Quan um F ee Pa icle exp(-iE +ip do ) and S a e P obabili y Pa 4
F ancesco R. Rugge i Hanwell, N.B. No . 19, 2025
In Pa 3, we s a ed wi h he no ion o a ee pa icle p obabili y linked o ene gy and
momen um conse a ion in wo body New onian elas ic sca e ing. In pa icula , we a gued ha
one needed a p obabili y which shows an equal weigh o any ou come (ei,ej) and (pi,pj)
(momen um ec o ) se which conse es ene gy and momen um. This led o a i s guess o
exp(iC1 E) and exp(iC2 p), bu in Pa 3, we a gued he momen um and ene gy a e de ined
h ough a Lo en z ans o ma ion o mo a es and such a ans o ma ion mus also a ec x, →
x’, ’. We concluded one canno c ea e any p obabili y based on p and E wi hou also conside ing
x and which ans o m acco ding o he same ans o ma ion. This hen allows one o c ea e a
Lo en z in a ian p obabili y.
He e we ake an opposi e app oach, i.e. one which does no in ol e p and E a p io i. I is well
known om classical physics ha P(x)dx=dx/L, whe e L is an a bi a y leng h, desc ibes a
uni o m dis ibu ion o pa icles a es . One may, howe e , iew hese pa icles om wo
ames, one mo ing wi h cons an and he o he wi h cons an - . This would seemingly
in oduce p obabili ies P(x1’, 1’, ) and P(x2’, 2’,- ) wi h no no ion o ene gy o momen um
p esen . I would seem ha he p oduc o he wo o an x1’=x2’ and 1’= 2’ should equal he
es ame esul . This sugges s an exp( (x, , )) solu ion, bu he e is no eason o ha e an e e
inc easing o dec easing exponen ial and so exp(i (x, , )) seems like a solu ion. The e is,
howe e , a p oblem, because gi en P(x, , =0), and such a solu ion is hen no P(x)= 1/L unless
P(x, , )=1, which we ejec as i assumes a mo ing pa icle is iden ical o one a es in e ms o
p obabili y.. In o he wo ds, he e a e wo p obabili ies o he es ame, P(x)=1/L and P(x, , =0)
= exp(i (x, , =0).
Gi en ha exp(i (x, , ) is based on Lo en z ames, one would also expec p obabili y o be
conse ed and no be c ea ed o des oyed simply because a pa icle wi h es mass is iewed
om a es ame o one mo ing a cons an speed. This seems o o ce one o use ela i is ic
a iables linked o a Lo en z in a ian , namely E,p as well as x, . As a esul , in Pa 3 we
a gued ha one canno conside a p obabili y o p,E wi hou conside ing x, as well and he e we
sugges ha one canno e en conside a p obabili y o x which is conse ed in di e en cons an
mo ing ames wi hou in oducing ,p and E. Thus, P(x, , =0) which seemed like a “P(x)” ype o
p obabili y in he es ame is eally a di e en p obabili y, namely one in ime exp(i (x, =0, )),
i.e. . o his o be Lo en z in a ian , one has exp(-iE +ipx) = exp(-iE ) o any x. In a sense, his
is like P(x)=1/L in ha each x ca ies he same weigh , bu he e is s ill he ime a iable and he
p obabili y ac ually changes in ime unlike P(x)=1/L. This seems o lead o a pa adox because
he gi en exp(-imocc ) one may o mally a gue ha exp(-px)=1 o p=0 which is he x pa o he
p obabili y, ye abo e we a gued ha in o de o ob ain his alue one should use
P(x1’, 1’ )P(x2’, 2’,- ) a x1’=x2’ and 1’= 2’. The di e ence be ween P(x)=1/L is ha he ime
measu emen has in ini e esolu ion, while exp(-imocc ) implies ha i does no because
o he wise in o ma ion abou x is he same. Thus, special ela i is ic conside a ions lead o he
no ion o unce ain y in ime (hba /E) and space hba /p, bu in he New onian scena io, hese do
no exis .
We a gue ha he ques ion hen becomes: Why would one wish o change om a special
ela i is ically ela ed p obabili y exp(-iE +ipx) which shows physical unce ain y uni s o ime
and space, o one which shows in ini e esolu ion o x and , h ough exp(-iE +ipx)exp(iE +ipx)?
