctuations cohérentes viennent s'ajouter aux fluctuations prédites par des identités thermodynamiques valides pour les systèmes classiques. Au zéro absolu, les fluctuations cohérentes sont les seules à subsister, et nous étudions dans ce cas leur lien avec l'entropie d'intrication. Nous montrons en particulier qu'une hypothèse de température effective modulée spatialement rend compte de la structure de l'intrication au sein d'un système à N corps, et montrons comment cette température peut être extraite des corrélations usuelles. Nos résultats permettent par ailleurs une compréhension affinée des transitions de phase quantiques. Nous montrons par exemple que la transition entre un isolant de Mott bosonique et un superfluide donne lieu à une singularité de l'entropie d'intrication induite par les fluctuations d'amplitude de la phase du condensat. Nous identifions enfin une longueur de corrélation pilotant les lois d'échelles des fluctuations cohérentes au sein de l'éventail critique avoisinant une transition de phase quantique du second ordre, et proposons une ouverture vers les potentielles applications métrologiques de ces fluctuations cohérentes exceptionnellement fortes en nous appuyant sur l'exemple du modèle d'Ising.
D erivation o f th e Lorentz Transform ation.Views o f E in s te in on Space, Time and Sim ^iltaneity.Minkowski' s 4-dim ensional a n a ly s is .In v a ria n ce o f Maxwell' s E quations under a L orentz T ran sfom atioE M echanics i n %>ecial R e la M v ity .The Doppler E ffe c t i n S p e c ia l R e la t iv it y .Transform ation o f T ensors in s p e c ia l R e l a t i v i t y .QÜANTIM THK)RY De B r o g lie Theory -Ssqpeiiment^l v e r i f ic a t io n .P ï l n d p l e s o f M echanics and O p tic s .^ Germat -Hamilton Comparison between th e Developm oit o f Theory in M echanics and O p tic s .Schrodinger* s Equation.S o lu tio n o f equation fo r t i e o s c i l l a t o r and t i e Hydrogen Atom.Meaning o f Wave and Gioup V e lo c ity .I k ic e r t'in ty R e la tio n .R ela tio n between H eisenberg and Schrodinger.P a u li' s I n c lu s io n o f %)in.H rac* s E quation.Dirac T h ^r y o f th e E lec tro n in th e Atom.In v a ria n ce o f th e D irac Equation.N eg a tiv e Energy S ta te s .if€c^x)ic& iind 0;jtics I r t h e s e tw o b r a n c h e s o f r^olenoe, t h e phenom ena, w h ich in th e c o u r s e o f t l r -e , f i r s t came t o be s t u d i e d , w ere t h o s e le r i; e s c a l e o r m a c r o s c o p ic e f f e c t s w h ich o b tr u d e thtansel^m s upon th e s e n s e e * T h is wae s o i n M e c h a n ic s, t h e s t u d y o f m a tte r in m o tio n and i n e q u ilib r iu m , w here b o d ie s a t l e a s t ccsm aT uble i n s i s e w it h t h e human body and m oving s lo w ly enough f o r t h e i r m o tio n t o be s t u d i e d , f i r s t came u n d er c o n s i d e r a t i o n , and i n L i g h t , w here th e atud^' o f th e m a c r o s c o p ic q u a l i t i e s l e d t o G s fx a e tr lc a l O p tic s* Laws w ere fo r m u la te d w h ich w ere i n a c c o r d w it h th e known phcRonena u n t i l f ; r t !i e r s tu d y b r o u g h t s m a ll s c a l e phenomena u n d er o b e e r v e t io r * H ere d i f f i c u l t i e s w ere fo u n d , b o th w here t h e b o d ie s w ere e x ti'e m e ly s m e ll and ivhere m o tio n was v e r y g r e a t* I t seem s u n d e r s ta n d a b le t h a t i f th e b e h a v io u r o f th e v e r y u io a lle s t p o r t io n s o f t h i n g s i s known, th e n l y a k in d o f a d d i t i o n i t may be p o s s i b l e t o f i n d t w la w f o r la r g e e f f e c t s and t h a t th e s t a t i s t i c a l la w s o o b ta iiw id may n o t show tW i n t r i c a c i e s o f th e l a u s f o r th e i n d i v i d u a l p a r t s * A ls o we c a n im a g in e t h a t v a r io u s p ^o ssib le la w s o f b e h a v io u r f o r th e i n d i v i d u a l s c a n le a d t o th e same s t a t i s t i c a l law* I n a d d i t i o n , i t h a s b egu n t o be u n d e r sto o d t h a t we c a n n o t make e x a c t m easurem ent on a l l a s p e c t s o f a p h y s ic a l, p rob lem a t t h e aaae t im e , aikl t h a t -1 « th o u g h , i f we a re c o n t e n t w it h a r o u g h ly c o r r e c t a n sw er we may 4 f i n d a c o n s i s t e n t f o r m u la t io n a law f o r t h e w h o le b e h a v io u r o f A a b o d y , y e t i t now seem s t h a t i f we c o n c e n t r a t e upon one p a r t i c u l a r a s p e c t and su c c e e d i n a n e x p la n a t io n o f t h a t , i t w i l l n o t be p o s s i b l e t o e x p la in o t h e r a s p e c t s i n th e same te r m s .I n b o th M ech a n ics and O p tic s we have a d u a l i t y i n t h a t