Constrution of Exat Solutions for the
Stern-Gerlah Eet
J. DazBulnes and I.S. Oliveira
CentroBrasileirode PesquisasFsias
RuaDr. XavierSigaud150,Riode Janeiro-22290-180, Brazil
Reeivedon1February,2001
Weobtainexatsolutionsfor theShrodinger-Paulimatrixequationfor aneutralpartileofspin
1/2 in a magneti eld with a eld gradient. The analytial wavefuntions are written on the
symmetry plane Y = 0, whih ontains the inident and splitted beams, in terms of the Airy
funtions. Thetime-evolutionoftheprobabilitydensities, j
+ j
2
and j j 2
,andtheeigenenergies
are alulated. These inlude a small ontribution from the eld gradient, , proportional to
(~) 2=3
,whihamountsto equal energydisplaements onboth magnetilevels. Theresults are
generalizedforspinS=3=2,andinthisasewefoundthatthem=1=2andm=3=2magneti
sublevelsareunequalysplittedbytheeldgradient,beingthediereneinenergyoftheorder0.4
MHz. Replaing realexperimentalparameterswe obtained aspatialsplitting ofthe spinup and
spindownstatesoftheorderz4mm,inaordanetoarealStern-Gerlahexperiment.
I Introdution
TheexperimentofStern-Gerlah,performedintherst
quarterofthe20thentury,isintroduedinbasi
quan-tum mehanistextbooksin orderto illustrate the
ex-istene of thespinof apartile [1 6℄. Inspiteof its
historialimportane,anditswideuseasan
experimen-tal paradigmforthedisussion oftheoneptof
mea-surementinquantummehanis,authorshavefailedto
exhibit a full, oreven partiular solution of the
prob-lem,thatis, thepartilesanalytialwavefuntionsand
their timeevolution,as well astheir eigenenergies. A
fairlylargenumberofreentworkshavebeenpublished
onthe subjet,as that of Batelaanet al: [7℄who
pro-posed an experimental setup where harged partiles
ould be separatedby their spins in a\Stern-Gerlah
apparatus",ontraditing theideasof Bohr andPauli
atthebeginningoftheentury;NiChormaietal:[8℄,
employinginterferometrytehniquesinaStern-Gerlah
experiment,investigatedpropertiesofneutralpartiles
inabeam;Hannoutetal:[9℄ontributedtothetheory
ofmeasurementsinquantum mehanisusingtheidea
ofaStern-Gerlah apparatus. Insomeother workson
thesubjet[10 13℄theroleplayedbytheeldgradient
remainsundetermined.
Inthispaper,weareonernedwithabeamof
neu-tral atoms of spin S = 1=2 and mass m whih
pene-tratestheeld regionalong thex-axis. Themagneti
eld insidethe magnetregionanbeapproximatedas
:
B(y;z)= yj+(B
0
+z)k (1)
where B
0
representsahomogeneousomponentof the
eld, and (>0) the eld gradient along the z
dire-tion. One notiesthat this eld satises theequation
rB =0,and alsothat itanbederivedfrom a
po-tentialvetorA= y(z+B
0
=2)i+(B
0 x=2)j.