We sugges ha exp(-iE +ipx) o exp(ipx) in ime-independen p oblems is a dynamic p obabili y
needed o desc ibe in e ac ions. One may see ha cos(px) and sin(px) may ha e posi i e and
nega i e alues, implying he emo al o p obabili y om one place and i s addi ion somewhe e
else. One he in e ac ion esul has been ob ained, one wishes o measu e spa ial esul s,
i.e.coun pa icles in bins in space. The e is no need o he in e ac ion in o ma ion and i may be
o mally emo ed using exp(iE -ipx) exp(-iE +ipx).
Two Body Elas ic Sca e ing
In Pa 3 we s a ed wi h he no ion o New onian elas ic wo-body sca e ing and sough a
p obabili y in p,E such ha any (ei,ej) (pi,pj) (momen um ec o ) ou come should ha e he same
weigh . This au oma ically o ces one o conside E and p, bu no x and . We i s pos ula ed:
exp(iC1 E) and exp(iC2 p) whe e p lies along he x-axis ((1))
The main poin o Pa 2 was o a gue ha E and p a e de ined by iewing a pa icle a es om
a ame mo ing wi h cons an speed - . A he same, x and o he es ame mus be
ans o med in o x’, ’. We no ed ha one may ha e wo ames wi h and - and ha o
conse e p obabili y one mus conside no only p,E, bu also x, o c ea e a Lo en z in a ian .
O he wise, o p obabili ies exp(iC2 p) and exp(iC2 (-p)) one may mix ames and c ea e AND
si ua ions which do no co espond o p obabili ies in he es ame.
He e we wish o s a wi h no conside a ion o p and E, bu only o he classical p obabili y
equa ion:
P(x)dx = dx/L whe e L is an a bi a y leng h ((2))
P(x)dx=dx/L
P(x)dx=dx/L is a uni o m dis ibu ion o pa icles a es . One may, howe e , iew his om wo
mo ing ames - and (bo h cons an speeds) o ob ain:
P(x, , ) and P(x, , - ) ((3))
The e is no no ion o p in ((3)). We a gue ha one may always apply an AND si ua ion o
p obabili ies and so:
P(x, , ) P(x, - ) = P(x) ((4))
As a esul , mo ion seems o c ea e a kind o squa e oo p obabili y o P(x)dx =dx/L.
The e is, howe e , an immedia e pa adox in ((4)) because he e is no eason why one canno
conside :
P(x, , =0) which should also gi e he in o ma ion o P(x) ((5))
How can one ha e wo P(x, , ) ac as a squa e oo p obabili y in ((4)) and a ull p obabili y in
((5))? The only way i seems is o P(x, , ) =1, i.e. o ha e no eloci y dependence. This
sugges s ha p obabili y o mo ing pa icles is exac ly he same as ha o one a es , bu we
a gue ha his does no make sense om he poin o iew o special ela i i y.
Special Rela i is ic Conside a ions
We ejec he no ion o P(x, , ) =1 and sea ch o a p obabili y which is Lo en z in a ian and
also con ains dynamics, i.e in o ma ion abou . In special ela i i y is ob ained h ough E and
p which a e pa o a 4- ec o and so we p opose he Lo en z in a ian p obabili y:
exp(-iE +ipx) (o exp(-iE +i p do ) ((5))
One may no ice immedia ely ha his in oduces E and p based uni s in ime and space,
o e iding he in ini e esolu ion associa ed wi h New onian ule s and clocks:
d = hba /E and dx = hba /p ((6))
Thus, he e is an inhe en physicali y, namely ((6)), associa ed wi h p obabili y linked o mo ion.
One does no simply ha e 1, bu he modulus is 1. One may no e ha se ing p=0 in ((5)) yields:
exp(-iE ) = exp(-i mocc ) ((7))
As a esul , he x po ion o p obabili y may seem o be iden ical o P(x)=1/L because hba /p
goes o in ini e, bu he e is s ill he ime esolu ion. Thus, special ela i i y is consis en wi h a
dynamical p obabili y which is no ha same as i s modulus o 1 and in oduces new physics.
This new physics is p esumably linked wi h in e ac ions because ((5)) con ains p which is
physically linked wi h impulse.
