Join Phase Space (x-p) P obabili y and F ee Pa icle Quan um Mechanics Pa 2
F ancesco R. Rugge i Hanwell, N.B. No . 22, 2025
In Pa 1, we a gued ha P(x)dx=dx/L (whe e L is an a bi a y leng h) applies o a pa icle a
es . I one iews such a pa icle om a ame mo ing a cons an - , i is mo ing. We a gued
ha in such a case, he ele an p obabili y is no simply linked o he posi ion x, bu also o he
momen um p a x because his deli e s an impulse hi . In pa icula , p obabili y is p esumably o
be used in a physical p oblem wi h in e ac ions and p obabili y and a dynamical pa icle equi es
a dynamical p obabili y desc ibing bo h x, and E,p. This led o exp(-iE +px) which ac ually
p edic s in e als in space hba /p and ime hba /E. In o he wo ds, a p obabilis ic iew poin
leads o hese esul s.
He e, we ask he ques ion: Wha i one simply wishes o use de e minis ic physics o desc ibe
a mo ing pa icle? A e all, New onian mechanics is de e minis ic and one does no eally need
o ocus on P(x)=1/L o i s modi ica ion o a mo ing pa icle. In such a case, a pa icle mo ing
wi h cons an (s a ing a x=0, =0) is desc ibed by x/ = bo h in New onian mechanics and
special ela i i y. The e is no unce ain y in x and he e which begs he ques ion: Why does i
appea in he p obabilis ic o malism o Pa 1? In pa icula , exp(-iE +ipx) de i ed in Pa 1
implies physical egions: dx = hba /p and d =hba /E and hese seem o be absen in x/ = and
his is a p oblem.
We sugges ha a p obabilis ic app oach is bu ied in he Lag angian o malism which leads o
he esul d/d = 0, i.e desc ibes he pa icle’s mo ion in space. This mo ion is ac ually x/ = , bu
his equa ion only p esen s a subse o he desc ip ion o a pa icle ha is mo ing wi h cons an
speed, e en in a de e minis ic Lag angian/Ac ion app oach. We ha e al eady poin ed his ou in
(1). We b ie ly explain i again he e, bu he eal ques ion is why does one need o conside he
Lag angian in o de o desc ibe x/ = ? In o he wo ds, he p obabilis ic app oach associa ed wi h
=x/ only eme ges in he Lag angian o malism using a ela i is ic ac ion L = -E +px, wi h x/ = .
Igno ing he ac ion and Lag angian leads o an absence o hba /p and hba /E and one is only
conce ned wi h x/ = . We no e ha he Lag angian is p ima ily linked o he esul : p = dL/d
pa ial and so a gue ha one is no simply in e es ed in mo ion (x/ = ), bu also in in e ac ion
because p is associa ed wi h an impulse hi . Thus, he Lag angian app oach is mo e han simply
a mechanism which desc ibes x, mo ion, i is associa ed wi h p and E ( h ough Hamil onian = E
= p -L.)
Using he Lag angian app oach sugges s ha i does no su ice o conside x and by
hemsel es when p and E a e p esen . One mus conside all ou a iables x, ,E,p which is he
same poin made in Pa 1. The physics o he p oblem dic a es ha i is no enough o simply
hink in e ms o x, , . The Lag angian app oach, which is cons uc ed o ul ima ely yield x/ = ,
when w i en in e ms o E, p, i.e. L = -E +px demons a es unce ain y in x, hba /p and , hba /E
as shown in (1). This unce ain y mus hen be w i en in e ms o a p obabili y which is Lo en z
in a ian and leads o exp(-iE +ipx). The poin is ha one would no e en conside a p obabili y i
one did no conside E,p and x, oge he as a comple e ea men o a mo ing pa icle.
In o he wo ds, x/ = is an incomple e desc ip ion o he s a e o a pa icle mo ing wi h
cons an speed, e en in a de e minis ic amewo k.. A i s his migh seem su p ising, bu i is
well known ha ene gy and p a e linked o in e ac ions. In special ela i i y, one does no
conside x, , alone and hen compu e p and E om hese, bu conside s ou ec o s and
Lo en z in a ian equa ions. I is possible o c ea e a Lo en z in a ian equa ion in ol ing all ou
s a e a iables x, ,E,p and i u ns ou ha his is he ela i is ic Lag angian which desc ibes
mo ion, so his app oach is mo e gene al han simply s a ing x/ = .
