EPR Paradox and the Physcal Meanng of an Experment n Quantum Mechancs Vesseln C Nonnsk vesselnnonnsk@verzonnet Abstract It s shown that there s one purely determnstc outcome when measurement s made on the state functon chosen by EPR to descrbe the combned two-partcle system the dstance between the two partcles s preserved the same Further, t s shown that, surprsngly, the Ψ -functon desgned accordng to QM leads to the followng paradox despte the fact that the two partcles move n opposte drectons, n tme the dstance between them becomes shorter and shorter As s known, Ensten, Podolsky and Rosen (EPR) [] observe two partcles whch have nteracted n the past (for < t < T) but the nteracton between them has ceased for tmes t > T Nevertheless, both partcles, even at tmes t > T EPR consder to be descrbed by one common state-functon, namely Ψ ( x, x) The authors have chosen to explore one partcular form of that state-functon, namely Ψ ( x, x ) = dpe ( x x x) p () Ths functon may be rewrtten as Ψ ( x, x ) = dpe ( x x ) p x p () + e xp and then e can be consdered as the orthonormal bass of an observable A pertanng to partcle (poston of partcle ) whle the exponents
( x x) p e + are the expanson coeffcents In other words, we may observe the above ntegral as the expanson of the Ψ -functon along the contnuous xp bass e When a measurement of the poston of partcle at tme t s carred out the Ψ -functon collapses to the followng functon (, ) δ ( ) δ ( ) f x x = x x x x x (3) and one mmedately sees from eq(3) that x = x + x (4) It can easly be seen also that f we carry out measurement of the momentum of the frst partcle at tme t the Ψ -functon collapses n such a way as to gve egenvalues p and p of the frst and the second partcle, respectvely Now, we may wsh to explore what the reproducblty would be of the mentoned measurement (collapse of the same functon Ψ ( x, x) ) at the same tme t when carryng out repettve measurements on t Obvously, each tme the operator A s appled on the functon Ψ ( x, x) an egenvalue of that operator wll be produced It should be noted, however, that, although the probablty (expectaton value) of obtanment of a gven egenvalue (from the set of the egenvalues of the operator A ) s easly wrtten as < A j >, the outcome from the process of a concrete measurement s completely unpredctable In other words, as a result of a gven measurement the obtanment of a gven egenvalue s completely random Interestngly, however, despte the fact that as a result of repettous measurements (always done at tme t) we wll obtan dfferent values of x (resp x ), the separaton between these two partcles at each measurement wll be one and the same x Ths wll not be so noteworthy f there was
not another fact concernng ths system of two partcles As mentoned, f we happen to measure the momentum A of partcle we wll each tme (after each successve measurement) obtan a dfferent value p of that momentum and, whch s even more remarkable, as mentoned, ths momentum wll be exactly equal n sze but opposte n sgn to the momentum of partcle In other words, despte the fact that the two partcles wll tend to move, at tme t, n opposte drectons and after each measurement these equal but opposng veloctes wll be of dfferent magntude (absolute value of the velocty wll dffer at each measurement), the separaton between the two partcles wll nvarably be equal to a constant x Thus, although we cannot predct the outcome of the poston or momentum measurement at tme t we can wth certanty predct that after each measurement the two partcles wll always be found at the same separaton x Paradoxcally, snce the tme t s arbtrary, the separaton between the two partcles wll be one and the same, namely x, no matter at what pont of tme we make the experment The above may appear lke an nterestng observaton but t does not seem to lack physcal meanng Collapsed functon s not a functon of tme and one may argue that nothng prevents t from havng the above property We want now to explore ths observaton a lttle further We want now to see what the tme-propagaton of the collapsed functon (, ) δ ( ) δ ( ) f x x = x x x x x (5) s For ths reason we may wrte the delta-functons explctly ( x x ) p ( x x x ) p dp' e (, ) = dpe f x x ' = dp A dp 'B = dp dp 'AB (6) 3