From the eld given in Eq.(1), one an write the
hamiltonian:
^
H=I ^
P 2
2m +
B ^
B(y;z) (2)
andthereforetheShrodinger-Paulimatrixequation:
^
H = I ~
2
2m r
2
+
B
[ y
2 +(B
0 +z)
3 ℄ =i~
t
(3)
whereIisthe22identitymatrixand
1 ;
2 and
3 the
Paulimatries. = (x;y;z;t)is thetwo-omponent
spinor:
=
+
(4)
Inwhatfollowswewill usethespinoromponents,
+
and ,denedthrough:
+ =
+
0
; =
0
(5)
Beforeproeeding,itisinstrutivetoonsidera
sim-plealulationoftheexpetedvalueforthepositionof
the partile on the z-axisin a instantof time t using
< ^
Z>
(t)=
Z
y
(t)
^
Z
(t)d
3
r (6)
SinethehamiltonianinEq.(2)isindependentoft,one
anwrite:
(t)=e
i ^
Ht=~
(0) (7)
andapplytheBaker-Hausdoridentity[5℄:
e ^
O
^
Ae ^
O
= ^
A+[ ^
O; ^
A℄+ 1
2! [
^
O;[ ^
O; ^
A℄℄+ 1
3! [
^
O;[ ^
O;[ ^
O; ^
A ℄℄℄+ (8)
Inthepresentase:
^
A= ^
Z and ^
O= i
~ (
^
P
x 2
2m +
^
P
y 2
2m 2
B
^
Y ^
S
y +
^
P
z 2
2m +2
B B
0 ^
S
z +2
B
^
Z ^
S
z )t
Fromtheseexpressionsoneobtains:
[ ^
O; ^
A℄ = ^
P
z
m t
[ ^
O;[ ^
O; ^
A℄℄ = 2
B t
2
m ^
S
z
[ ^
O;[ ^
O;[ ^
O; ^
A℄℄℄ = (2
B )
2
t 3
m~ ^
Y ^
S
x
[ ^
O;[ ^
O;[ ^
O;[ ^
O; ^
A℄℄℄℄= (2
B )
3
t 4
m~ 2
^
Y 2
^
S
z +
(2
B )
3
2
B
0 t
4
m~ 2
^
Y ^
S
z +
+ (2
B )
3
t 4
m~ 2
^
Z ^
Y ^
S
y (2
B )
2
t 4
m 2
~ ^
P
y ^
S
x ;et:
Consideringthe\redued"hamiltonian, ^
H
r
,atingonthewavefuntionsontheplaneY =0:
^
H
r =
^
P
x 2
2m +
^
P
z 2
2m +2
B B
0 ^
S
z +2
B
^
Z ^
S
z
theproblemisgreatlysimpliedforallthetermsoforderhigherthan2vanish. Wewillallthese funtions:
1
(x;z;t)
+
(x;0;z;t) and
2
(x;z;t) (x;0;z;t)
from whih oneanalulate theexpeted valueofZ. Weobtainthenalresult:
< ^
Z >
1;2 (t)=<
^
Z>
1;2
(0)+<v^
z >
1;2 (0)t
B t
2
m <
^
S
z >
1;2
(0) (9)
d
thisresultisinaordanewiththetheoremof
Ehren-fest[5℄. Thus,ifthebeamhasbeenpreviouslypolarized
alongthez-axis,byenteringtheStern-Gerlah
appara-tusitwillbesubjettoaforeequalto+
B
for
spin-down partiles (< ^
S
z
> (0) = 1=2), and
B for
spin-up partiles (< ^
S
z
>(0)=+1=2). Consequently,
thebeamissplittedbytheeldgradient,andthe
parti-lesareseparatedbytheirspindiretion onthez-axis.
Inthenextsetioneq.(3)issolvedontheplaneY =0,
andtheexatwavefuntionsareobtained. InsetionIII
thestationarywavefuntionaredeterminedalongwith
theorrespondingeigenenergies. The ase S = 3=2is
disussedinSe.IVandtheenergyandspatialsplittings
II Exat Solutions on Y = 0
On this setion we will derive exat solutions for the
Stern-Gerlah eet on a symmetry plane. We will
onsider a magneti eld with eld gradient dierent
from zero for x > 0 (diretion of the inident
parti-le) and equal to zero for x 0. Thetime-evolution
ofthepartileswavefuntionsrossingaStern-Gerlah
apparatusanbeobtainedanalytiallyonthe
symme-try plane Y = 0, on whih the two equations in (3)
beomedeoupled:
~ 2
2m
2
1
x 2
+
2
1
z 2
+
B (B
0
+z)
1 =i~
1
t
(10)
~ 2
2m
2
2
x 2
+
2
2
z 2
B (B
0
+z)
2 =i~
2
t
(11)
d
Onthebasisoftheresultsofthepreeedingsetion,one
anexpetthatthesolutionsoftheaboveequationswill
ontain afuntion dependent on thevaribles z and t,
representing the separation of the beam along the
z-axis. Besides, Berry [14℄ showed that the solutionsof
theShrodingerequationofafree-partileanbe
writ-tenasproduts of AiryAi funtions byomplex
expo-nentialfuntionswhosesquaremodulusevolveswithout
deformation. Thefat that Eqs.(10)and (11)ontain
the oordinate z suggeststhat the same type of Airy
funtionsanbefoundhere.
