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Polarization of Triplet P, Photons in Asymmetrical Decay

G. Faraoi, A. R. Pe Nnisi

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

I n order to find out new exper iments useful to t es t q u a n t u m predic t ions for the polar izat ion correlat ion of the decay p roduc t s of a composi te sys tem, we have recent ly (1,~) proposed a new m e a s u r e m e n t us ing the th ree annih i la t ion pho tons of t r iplet P , in S-state . The proposed expe r imen t was concerned wi th the m u t u a l polar izat ion of a pa i r of pho tons ana lysed b y ideal p la tes in two typ ica l a symmet r i ca l cases shown in fig. 1, where the pos i t ron ium a t o m is supposed to decay into two pho tons wi th the same m o m e n t u m , while the th i rd one is de te rmined b y the e n e r g y m o m e n t u m conservat ion. I n bo th si tuations the joint p robab i l i t y of detect ing pho tons To . 1 and To . 2 polarized along definite directions was der ived according to the hypo theses which are now briefly recalled: the first hypothes is was following the convent ional lines of q u a n t u m theo ry b y assuming t h a t the initial po lar iza t ion corre lat ion of the emi t t ed pho tons is indefinitely ma in t aned , so t h a t i t r emains

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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.

•Jan 1, 2007•0 citations

Polarization correlations in the two–photon decay of hydrogen–like ions

F. Fratini, A. Surzhykov

arXiv:1006.4799•Hyperfine Interactions•Apr 14, 2011•7 citations

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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

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a Wave-Radiation Model for the Onset of Dissipation in Superfluid Helium at the Roton Critical Velocity.

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in g the o n s e t of d is s ip a tio n at th e ro to n c r i t ic a l v e lo c ity in s u p e r flu id h e liu m .T h is is a c la s s ic a l , c o r r e s p o n d e n c e -p r in c ip le -lim it po in t of v iew and is q u ite an alo g o u s to th e c la s s ic a l d e s c r ip tio n that i s w idely a c c e p te d fo r u n d e rs ta n d in g the e m is s io n of photons by an e le c t r i c c h a rg e th a t m o v e s at a c o n s ta n t v e lo c ity g r e a t e r th a n the sp e e d of lig h t in a d ia le c tr ic m e d iu m .A fte r f i r s t d e riv in g a th e o ry f o r th e w ave e x c ita tio n s and b o u n d a ry -v a lu e p ro b le m s of the c o rre s p o n d e n c e p rin c ip le lim it fo r a co n d en se d -b o so n s y s te m , th is p a p e r show s th a t a w ave r a d i a tio n field f o r m s a ro u n d a fo re ig n body m o v in g th ro u g h the s u p e r fluid m e d iu m w hen the fo re ig n body m o v e s f a s t e r th an the m in im u m p h a se v e lo c ity , {w/k )m jn » w ave e x c ita tio n s in the m e d iu m .M o re o v e r, the sp a tia l d e p e n d e n c e of th is w ave fie ld and the lo s s of e n e rg y r a d ia tin g th ro u g h it a r e c a lc u la te d a s fu n ctio n s of the f r a c tio n a l in c r e a s e of the fo re ig n b o d y 's v e lo c ity above the th r e s h o ld v e lo c ity In s p ite of a p p ro x im a tio n s m a d e in th e c a lc u la tio n s , th e r e s u l t s in d ic a te th a t it is im p o s s ib le , e x p e r i m e n ta lly , to f o r c e an io n c o m p lex to m ove th ro u g h the s u p e rflu id c o m ponent o f h e liu m a t a v e lo c ity o b s e r v a b ly g r e a t e r th an th e ro to n c r i t i c a l v e lo c ity .ACKNOWLEDGMENTS I would lik e to thank P r o f e s s o r R. L. M ills fo r h is c o n trib u tio n s to m y g ra d u a te ed u catio n and fo r h is p e rc e p tiv e c o u n s e l w hile th is r e s e a r c h w a s b e in g c o n d u cted , e s p e c ia lly r e g a r d in g th e re s o lu tio n of s e v e r a l m a th e m a tic a l d iffic u ltie s w hen no so lu tio n s s e e m e d to e x is t.T h e N a tio n a l A e ro n a u tic s and Space A d m in is tr a tio n i s th an k ed f o r i t s fin a n c ia l su p p o rt d u rin g the y e a