Thus, del a x = hba /p and del a = hba /E appea in bo h he p obabilis ic app oach o Pa 1
and he de e minis ic one o he Lag angian used he e.
A P io i P obabili y App oach o Pa 1
In Pa 1, we s a ed wi h an a p io i p obabilis ic desc ip ion o a pa icle a es somewhe e in
a leng h L, i.e.
P(x)dx = dx/L ((1))
Al hough ((1)) is well-known, one migh a gue ha in de e minis ic physics one would simply
gi e he x posi ion o he pa icle which holds o all . The e would be no need o a p obabili y
equa ion ((1)).
In Pa 1, ou en i e a gumen was based on inding a p obabili y which applied o a
desc ip ion o he pa icle in ((1)) when iewed om a mo ing ame. In o he wo ds, one
p obabilis ic app oach led o ano he . We a gued ha o a mo ing pa icle, posi ion is no
su icien because he pa icle in e ac s by deli e ing an impulse hi . A comple e physical
desc ip ion would be: Wha is he p obabili y o ind he pa icle a x and ha e i deli e an
impulse hi o p? This sugges s a link be ween x,p, E, ins ead o x, , simply desc ibing he
whole pic u e. The ela ion is gi en by he Lo en z in a ian ;
In a ian = -E + px ((2))
We hen a gued o a Lo en z in a ian p obabili y exp(-iE +ipx) whose modulus p ese ed
P(x)=1/L and P( )=1/T.
As no ed abo e, his analysis is based on an a p io i p obabili y scheme ((1)). One migh a gue
ha he e is no need o ((1)) i one uses de e minis ic physics. In o he wo ds, x/ = does no
seem o imply a p obabili y exp(-iE +ipx) a all. Mo eo e , exp(-iE +ipx) implies physical in e als
hba /p =del a x and hba /E= del a and his do no seem o be a all p esen in he equa ion
x/ = .
Lag angian Fo malism
The Lag angian o malism is an app oach which allows one o ob ain New on’s second
equa ion:
dp/d = Fo ce which ul ima ely becomes a di e en ial equa ion in x wi h d/d ’s and allows one o
ind x( ) ((3))
Speci ically: d/d dL/d pa ial - dL/dx = 0 ((4)) wi h L=T-V usually wi h T = -mo sq (1- /cc)
in he ela i is ic case o .5mo in he non ela i is ic case.
We poin ou , howe e , ha he e seems o be some hing mo e in ol ed in a Lag angian
scheme because:
dL/d pa ial = p = momen um ((5))
((5)) is an equa ion in ol ing momen um which deli e s an impulse hi . Fu he mo e, New on’s
second law dp/d = Fo ce is all abou o ce as well. Thus, one begins wi h a o mula ega ding
o ce and a i es a one which simply desc ibes x( ). This sugges s ha he esul o he o ce on
he pa icle is a ajec o y based on x( ).
As a esul , a pa icle which mo es wi h a cons an speed should be desc ibed by:
x/ = ((6))
The e is no p obabili y in ((6)) and no hba /p and hba /E, he e is no appea ance o p and E.
I is known, howe e , ha : p = dL/d pa ial ((7a)) and E=Hamil onian = p - L ((7b))
wi h L= -mo sq (1- /cc) ((8))
Thus, he Lag angian and Hamil onian a e in ima ely linked wi h p and E. This leads one o a
conside a ion o special ela i i y in which one has:
Lo en z in a ian : -E +px ((9))
Fu he mo e, using x/ = (x=0, =0 ini ial poin s)
L = -mo sq (1- ) = -E +px (wi h c=1, x/ = ) ((10))
L is called he ac ion, and i is he a ia ion o he ac ion which yields he di e en ial equa ion in
x and d/d ’s which ul ima ely gi es x( ).
I seems one canno sepa a e E,p om x, in L , he ac ion e en hough he ul ima e solu ion o
he Lag ange equa ion d/d dL/d pa ial - dL/dx = 0 yields x/ = which is a simple solu ion. We
sugges ha his di e en ial equa ion “loses” some in o ma ion o he p oblem con ained in L , as
i ollows om L and L i sel con ains in o ma ion abou p and E h ough ((7a)) ((7b)). We a gue
ha L is mo e gene al han he di e en ial equa ion con aining L and he esul x/ = . In o he
wo ds, x/ = is a subse o he in o ma ion desc ibing he mo ing pa icle. A ull desc ip ion mus
accoun o E,p,x, wi h x/ = and his desc ip ion is gi en by -E +px= L . This Lo en z in a ian
o m which includes p,E,x, o a mo ing pa icle seems o ep esen a ull desc ip ion, we a gue.
