Construction of Exact Solutions for the
Stern-Gerlach Eect
J. DazBulnes and I.S. Oliveira
CentroBrasileirode PesquisasFsicas
RuaDr. XavierSigaud150,Riode Janeiro-22290-180, Brazil
Receivedon1February,2001
Weobtainexactsolutionsfor theSchrodinger-Paulimatrixequationfor aneutralparticleofspin
1/2 ina magnetic eld with a eld gradient. The analytical wavefunctions are written on the
symmetry plane Y = 0, which contains the incident and splitted beams, in terms of the Airy
functions. Thetime-evolutionoftheprobabilitydensities, j
+ j
2
andj j 2
,andtheeigenenergies
are calculated. These include a small contribution from the eld gradient, , proportional to
(~) 2=3
,whichamountsto equal energydisplacements onboth magneticlevels. Theresults are
generalizedforspinS=3=2,andinthiscasewefoundthatthem=1=2andm=3=2magnetic
sublevelsareunequalysplittedbytheeldgradient,beingthedierenceinenergyoftheorder0.4
MHz. Replacing realexperimentalparameterswe obtained aspatialsplitting ofthe spinup and
spindownstatesoftheorderz4mm,inaccordancetoarealStern-Gerlachexperiment.
I Introduction
TheexperimentofStern-Gerlach,performedintherst
quarterofthe20thcentury,isintroducedinbasic
quan-tum mechanicstextbooksin orderto illustrate the
ex-istence ofthe spinof aparticle [1 6]. Inspiteof its
historicalimportance,anditswideuseasan
experimen-tal paradigmforthediscussion oftheconceptof
mea-surementinquantummechanics,authorshavefailedto
exhibit a full, oreven particular solution of the
prob-lem,thatis, theparticlesanalyticalwavefunctionsand
their timeevolution,as well astheir eigenenergies. A
fairlylargenumberofrecentworkshavebeenpublished
onthe subject,as that of Batelaanet al: [7]who
pro-posed an experimental setup where charged particles
could be separatedby their spins in a\Stern-Gerlach
apparatus",contradicting theideasof Bohr andPauli
atthebeginningofthecentury;NicChormaicetal:[8],
employinginterferometrytechniquesinaStern-Gerlach
experiment,investigatedpropertiesofneutralparticles
inabeam;Hannoutetal:[9]contributedtothetheory
ofmeasurementsinquantum mechanicsusingtheidea
ofaStern-Gerlach apparatus. Insomeother workson
thesubject[10 13]theroleplayedbytheeldgradient
remainsundetermined.
Inthispaper,weareconcernedwithabeamof
neu-tral atoms of spin S = 1=2 and mass m which
pene-tratestheeld regionalong thex-axis. Themagnetic
eld insidethe magnetregioncan beapproximatedas
:
B(y;z)= yj+(B
0
+z)k (1)
where B
0
representsahomogeneouscomponentof the
eld, and (>0) the eld gradient along the z
direc-tion. One noticesthat this eld satises theequation
rB =0,and alsothat itcanbederivedfrom a
po-tentialvectorA= y(z+B
0
=2)i+(B
0 x=2)j.
From the eld given in Eq.(1), one can write the
hamiltonian:
^
H=I ^
P 2
2m +
B ^
B(y;z) (2)
andthereforetheSchrodinger-Paulimatrixequation:
^
H = I ~
2
2m r
2
+
B
[ y
2 +(B
0 +z)
3 ] =i~
@
@t
(3)
whereIisthe22identitymatrixand
1 ;
2 and
3 the
Paulimatrices. = (x;y;z;t)is thetwo-component
spinor:
=
+
(4)
Inwhatfollowswewill usethespinorcomponents,
+
and ,denedthrough:
+ =
+
0
; =
0
(5)
Beforeproceeding,itisinstructivetoconsidera
sim-plecalculationoftheexpectedvalueforthepositionof
the particle on the z-axisin a instantof time t using
< ^
Z>
(t)=
Z
y
(t)
^
Z
(t)d
3
r (6)
SincethehamiltonianinEq.(2)isindependentoft,one
canwrite:
(t)=e
i ^
Ht=~
(0) (7)
andapplytheBaker-Hausdoridentity[5]:
c
e ^
O
^
Ae ^
O
= ^
A+[ ^
O; ^
A]+ 1
2! [
^
O;[ ^
O; ^
A]]+ 1
3! [
^
O;[ ^
O;[ ^
O; ^
A ]]]+ (8)
Inthepresentcase:
^
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 ;etc:
Consideringthe\reduced"hamiltonian, ^
H
r
,actingonthewavefunctionsontheplaneY =0:
^
H
r =
^
P
x 2
2m +
^
P
z 2
2m +2
B B
0 ^
S
z +2
B
^
Z ^
S
z
theproblemisgreatlysimpliedforallthetermsoforderhigherthan2vanish. Wewillcallthese functions:
1
(x;z;t)
+
(x;0;z;t) and
2
(x;z;t) (x;0;z;t)
from which onecancalculate theexpected 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
thisresultisinaccordancewiththetheoremof
Ehren-fest[5]. Thus,ifthebeamhasbeenpreviouslypolarized
alongthez-axis,byenteringtheStern-Gerlach
appara-tusitwillbesubjecttoaforceequalto+
B
for
spin-down particles (< ^
S
z
> (0) = 1=2), and
B for
spin-up particles (< ^
S
z
>(0)=+1=2). Consequently,
thebeamissplittedbytheeldgradient,andthe
parti-clesareseparatedbytheirspindirection onthez-axis.