•Jan 1, 1967•0 citations

Polarization correlations in the two–photon decay of hydrogen–like ions

F. Fratini, A. Surzhykov

Springer eBooks•Jan 1, 2011•0 citations
I n order to find out new exper iments useful to t es t q u a n t u m predic t ions for the polar izat ion correlat ion of the decay p roduc t s of a composi te sys tem, we have recent ly (1,~) proposed a new m e a s u r e m e n t us ing the th ree annih i la t ion pho tons of t r iplet P , in S-state . The proposed expe r imen t was concerned wi th the m u t u a l polar izat ion of a pa i r of pho tons ana lysed b y ideal p la tes in two typ ica l a symmet r i ca l cases shown in fig. 1, where the pos i t ron ium a t o m is supposed to decay into two pho tons wi th the same m o m e n t u m , while the th i rd one is de te rmined b y the e n e r g y m o m e n t u m conservat ion. I n bo th si tuations the joint p robab i l i t y of detect ing pho tons To . 1 and To . 2 polarized along definite directions was der ived according to the hypo theses which are now briefly recalled: the first hypothes is was following the convent ional lines of q u a n t u m theo ry b y assuming t h a t the initial po lar iza t ion corre lat ion of the emi t t ed pho tons is indefinitely ma in t aned , so t h a t i t r emains
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.
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
in g the o n s e t of d is s ip a tio n at th e ro to n c r i t ic a l v e lo c ity in s u p e r flu id h e liu m .T h is is a c la s s ic a l , c o r r e s p o n d e n c e -p r in c ip le -lim it po in t of v iew and is q u ite an alo g o u s to th e c la s s ic a l d e s c r ip tio n that i s w idely a c c e p te d fo r u n d e rs ta n d in g the e m is s io n of photons by an e le c t r i c c h a rg e th a t m o v e s at a c o n s ta n t v e lo c ity g r e a t e r th a n the sp e e d of lig h t in a d ia le c tr ic m e d iu m .A fte r f i r s t d e riv in g a th e o ry f o r th e w ave e x c ita tio n s and b o u n d a ry -v a lu e p ro b le m s of the c o rre s p o n d e n c e p rin c ip le lim it fo r a co n d en se d -b o so n s y s te m , th is p a p e r show s th a t a w ave r a d i a tio n field f o r m s a ro u n d a fo re ig n body m o v in g th ro u g h the s u p e r fluid m e d iu m w hen the fo re ig n body m o v e s f a s t e r th an the m in im u m p h a se v e lo c ity , {w/k )m jn » w ave e x c ita tio n s in the m e d iu m .M o re o v e r, the sp a tia l d e p e n d e n c e of th is w ave fie ld and the lo s s of e n e rg y r a d ia tin g th ro u g h it a r e c a lc u la te d a s fu n ctio n s of the f r a c tio n a l in c r e a s e of the fo re ig n b o d y 's v e lo c ity above the th r e s h o ld v e lo c ity In s p ite of a p p ro x im a tio n s m a d e in th e c a lc u la tio n s , th e r e s u l t s in d ic a te th a t it is im p o s s ib le , e x p e r i m e n ta lly , to f o r c e an io n c o m p lex to m ove th ro u g h the s u p e rflu id c o m ponent o f h e liu m a t a v e lo c ity o b s e r v a b ly g r e a t e r th an th e ro to n c r i t i c a l v e lo c ity .ACKNOWLEDGMENTS I would lik e to thank P r o f e s s o r R. L. M ills fo r h is c o n trib u tio n s to m y g ra d u a te ed u catio n and fo r h is p e rc e p tiv e c o u n s e l w hile th is r e s e a r c h w a s b e in g c o n d u cted , e s p e c ia lly r e g a r d in g th e re s o lu tio n of s e v e r a l m a th e m a tic a l d iffic u ltie s w hen no so lu tio n s s e e m e d to e x is t.T h e N a tio n a l A e ro n a u tic s and Space A d m in is tr a tio n i s th an k ed f o r i t s fin a n c ia l su p p o rt d u rin g the y e a