Quantum mechanical description of Stern-Gerlach experiments

Quan um mechanical desc ip ion o S e n-Ge lach expe imen s G. Po el,1F. Ba anco,2S. C uz-Ba ios,3,1 and J. Gómez-Camacho1 1Depa amen o de Física A ómica, Molecula y Nuclea , Apa ado Pos al 1065, 41080 Se illa, Spain 2Depa amen o de Física Aplicada 3, E.S.I. Isla de la Ca uja, Se illa, Spain 3Depa amen o de Física Aplicada 1, E.U.P. Vi gen de Á ica, Se illa, Spain 共Recei ed 21 Sep embe 2004; published 27 May 2005兲 The mo ion o neu al pa icles wi h magne ic momen s in an inhomogeneous magne ic ield is desc ibed in a quan um mechanical amewo k. The alidi y o he semiclassical app oxima ions which a e gene ally used o desc ibe hese phenomena is discussed. App oxima e exp essions o he e olu ion ope a o a e de i ed and compa ed o he exac calcula ions. Focusing and spin- lip phenomena a e p edic ed. The eliabili y o S e n- Ge lach expe imen s o measu e spin p ojec ions is assessed in his amewo k. DOI: 10.1103/PhysRe A.71.052106 PACS numbe 共s兲: 03.65.Sq, 03.65.Ta, 03.65.Wj, 03.65.Nk I. INTRODUCTION The S e n-Ge lach expe imen consis s in aking a beam o pa icles ha ha e a neu al elec ic cha ge, bu a ini e magne ic momen , and passing hem h ough an inhomoge- neous magne ic ield. The obse ed esul is ha he pa icles de lec di e en ly depending on he spin p ojec ion along he magne ic ield. So, by measu ing he de lec ion, one can in e he alue o he spin p ojec ion o he pa icles along he di ec ion o he magne ic ield. The desc ip ion o his phe- nomenon is done wi h he ollowing assump ions. 共i兲The spin p ojec ion along he zaxis, aken along he magne ic ield a he cen e o he beam, is conse ed. 共ii兲Pa icles wi h di e en spin p ojec ions along he z axis, as hey go h ough he inhomogeneous magne ic ield, su e a o ce in he zdi ec ion ha is gi en by he p oduc o he magne ic momen imes he g adien o he ield imes he spin p ojec ion. This is wha we will call he ex book desc ip ion o he S e n-Ge lach expe imen 关1–4兴. Thus, conside ing he pa - icle posi ion as a poin e and he spin p ojec ion as he quan- um p ope y o be measu ed, he S e n-Ge lach se up is as- socia ed wi h a measu emen ope a o on he spin s a e which has as eigen alues he spin p ojec ions along he zaxis. Un- de he ex book desc ip ion, he S e n-Ge lach expe imen co esponds o an “ideal” measu emen , in he sense o on Neumann 关5兴, because he quan um s a e is no modi ied by he measu emen p ocess when i is an eigens a e o he mea- su ing appa a us. Besides, i is “comple ely eliable,” in he sense discussed in 关6兴, because he posi ion is comple ely co ela ed wi h he spin p ojec ion. Howe e , when he expe imen is in es iga ed in mo e de ail, he si ua ion becomes mo e complica ed. As he mag- ne ic ield has ze o di e gence, hen i is no possible o ha e a g adien o he ield only in one di ec ion. This p oduces e ms in he Hamil onian ha can change he spin o he inciden pa icle. A de ailed in es iga ion o hese e ec s was made in a ecen publica ion 关7兴, making use o he concep o cohe en in e nal s a es 关8兴in a semiclassical app oach. In his app oach, i is shown ha he quan um mechanical wa e unc ion which desc ibes he mo ion o a sys em wi h in e - nal deg ees o eedom can be app oxima ed by a single a- jec o y only o ce ain in e nal s a es which a e called co- he en in e nal s a es. These in e nal s a es e ol e in ime acco ding o an e olu ion ope a o which is de e mined by he in e ac ion e alua ed along he ajec o y. The cohe en in e nal s a es, in he case o he S e n-Ge lach expe imen s, a e s a es wi h de ini e p ojec ion along he di ec ion o he magne ic ield. This di ec ion may a y depending on he posi ion o he pa icle, because he magne ic ield is no homogeneous. The main esul o 关7兴is ha , indeed, when a beam o pa icles goes h ough a S e n-Ge lach magne , he di e en spin p ojec ions de ia e depending on he spin p ojec ion. Howe e , when he size o he beam is no e y small com- pa ed o he ange o inhomogenei y o he magne ic ield, addi ional e ec s occu . 共i兲The e is a ocusing e ec , so ha he pa icles de ia - ing in he di ec ion in which he ield dec eases end o ocus, while hose going in he di ec ion o inc easing ield end o de ocus. 