Inthenextsectioneq.(3)issolvedontheplaneY =0,
andtheexactwavefunctionsareobtained. InsectionIII
thestationarywavefunctionaredeterminedalongwith
thecorrespondingeigenenergies. The case S =3=2 is
discussedinSec.IVandtheenergyandspatialsplittings
II Exact Solutions on Y = 0
On this section we will derive exact solutions for the
Stern-Gerlach eect on a symmetry plane. We will
consider a magnetic eld with eld gradient dierent
from zero for x > 0 (direction of the incident
parti-cle) and equal to zero for x 0. Thetime-evolution
oftheparticleswavefunctionscrossingaStern-Gerlach
apparatuscanbeobtainedanalyticallyonthe
symme-try plane Y = 0, on which the two equations in (3)
becomedecoupled:
c
~ 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
Onthebasisoftheresultsofthepreceedingsection,one
canexpectthatthesolutionsoftheaboveequationswill
contain afunction dependent on thevaribles z and t,
representing the separation of the beam along the
z-axis. Besides, Berry [14] showed that the solutionsof
theSchrodingerequationofafree-particlecanbe
writ-tenasproducts of AiryAi functions bycomplex
expo-nentialfunctionswhosesquaremodulusevolveswithout
deformation. Thefact that Eqs.(10)and (11)contain
the coordinate z suggeststhat the same type of Airy
functionscanbefoundhere.
We therefore propose as possible solutions of (10)
and(11)thefollowingmultiparameterfunctions
1 and
2 :
1
(x;z;t)=F[a(z+bt 2
)]e ictz
e (i=~)(p
x x ~!
+ t)
(12)
and
2
(x;z;t)=G[ ~a( z+ ~
bt 2
)]e i~ctz
e
(i=~)(pxx ~! t)
(13)
Wherep
x
iseigenvalueof ^
P
x
,whichisinturnconserved.
Replacing (13)and (12)in(11)and (10),respectively,
onends that the functions F and Gsatisfy theAiry
equation[15]onlyiftheparametersassumethe
follow-ingvalues:
c
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)
c= 2mb
~
; ~c= 2m
~
b
~
(16)
b= ~
b=
B
4m
(17)
One notices that with this set of parameters, a and ~a are positive, a necessarycondition for j
1 (z;t)j
2
and
j
2 (z;t)j
2
to represent functions which moveappart with time. Explicitly, 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)
Onenoticesthatthesesolutionsarecorrectlyreducedtofreeparticleplanewavesintheregionoutsideofeld,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)
Thesefunctions,however,stilldonotrepresentphysicallycorrectsolutionstotheproblem. Firstonenotesthat
theAiryAifunctionsoscillatealongthez-axiswithoutdecayingforlargevaluesofz. Togetridoftheseoscillations
andobtainaproperwavefunction,wesimplymultiplythesolutions(18)and(19)byHeaviside'sstepfunctions,,
which\truncate" theoscillationsattherstzero(
o
). Secondly, weintroduceadisplacement
o
= 1:0188(rst
maximumofAi) inordertomakethetwofunctions coincideat t=0. With this,thenalcorrect time-dependent
wavefunctionsbecome:
1
(x;z;t)=Ai
a(z+(
o
=a)+bt 2 ) z ( o
=a)+(
o
=a)+bt 2 e i(czt) 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(czt) e (i=~)[p x x ~w t]
(23)
Itmustbeemphesizedthatthefunctions(22)and(23)arealsosolutions ofthe samesetof equations(10)and
(11),with thesame setofparameters(14),(16)and(17),but ~w
+
and~w , whicharenowgivenby:
~! + = 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,respectively. Thespin-upand spin-downprobabilitydensitiesbecometherefore:
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 produceraplotofthefunctions we
de-ne anunit oflength,,asfollow:
2
= Z
y
(x;z;t)(X 2
+Z 2
) (x;z;t)dxdz
Intermsof,otherphysicalquantitiesbecome:
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"state separatefrom the\spin-down"asthe
Figure.1Timeevolutionofthespin-upandspin-down
prob-ability densitiesas particlesythroughthe magnetregion
for (a) t = 0:0001, (b) t = 0:03 and (c) 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)
Replacingsolutionsofthetype:
1
(x;z)=e (i=~)pxx
R (z) (31)
one obtainsthefollowingequationforthefunction 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)
wich is the Airy equation. The solutionof Eq.(29) is
therefore:
c
1
(x;z)=Ai( 1=3
z+
o )
z
o
1=3
+
o
e (i=~)p
x x
(37)
Wherethestepfunctionhasbeenintroduced,asdiscussedinthepreceedingsection. Theeigenenergyassociated
tothisfunctions 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)
Weseethattheeectoftheeldgradientistoproduce
a small positive displacement on the magnetic levels,
and consequently, theenergy dierence E
1 E
2 does
notdependon.