This idea is simila o he one we made in Pa 1 in which we a gued ha i is no su icien o
alk abou and x o a mo ing pa icle, one mus also include E and p because hese exis and
a ec he in e ac ions o he pa icle. The e is no poin no ing ha a pa icle is a x om an
in e ac ion poin o iew unless one knows p, i.e. he impulse hi one is o ecei e.
I one examines he speci ics as done in (1), one sees ha o :
L = -E +px one has del a x = cons an /p and del a = cons an /E ((11))
E en hough he e is a de e minis ic ajec o y x/ = , he e is also unce ain y in he de e minis ic
ac ion/Lag angian o malism (which was supposed o be en i ely de e minis ic).
To desc ibe his p obabili y, one mus make use o ((11)) oge he wi h he no ion o no special
weigh s o x and and his yields:
exp(-iE +ipx) ((12))
As he in e als a e p esen and he magni ude shows no p e e en ial weigh o x and . Thus,
he esul ((12)) which eme ged om p obabilis ic a gumen s in Pa 1, ac ually eme ge om
de e minis ic conside a ions o L (ac ion) because i is no enough o conside only x( ) which
ollows o m d/d dL/d pa ial - dL/dx = 0 which seems o con ain a subse o he in o ma ion
p esen in L . Thus, L seems o con ain mo e in o ma ion han x( ) and his in o ma ion appea s
o be physically ele an . Fu he mo e, i gi es ise o a p obabili y which is no p esen in he
de e minis ic solu ion x/ = , sugges ing ha he e a e p oblems in na u e which equi e such a
p obabili y, such as 1-dimensional e lec ion- e ac ion om an n1-n2 index o e ac ion junc ion.
Conclusion
In Pa 1, we a gued o he exis ence o a p obabili y exp(-iE +ipx) o a mo ing pa icle based
solely on p obabili y a gumen s. We s a ed wi h a pa icle a es in a egion L and asked: Wha
p obabili y pe ains o he pa icle when seen om a ame mo ing wi h cons an - ? We a gued
ha om a physical poin o iew, i is no su icien o know ha he pa icle is a x, one needs o
know p as well because his is ela ed o he impulse hi one ecei es (and a simila a gumen
o E, ). In o he wo ds, one has a dynamical p obabili y which may be used in dynamical
p oblems which exhibi p obabili y.
He e we no e ha adi ionally one desc ibes a de e minis ic pa icle wi h cons an speed by
x/ = . The e hen is no p obabili y and hence no exp(-iE +ipx) and no hba /p, hba /E. This begs
he ques ion: Why a e he e wo schemes? Pa 1 is based on p obabilis ic no ions and
p esumably sugges s physical in e als del a x = hba /p and del a = hba /E, bu whe e a e
hese in x/ = ? I hey a e physical, hey should be p esen , bu do no appea .
We a gue he e ha his seems o sugges ha x/ = is a subse o he in o ma ion needed o
ully desc ibe a pa icle mo ing a cons an speed, e en in a de e minis ic app oach. We
sugges ha L, he Lag angian, yields p = dL/d pa ial and Hamil onian = E = p -L and so
s a ing wi h L and L ( he ac ion) one has much mo e in o ma ion han wha one ob ains om
d/d dL/d pa ial - dL/d =0. which ul ima ely yields x/ = . One mus conside he ull L, L
in o ma ion and as shown in (1), L = -E +px o x/ = in he ela i is ic (and non ela i is ic
cases). This means ha e en in he de e minis ic Lag angian/Ac ion app oach one has he
unce ain y o p obabili y hba /p = del a x and hba /E = del a . Thus, he de e minis ic app oach
also poin s o hese physical unce ain y egions and exp(-iE +ipx).
Re e ences
1. Rugge i, F ancesco R. Classical F ee Pa icle Lag angian, Special Rela i i y, Quan um
Mechanics (p ep in , zenodo, 2022)
