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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

(2)

< ^

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

(3)

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

(4)

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

(5)

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

(6)

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

(7)

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.

References

[1] A. Bohm, Quantum Mechanics Foundations and

Ap-plications,Springer-Verlag,NewYork(1993).

[2] D.I.Blokhintsev,QuantumMechanics,D.Reidel

Pub-lishingCompany,Holland(1964)

[3] P.R.Holland, The Quantum Theory of Motion,

Cam-bridgeUniversityPress.Cambridge(1995)

[4] R.G.Winter,Quantum Physics,Sec.2.9, 8.1and 8.2,

WadsworthPublishingCompany,California(1979).

[5] F. Schwabl, Quantum Mechanics, Springer, Berlin

(1995).

[6] J.L. Martin, Basic Quantum Mechanics, Oxford

Sci-encePublications,Oxford(1982).

[7] H. Batelaan, T.J. Gay,and J.J Schwendiman, Phys.

Rev.Lett.79(23),4517(1997).

[8] S.NicChormaic,Ch.miniatura,O.Giorceix,B.Viaris

de Lesegno, J. Robert, S. Feron, V. Lorent, J.

Rein-hardt,J.BaudonandK.Rubin,Phys.Rev.Lett.72(1),

1(1994).

[9] M. Hannout, S.Hoyt, A. Kryowonos and A. Widon,

Am.J.Phys.66(5),377(1998).

[10] D.E.Platt,Am.J.Phys.60(4),306(1992).

[11] M.F Barros, J. Andrade and M.H. Andrade, Ann.

Fond.LouisdeBroglie 12(2),285(1987).

[12] M.F Barros, J. Andrade and M.H. Andrade, Ann.

(8)

[13] M.I.Shirokov, Ann.Fond.LouisdeBroglie21(4),391

(1996).

[14] M.V. Berry, N.L. Balazs, Am. J. Phys. 47(3), 264

(1979).

[15] N. Bleistein and R. Handelsman, Asymptotic

Expan-sionsofIntegrals,DoverPublications,NewYork(1986)

[16] J.M. Gracia-Bonda, Trevor W. Marshall and Emilio

Santos, Phys.Lett.A183,19(1993).

[17] C.W. Muller and F.W. Metz, J. Phys. A 27, 3511

(1994).

[18] J. Daz Bulnes, Construc~ao de Soluc~oes Exatas e

Aproximadas para o Efeito Stern-Gerlach, MSc.

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In this work, using differential Galois theory, we study the spectral problem of the one-dimensional Schr¨ odinger equation for rational time dependent KdV potentials.. In

Motivated by the previous results of Fabricio Maci` a and Gabriel Rivi` ere on the dynamics of the Schr¨ odinger equation associated with small perturbations of completely

As we have seen, even though the addition of a cosmological constant to Einstein’s eld equations may be the simplest way to obtain acceleration, it has its caveats. For this rea-

In this Thesis we study optimal constants in Hardy inequalities for Schr¨ odinger opera- tors with quadratic singular potentials, and we describe their contribution to the study

IBVP associated to the magnetic Schr¨ odinger operator L A,q with full data In the full data case and in the presence of a magnetic potential, special solutions adapted to the

In order to model the linear fermionic Lindblad operators introduced in equation (5) we modify this construction slightly by considering a different class of system-ancilla coupling