We therefore propose as possible solutions of (10)
and(11)thefollowingmultiparameterfuntions
1 and
2 :
1
(x;z;t)=F[a(z+bt 2
)℄e itz
e (i=~)(p
x x ~!
+ t)
(12)
and
2
(x;z;t)=G[ ~a( z+ ~
bt 2
)℄e i~tz
e
(i=~)(pxx ~! t)
(13)
Wherep
x
iseigenvalueof ^
P
x
,whihisinturnonserved.
Replaing (13)and (12)in(11)and (10),respetively,
one nds that the funtions F and Gsatisfy theAiry
equation[15℄onlyiftheparametersassumethe
follow-ingvalues:
a=
2m(
B
2mb)
~ 2
1=3
; ~a= "
2m(
B
2m
~
b)
~ 2
#
1=3
(14)
~!
+ =
p
x 2
2m +
B B
0
; ~! = p
x 2
2m
B B
0
(15)
= 2mb
~
; ~= 2m
~
b
~
(16)
b= ~
b=
B
4m
(17)
One noties that with this set of parameters, a and ~a are positive, a neessaryondition for j
1 (z;t)j
2
and
j
2 (z;t)j
2
to represent funtions whih moveappart with time. Expliitly, thenon-normalized solutionsof (10)
and(11)are:
1
(x;z;t)=Ai h
(
B m
~ 2
) 1=3
(z+(
B
4m )t
2
) i
e
i( B=2~)zt
e (i=~)[p
x x ((p
x 2
=2m)+
B B
0 )t℄
(18)
2
(x;z;t)=Ai h
(
B m
2 )
1=3
( z+(
B
)t 2
) i
e i(
B =2~)zt
e
(i=~)[pxx ((px 2
=2m) BB0)t℄
(19)
Onenotiesthatthesesolutionsareorretlyreduedtofreepartileplanewavesintheregionoutsideofeld,that
is, x0. Thetime-evolutionoftheprobabilitydensityisproportionaltothesquaredmodulusof
1 and 2 : j 1 (z;t)j
2 = Ai h ( B m ~ 2 ) 1=3
(z+( B 4m )t 2 ) i 2 (20) j 2 (z;t)j
2 = Ai h ( B m ~ 2 ) 1=3
( z+( B 4m )t 2 ) i 2 (21)
Thesefuntions,however,stilldonotrepresentphysiallyorretsolutionstotheproblem. Firstonenotesthat
theAiryAifuntionsosillatealongthez-axiswithoutdeayingforlargevaluesofz. Togetridoftheseosillations
andobtainaproperwavefuntion,wesimplymultiplythesolutions(18)and(19)byHeaviside'sstepfuntions,,
whih\trunate" theosillationsattherstzero(
o
). Seondly, weintrodueadisplaement
o
= 1:0188(rst
maximumofAi) inordertomakethetwofuntions oinideat t=0. With this,thenalorret time-dependent
wavefuntionsbeome:
1
(x;z;t)=Ai
a(z+(
o
=a)+bt 2 ) z ( o
=a)+(
o
=a)+bt 2 e i(zt) e (i=~)[pxx ~w+t℄ (22) and 2
(x;z;t)=Ai
a( z+(
o
=a)+bt 2 ) z ( o
=a)+(
o =a)+t
2 e i(zt) e (i=~)[p x x ~w t℄
(23)
Itmustbeemphesizedthatthefuntions(22)and(23)arealsosolutions ofthe samesetof equations(10)and
(11),with thesame setofparameters(14),(16)and(17),but ~w
+
and~w , whiharenowgivenby:
~! + = p x 2 2m + B B 0 o 2 ( B 2 2 ~ 2 m ) 1=3 (24) ~! = p x 2 2m B B 0 o 2 ( B 2 2 ~ 2 m ) 1=3 (25)
Thedomainsof
1 and
2 are<
3
f(x;z;t)=z=(
o =a) (
o =a) bt
2
gand< 3
f(x;z;t)=z= (
o =a)+( o =a)+ bt 2
g,respetively. Thespin-upand spin-downprobabilitydensitiesbeometherefore:
j'
1 (z;t)j
2 =fAi ( B m ~ 2 ) 1=3 z+ o + 1 4 ( B 2 2 m~ ) 2=3 t 2 h z o ( B m ~ 2 ) 1=3 + o ( B m ~ 2 ) 1=3 +( B 4m )t 2 i g 2 (26) and j' 2 (z;t)j