This begs he ques ion: Why should one hen e en conside an equa ion like P(x)dx=dx/L i
he e is no in ini e esolu ion o x? We sugges ha he answe is linked o coa se g aining o
emo e he e ec s o in e ac ion. As an example, one may conside wo-sli in e e ence o e en
a bound s a e:
W(x)=Sum o e p a(p)exp(ipx) ((8))
One may no e ha W(x) is bo h posi i e and nega i e, bu coun ing pa icles in spa ial bins
canno yield a nega i e numbe . The nega i e numbe in ((8)) is ine o in e ac ion calcula ions
because i shows p obabili y being emo ed om one place and added o ano he . Fu he mo e,
one canno simply ake he absolu e alue o ((8)) because his is no a con inuous unc ion.
One equi es an app oach which emo es he in e ac ional ea u e associa ed wi h hba /p. To
ind his we e u n o ((4)), bu now use he Lo en z in a ian o m:
exp(iE -ipx) exp(-iE +ipx) = 1 ((9))
exp(-iE +ipx) shows he Lo en z in a ian p obabili y needed o p obabilis ic in e ac ion
calcula ions, bu his is a dynamic kind o p obabili y. The modulus ((9)) is eally a subse o he
in o ma ion which may be all ha is needed i one is simply coun ing pa icles in bins in space.
The e is no quan um in e ac ion assumed o be occu ing in his coun ing app oach and so he
coa se g ained app oach o ((9)), i.e. using he modulus ins ead o he en i e p obabili y is used.
Fo mally, his has he o m o mul iplying a p obabili y o mo ion o he igh wi h mo ion o he
le . Hence o a bound s a e, one conside s spa ial densi y o be:
P(x) = W*(x)W(x) ((10))
We no e (1) also conside s W(x) o be a p obabili y and de ines W*(x)W(x) a p io i wi hou
e e ence o special ela i i y. We sugges ha special ela i i y is key o he quan um ee
pa icle wa e unc ion.
Conclusion
In conclusion, in Pa 3 we s a ed wi h a p obabili y based on p and E which desc ibes equal
weigh s o any (ei,ej) (pi,pj) (momen um ec o ) ou come se in a New onian elas ic 2-body
sca e ing in e ac ion. This led o a guess o exp(iC1 E) and exp(iC2 p) as p obabili ies, bu we
a gued ha i is special ela i i y (Lo en z ans o m) which de ines E and p and his same
ans o m mus ac on x and . In o de o ha e conse ed p obabili y, we a gued in Pa 2 ha
one mus include x and in a P(p)P(E) exp ession. This led o exp(-iE +ipx), i.e. a Lo en z
in a ian o m.
He e we s a wi h P(x)dx=dx/L, he classical spa ial esul and no p o E appea s as his
applies o s a iona y pa icles. One may iew his se om a ame mo ing wi h cons an o -
and de ine P(x, , ) and P(x, ,- ). Fo mally, one would expec ha P(x, , )P((x, - ) = 1, i.e. he
P(x) esul , bu his is a pa adox because P(x, , =0) should also yield P(x). To esol e his
pa adox, we a gue ha i one conside s ames, hen one should use special ela i i y and
c ea e a Lo en z in a ian p obabili y. We ejec he idea o a p obabili y ha is 1 o a pa icle in
mo ion and see ha special ela i i y allows o he solu ion exp(-iE +ipx). This au oma ically
in oduces esolu ion in ime (hba /E) and space (hba /p) which is no ound in New onian
mechanics. New onian mechanics, howe e , is an app oxima ion. As a gued in Pa 2, he ull
alues o p and E ollow om special ela i i y and so we wish o conside Lo en z in a ian
p obabili ies. We no e ha hba /E and hba /p ha e consequences o in e ac ions and ha
exp(-iE +ipx) should be used when desc ibing in e ac ions. We also no e ha he modulus o
exp(-iE +ipx) is 1 and i one is no in e es ed in an in e ac ion, he modulus squa ed in o ma ion
is good enough. In o he wo ds, one may o mally emo e he in e ac ion in o ma ion by using
exp(iE -ipx)exp(-iE +ipx) lea ing only coa se-g ained in o ma ion, we a gue. This e en holds in
he es ame in which one has exp(-i mocc ).
Re e ences
1. An onakos, Cha alampos The Quan um Wa e unc ion as a Complex P obabili y
Dis ibu ion (2025)
h ps://a xi .o g/h ml/2502.10523 3