To see the development n tme of above functon we apply the propagator H t e where the Hamltonan corresponds to a free partcle (e nteracton between partcles s zero, as requred by EPR), e H p = = m m x (7) Propagator ncludng the postons of the two ndvdual partcles, nstead of center-of-mass propagator yelds the same result Let us now expand the propagator ( ) = H t e H t + t (8) m x And now we are ready to apply the propagator (n ts expanded form) on the functon f( x, x) = dp dp'ab (notce the defnton of A and B above): It can easly be shown that t s equal to: (Detals) ( H t) dp dp 'AB (9) t( p p') m dp dp' e + AB Above expresson s the propagated functon f( x, x, t ) whch has the explct form 4
( x x p) p ( x x x p') p' dp' α + α dpe e () (Detals) Last ntegral, however, mmedately yelds the followng delta-functons and δ δ ( x x α p) From these delta-functons t follows that and from where we have () ( x x x α p ) + () ' x = x + α p (3) x = x + x p (4) α ' x + p = x + x p (5) α α ' and, respectvely, because p' = p, x = x + x (6) As s seen from the above defnton of α, namely ths α s the parameter that s tme-dependent t turns out that the tme-dependent parameter s elmnated when applyng the propagator Ths means that n tme the collapsed functon (at any moment of tme) wll be only coordnate- 5
dependent In other words, the collapsed functon (at any moment of tme) wll look exactly as the functon whch we were observng to collapse at a gven tme t that s, even before applyng the propagator Thus, t appears that n tme the dstance between the two partcles wll be always one and the same despte the fact that the two partcles move n opposte drectons One may argue that when collapsed state functon s propagated the condton p' = p may not be vald anymore Instead, a more general condton would apply, namely p' = np, where n= const In such a case we wll have and therefore or x + α p = x + x + α np (7) x x = x + α ( n ) p (8) x x = x + t( n ) p m (9) whch s x x = x t( n ) p m () Thus, t appears that as the tme progresses and the partcles move n opposte drectons from one another, the dstance between them gets shorter and shorter The state functon chosen by EPR to descrbe the combned two-partcle system s specally desgned by them to show that the Ψ -functons used n 6
QM do not descrbe completely the state of the system From the above argument t follows further that such specally desgned functon (and EPR have all the rght to desgn any functon they need to prove ther pont) appears to lead to conclusons that seem to even lack physcal meanng One possble way of lookng at t, n tryng to resolve ths probable problem, s to consder that after collapse the two partcles fall on a sphere where ther veloctes can be of opposte drectons whle stll mantanng the same dstance Further, n tme ths may be a shrnkng sphere whereby these partcles stll can mantan somehow ther opposte veloctes whle the dstance between them gets closer and closer Ths pcture, however, does not seem to be supported by expermental evdence One last pont despte the generally probablstc character of the outcome of measurement n QM, there appears to be an unsuspected determnstc outcome n ths partcular set up No matter how uncertan, n general, the outcome regardng the state of the system may be after the experment, there s somethng whch we know for sure the dstance between partcles wll reman the same as a result of experments of the type descrbed Ths seems to be an nterestng determnstc feature of QM whch probably s worth explorng further Acknowledgment The author wshes to acknowledge the frutful dscussons wth Prof Frank McLafferty of Long Island Unversty, NY References Ensten A, Podolsky B, Rosen N, Phys Rev, 47, 777-78 (935) 7
I As a helpful step, let us frst of all observe what the second dervatve of AB wll look lke: A B AB = B A + = x x x x Now we are ready to do the dervaton properly A A B B B+ + A x x x x ( ) dp dp ' ( p AB + pp ' AB + p ' AB = dp dp '( ( p + p ') AB) Then, the whole expresson wll be Now, once we have the whole expresson we can restore the exponent dp dp ' t ( p + p ') AB m t( p p') m dp dp ' e + AB 8
II Denote t = α() t = α m, then we have dp ( ') ' p + p AB dp e α and substtutng A and B we have ( ') px ( x ) p ' '( x x x dp dp e α + e e ) p p whch may also be wrtten as α p α p' α pp' px ( x ) p '( x x x ) dp dp' e e e e e or as αp αpp' p ( x x ) p '( x x x ) αp' dp ' dpe e e e e respectvely ' ( ) '( ) ' αp αpp + p x x p x x x αp dp ' dpe e Now, because whch s also p = p' (and therefore αp αpp' = αp + αp = αp ) we may wrte the above expresson as p x x + αp p'( x x x ) αp' dp ' dpe e p ' x p' x p' x p' px px +αp α dp ' dpe e = px px p dp ' dpe e α p' x p' x p' x + αp' 9