2 =fAi ( B m ~ 2 ) 1=3 z+ o + 1 4 ( B 2 2 m~ ) 2=3 t 2 h z o ( B m ~ 2 ) 1=3 + o ( B m ~ 2 ) 1=3 +( B 4m )t 2 i g 2 (27) d
Inorderto produeraplotofthefuntions we
de-ne anunit oflength,,as follow:
2
= Z
y
(x;z;t)(X 2
+Z 2
) (x;z;t)dxdz
Intermsof,otherphysialquantitiesbeome:
x; z in units of \"
t in units of \(m 2
)=~"
in units of \~ 2 =(m 2 B 2 )"
=t in units of \~=(m 2 )" 2 =x 2
in units of \1= 2
"
remaining arbitrary. Fig. 1shows j'
1 (z;t)j
2
and
j'
2 (z;t)j
2
for=800. Itis apparentthatthe
\spin-up"stateseparate from the\spin-down"asthe
Figure.1Timeevolutionofthespin-upandspin-down
prob-ability densitiesas partilesythroughthe magnetregion
for (a) t = 0:0001, (b) t = 0:03 and () t = 0:05 and
~=1;m=1;B =1and=800.
III Eigenenergies
Theeigenenergiesof thesystemaregivenbythe
solu-tionsofthestationaryequation:
^
H =E (28)
where ^
H is given by (2). On the plane Y = 0 one
obtains:
1
2m (
^
P 2
x +
^
P 2
z )+
B (B
0 +z)
1 =E
1
(29)
1
2m (
^
P 2
x +
^
P 2
z
)
B (B
0 +z)
2 =E
2
(30)
Replaingsolutionsofthetype:
1
(x;z)=e (i=~)pxx
R (z) (31)
oneobtainsthefollowingequationforthefuntion R :
R 00
(z
1
)R=0 (32)
where:
= 2m
B
~ 2
(33)
and
1 =
2m
~ 2
[E p
x 2
2m
B B
0
℄ (34)
Deningthenewvarible through:
= 1=3
z
1
2=3
(35)
wendthenewequation:
R 00
() R()=0 (36)
wih is the Airy equation. Thesolution of Eq.(29) is
therefore:
1
(x;z)=Ai( 1=3
z+
o )
z
o
1=3
+
o
e (i=~)p
x x
(37)
Wherethestepfuntionhasbeenintrodued,asdisussedinthepreeedingsetion. Theeigenenergyassoiated
tothisfuntions is:
E
1 =
p
x 2
2m +
B B
o
o (
B 2
2
~ 2
2m )
1=3
(38)
Similarly,areobtainfor
2 :
2
(x;z)=Ai( 1=3
z+
o )
z
o
1=3
+
o
e (i=~)p
x x
d
witheigenenergy:
E
2 =
p
x 2
2m
B B
o
o (
B 2
2
~ 2
2m )
1=3
(40)
Weseethattheeetoftheeldgradientistoprodue
a small positive displaement on the magneti levels,
and onsequently, theenergy dierene E
1 E
2 does
notdependon.
IV Solutions for S = 3=2
Onthissetionwegeneralizetheresultsofthe
preeed-ingsetion to the ase S =3=2. The hamiltonian for
aneutralpartilewithzeroorbitalangularmomentum
andarbitraryspin ^
S inamagnetieldBis:
^
H= ^
P 2
2m I+
2
B
~ ^
S:B (41)
The orresponding Shrodinger-Paulimatrix
equa-tionbeomes:
^
H = I ~
2
2m r
2
+
B
[ y
2 +(B
0
+z)
3 ℄ =i~
t
(42)
where
i
,with i=1;2;3,arethespinmatries. ForS=3=2,overY =0wehavefourdeoupledequations:
~ 2
2m
2
3
x 2
+
2
3
z 2
+3
B (B
0
+z)
3 =i~
3
t
(43)
~ 2
2m
2
2
x 2
+
2
2
z 2
+
B (B
0
+z)
2 =i~
2
t
(44)
~ 2
2m
2
1
x 2
+
2
1
z 2
B (B
0
+z)
1 =i~
1
t
(45)
~ 2
2m
2
0
x 2
+
2
0
z 2
3
B (B
0
+z)
0 =i~
0
t
(46)
where
1
,et., aretheomponentsofthefour-dimensionalspinor.