t h e r e i s b o th a c o r p u s c u la r and wave a s p e c t o f e a c h t o b e c o n s id e r e d .The r e a l d ev elo p m en t o f b o th M eciian ics and O p tic s a s s c i e n c e s i n t h e s y s t e m a t ic form i n w h ich we u n d e r sta n d them , b egan i n th e 1 7 th c e n t u r y , th o u g h lo n g b e f o r e t h a t tim e t h e r e w ere t h e o r i e s o f M a tte r and o f L i g h t , some o f w h ich b e a r d i s t i n c t r e sem b la n c e t o v ie w s now h e l d .O f t h e s e we may n o te th e A tom ic th e o r y due t o D em o critu s o f th e 5 t h c e n tu r y B .C .a c c o r d in g t o w h ic h , a l l s u b s t a n c e s , s o l i d s , l i q u i d s , g a s e s o r l i g h t w ere c o r p u s c u la r i n n a tu r e and t h e i r b e h a v io u r became e x p l i c a b l e by th e m o tio n s and i n t e r a c t i o n s o f t h e s e 'a to m s * .The c o r p u s c le s th e m s e lv e s w ere i n d i v i s i b l e and w ere th e u lt im a t e u n i t s o f n a t u r e .I t i s t o be s e e n t h a t a s i n g l e u n d e r ly in g p r i n c i p l e w as p o s t u la t e d f o r th e b e h a v io u r o f b o th M a tter and L ig h t a t t h i s t im e .I n t h e 3 r d c e n tu r y B .C .A rch im ed es, i n h i s s tu d y o f m ech a n ics was m a in ly co n ce rn ed w it h S t a t i c s and h e w as
If we consider ( ) 2,kr tΨ G to be the probability density for finding, at time t, particle 1 at particle k at and particle N at ,,1 … Gr …G ,kr ,Nr G a normalization condition analogous to Eq.(29.2) holds as (35.2) 1V* =ΨΨ∫ d The integral is defined in a space with 3N dimensions called the configuration space of the system, where a configuration point is given by 3N position coordinates 1 1 1, , , , , , , , , ,k k k N N N ,x y z x y z x y z… … and the volume element is NNNkkkNk dzdydxdzdydxdzdydxdddd ………… 1111 VVVV == Physical observables of the system will be represented by Hermitian operators, obtained from the classical representation in terms of the position and momentum variables as indicated by Postulate 2. Thus the many-particle Hamiltonian results from Eq.(2.32) as ( ) (∑ ∑ = ≠ ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ++∇−= k rrUtrU m H ,, 2 ˆ GGG= ) ) (35.3) where only acts on the coordinates of the kth particle, of mass The one-particle potential energy is due to external fields where this particle moves and k∇ .km( trU kk ,G( )jkkj rrU GG , gives the mutual interaction between the particles k and j. It is assumed that the position and momentum operators of different particles commute, and so do all the operators describing single-particle observables if they refer to different particles. The evolution of a many-particle system will be then formulated as a generalization of Eq.(30.6) in terms of given by Eqs.(35.1) and (35.3) HˆandΨ Ψ=∂ Ψ∂ H t i ˆ= (35.4) If the Hamiltonian (35.3) does not depend explicitly on the time, it is usual to express the time dependence of the many-particle state function by means of the evolution operator defined as ( )tUˆ ( ) ( ) ( )NkNk rrrtUtrrr G…G…GG…G…G ,,,,ˆ,,,,, 11 αΨ=Ψ (35.5) which is formally derived from Eq.(35.4) as ( ) ( ) ( )ˆˆ ˆU ti U t i HU t t t ˆα α α ∂∂ ⎡ ⎤Ψ = Ψ =⎣ ⎦∂ ∂= = Ψ It follows that (35.6) ( ) =/ˆˆ tHietU −= which is the analogue of Eq.(30.12). The evolution operator is unitary, Eq.(II.22), since ( ) ( )†ˆ ˆU t U t Iˆ= (35.7) Substituting Eq.(35.5), one obtains from Eq.(35.4) that the stationary states of the Nparticle systems are given by the energy eigenvalue equation ( ) ( )1 1ˆ , , , , , , , ,k N k NH r r r E r r rα α αΨ = ΨG G G G G G… … … … (35.8) The solution to the N-particle problem is facilitated if we take into account the symmetries of many systems of physical interest, which allow us to define constants of motion and criteria for simplifying the form (35.3) of the Hamiltonian operator. An alternative to complex explicit calculations is to associate with specific symmetries time independent unitary operators which transform the state functions into so that Eq.(35.4) gives ˆ ,U Ψ ˆ ,U Ψ ( ) ( )†ˆ ˆ ˆ ˆ ˆ ˆ ˆi U i U UH UHU Ut t∂ ∂ΨΨ = = Ψ = Ψ∂ ∂= = ˆ This is again a Schrödinger equation for the state Uˆ Ψ with the new Hamiltonian †ˆ ˆ ˆ ˆH UHU′ = If the Hamiltonian remains unchanged by the transformation, we have (35.9) †ˆ ˆ ˆ ˆ ˆ ˆor , 0H UHU U H⎡ ⎤= ⎣ ⎦ = According to the evolution equation (29.43), this implies that so that U is a constant of motion. If the unitary transformation which leaves the Hamiltonian invariant, is a function of an observable, its constancy implies that the given observable is a constant in any state of the system.
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.