IV Solutions for S = 3=2
Onthissectionwegeneralizetheresultsofthe
preceed-ingsection to the case S =3=2. The hamiltonian for
aneutralparticlewithzeroorbitalangularmomentum
andarbitraryspin ^
S inamagneticeldBis:
^
H= ^
P 2
2m I+
2
B
~ ^
S:B (41)
The corresponding Schrodinger-Paulimatrix
equa-tionbecomes:
c
^
H = I ~
2
2m r
2
+
B
[ y
2 +(B
0
+z)
3 ] =i~
@
@t
(42)
where
i
,with i=1;2;3,arethespinmatrices. ForS=3=2,overY =0wehavefourdecoupledequations:
~ 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
,etc., arethecomponentsofthefour-dimensionalspinor.
d
Following the same proceedure as in the previous
sectionweobtainthefollowingenergies:
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)
Themaindierenceinrespecttothepreviouscase,
is theunequaldisplacementsofthemagneticlevelsfor
m= 1=2 and m= 3=2. This situation canbe
ex-ploitedtomeasuretheeectsoftheeldgradientona
beam,asshownbelow.
IV.1Energy Splitting
Inthissectionweevaluatethemagneticofthe
con-tribution of the eld gradient to the energy splitting.
Let us assume the following values for the physical
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 thedierence in energy
betweentheselevelsinunits offrecuency:
=(E
1 E
o
)=h0;4 MHz
In a real experimental situation we can superimpose
tothestaticelds, aradiofrequencyeldtuned to
fre-quencies at this range and change the populations of
themagneticlevels.
Otherordersofmagnitudesare:
p 2
x
2m
=0:2022 eV
B B
o
=0:578810 4
eV
V Spatial splitting
ForthecaseS =1=2oneobtainthe accelerationv_ by
makingtheargumentoftheAiryfunction,eq.(22),
con-stant andderivating in respectto time. Fromthis we
have:
_
v= 2b= (
B =2m)
CallinglthelengthofthemagnetandLthelength
betweenthemagnet andthedetector, thetotal
devia-tionoftheparticlealongthez axisisgivenby:
z=(
B
4m )
l 2
v 2
x +(
B
2m )
lL
v 2
x
Replacingthe valuesoftheconstantsgivenaboveand
makingl=L=0:2m,wendz4mm,whichisthe
correctorderofmagnitudeforthesplittingobservedin
areal Stern-Gerlachexperiment.
VI Discussions and Conclusions
Onthis paperweobtainedanalyticalsolutionsforthe
problem of a neutral particle with spin S in a static
magneticeldwitheldgradient. Thesesolutionswere
builtfrom oftheAiry functions,which arein turn
so-lutions of the Schrodinger equation in the symmetry
plane Y = 0. By choosing adequately the
parame-ters involved,weobtainedadiscrete energyspectrum,
andspinupandspindownwavefunctionswhichtravels
apartwithtime,asinarealStern-Gerlachexperiment.
Noapproximationhasbeenmadeonthemagnitudeof
theeldgradient,. Thiscontrastswiththeusual
pro-ceedure (and physically incorrect) of making B
0
[6;16]. Results were obtained for the cases S = 1=2
and S = 3=2. On this last case we showed that the
magnetic energy levelsare unequally displaced by the
eld gradient, and that the splitting is on the KHz
-MHz frequencyrange. Thecalculatedspatialsplitting
isin accordancewith whatisobservedin areal
Stern-Gerlac! h experiment.
As a nal remark we mention that Eqs (10) and
(11)predictanotherinterestingfeature,thefactthatin
arealStern-Gerlachexperimenttheseparationofspin
up and spindown particlesis not complete, asstated
in [17]. We found further soltions of those equations
whichshowanadmixtureofspinstatewavestravelling
in opositedirection[18].
Acknowledgments
Oneof us, JDB, is thankful to Prof. H.G. Valqui
(UNI,Lima)forusefulsugestionsconcerningsome
top-icsofthiswork.
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