d
Following the same proeedure as in the previous
setionweobtainthefollowingenergies:
E
3 =
p
x 2
2m +3
B B
o
o (
9
B 2
2
~ 2
2m )
1=3
(47)
E
2 =
p
x 2
2m +
B B
o
o (
B 2
2
~ 2
2m )
1=3
(48)
E
1 =
p
x 2
2m
B B
o
o (
B 2
2
~ 2
2m )
1=3
(49)
E
0 =
p
x 2
2m 3
B B
o
o (
9
B 2
2
~ 2
2m )
1=3
(50)
Themaindiereneinrespettothepreviousase,
is theunequaldisplaementsofthemagnetilevelsfor
m= 1=2 and m= 3=2. This situation anbe
ex-ploitedtomeasuretheeetsoftheeldgradientona
beam,asshownbelow.
IV.1Energy Splitting
Inthissetionweevaluatethemagnetiofthe
on-tribution of the eld gradient to the energy splitting.
Let us assume the following values for the physial
quantities involved:
v=600 m=s
m=1:810 25
Kg
B
=9:2740810 24
J=T
h=6:6260710 34
J:s
B
o =1 T
=10 3
Withthesevalueswendthefollowingvalueforthe
levelsE
0 andE
3 :
( 9
B 2
2
~ 2
2m )
1=3
=0:141110 8
eV
andforE
1 eE
2 :
(
B 2
2
~ 2
2m )
1=3
=0:183210 8
eV
Itis interesting to express thedierene in energy
betweentheselevelsinunits offreueny:
=(E
1 E
o
)=h0;4 MHz
In a real experimental situation we an superimpose
tothestatields, aradiofrequenyeldtuned to
fre-quenies at this range and hange the populations of
themagnetilevels.
Otherordersofmagnitudesare:
p 2
x
2m
=0:2022 eV
B B
o
=0:578810 4
eV
V Spatial splitting
FortheaseS =1=2oneobtainthe aelerationv_ by
makingtheargumentoftheAiryfuntion,eq.(22),
on-stant andderivating in respetto time. Fromthis we
have:
_
v= 2b= (
B =2m)
CallinglthelengthofthemagnetandLthelength
betweenthemagnet andthedetetor, thetotal
devia-tionofthepartilealongthez axisisgivenby:
z=(
B
4m )
l 2
v 2
x +(
B
2m )
lL
v 2
x
Replaingthe valuesoftheonstantsgivenaboveand
makingl=L=0:2m,wendz4mm,whihisthe
orretorderofmagnitudeforthesplittingobservedin
areal Stern-Gerlahexperiment.
VI Disussions and Conlusions
Onthis paperweobtainedanalytialsolutionsforthe
problem of a neutral partile with spin S in a stati
magnetieldwitheldgradient. Thesesolutionswere
builtfrom oftheAiry funtions,whih arein turn
so-lutions of the Shrodinger equation in the symmetry
plane Y = 0. By hoosing adequately the
parame-ters involved,weobtainedadisrete energyspetrum,
andspinupandspindownwavefuntionswhihtravels
apartwithtime,asinarealStern-Gerlahexperiment.
Noapproximationhasbeenmadeonthemagnitudeof
theeldgradient,. Thisontrastswiththeusual
pro-eedure (and physially inorret) of making B
0
[6;16℄. Results were obtained for the ases S = 1=2
and S = 3=2. On this last ase we showed that the
magneti energy levelsare unequally displaed by the
eld gradient, and that the splitting is on the KHz
-MHzfrequenyrange. Thealulatedspatialsplitting
isin aordanewith whatisobservedin areal
Stern-Gerla! h experiment.
As a nal remark we mention that Eqs (10) and
(11)preditanotherinterestingfeature,thefatthatin
arealStern-Gerlahexperimenttheseparationofspin
up and spindown partilesis not omplete, asstated
in [17℄. We found further soltions of those equations
whihshowanadmixtureofspinstatewavestravelling
in opositediretion[18℄.
Aknowledgments
Oneof us, JDB, is thankful to Prof. H.G. Valqui
(UNI,Lima)forusefulsugestionsonerningsome
top-isofthiswork.
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