共ii兲The e a e some pa icles wi h a gi en spin p ojec ion which de ia e as hose wi h a di e en spin p ojec ion. So he S e n-Ge lach se up is no , e en in heo y, a “comple ely eliable” measu ing appa a us. 共iii兲The e a e some pa icles, wi h a de ini e spin p ojec- ion along he quan iza ion axis, which change he spin p o- jec ion as hey go h ough he magne . So he S e n-Ge lach se up is no an “ideal” measu emen appa a us, as successi e measu emen s will no gi e exac ly he same esul s. This is wha we will call he semiclassical desc ip ion o he S e n-Ge lach expe imen . No e ha i we associa e he pa icle posi ion a e he magne as a “poin e ,” which gi es he esul o he measu emen o he spin p ojec ion along he zaxis, hen we conclude ha , in he semiclassical desc ip- ion, he S e n-Ge lach expe imen is no an ideal measu e- men , because i can al e he spin p ojec ion, o a comple ely eliable one, because he posi ion is no always co ela ed wi h he spin p ojec ion. These conclusions we e ob ained in a semiclassical amewo k, in which he mo ion o he pa icles was de- sc ibed by classical ajec o ies which depended in he spin p ojec ion along he magne ic ield ha hey encoun e ed. Ou mo i a ion he e is o see whe he he same conclusions hold when he ull quan um mechanical p oblem is consid- e ed. In Sec. II we o mula e he ime-dependen quan um PHYSICAL REVIEW A 71, 052106 共2005兲 1050-2947/2005/71共5兲/052106共10兲/$23.00 ©2005 The Ame ican Physical Socie y052106-1 mechanical p oblem o a wa e packe going h ough a S e n- Ge lach magne and discuss he alidi y o he ex book and semiclassical app oaches. In Sec. III we p esen he nume i- cal solu ion o he quan um mechanical p oblem. In Sec. IV we in es iga e se e al analy ic app oxima ions o he p ob- lem, conside ing he alidi y o he concep o cohe en in- e nal s a es. In Sec. V we discuss he in e p e a ion o S e n- Ge lach expe imen s as measu emen s de ices. Sec. VI is o a summa y and conclusions. II. QUANTUM MECHANICAL FORMULATION We wan o in es iga e he e ec o an inhomogeneous magne ic ield on he e olu ion o a quan um wa e packe . The si ua ion ha we will conside is a magne ic ield ha has componen s in he Xand Zdi ec ions, bu no in he Y di ec ion. This magne ic ield has a leng h L, and i can be w i en as B ជ =共B0+B1Z兲u ជ z−B1Xu ជ x,0艋Y艋L.共1兲 We use he capi al le e s X,Y,Z,T o ep esen magni udes wi h dimensions. Lowe case x,y,z, co espond o dimen- sionless quan i ies. We neglec bo de e ec s a ound Y=0 o Y=L. No e ha his ield ul ills ⵜB ជ =0 and also ⵜ⫻B ជ =0,as should be expec ed o a magne ic ield in he egion whe e he e a e no cu en s. These condi ions we e no ul illed in he case discussed in ex books such as 关1–3兴. The Hamil onian which desc ibes a non ela i is ic neu al pa icle which en e s in his ield is gi en by H=PX 2+PY 2+PZ 2 2M− ␮ B ជ ·I ជ ,共2兲 whe e ␮ is he magne ic momen and I ជ is he spin ope a o . We conside now a wa e packe 兩⌿共T兲;m0典which en e s in o his ield. Ini ially, he wa e packe can be cha ac e ized in coo dina e space as a Gaussian which is mo ing in he y di ec ion, while he ini ial spin p ojec ion along he Zaxis is m0: 具XYZ,m兩⌿共T=0兲;m0典=Nexp 冉 −X2+Y2+Z2 2 ␴ 2 冊 ⫻exp共ikyY兲 ␦ 共m,m0兲.共3兲 No e ha , neglec ing he e ec s o he bo de , he Ycompo- nen o he wa e unc ion is no a ec ed by he in e ac ion. Bo de e ec s will be ele an when he ansi ime, which is he ime ha he pa icle akes o go om he si ua ion in which he ield anishes 共B ជ =0兲 o he si ua ion whe e he ield is s a iona y 共B ជ ⯝B0u ជ z兲, is sho compa ed o he p e- cession ime o he spin in he magne ic ield. An es ima e o he ansi ime is =B0/共B1 兲. The p ecession ime is p =ប/共B0 ␮ 兲. I can be seen ha , o ealis ic cases, he ansi ime, al hough sho compa ed o he ime ha he beam spend wi hin he magne , is always la ge han he p ecession ime. So he spins o he pa icles o he beam ha e he ime o adap o he magne ic ield in which hey en e . The wa e unc ion is gi en by a wa e packe ha can be ac o ized in o a Ycomponen and an 共X,Z兲componen . The Ycomponen will e ol e eely inside he magne , because he Hamil onian does no ha e any in e ac ion e m which depends on Y, once ha he bo de e ec s 共occu ing a Y =0 and Y=L兲a e ound o be negligible. No e ha he wa e packe will s ay wi hin he magne ic ield du ing a ime ␶ =L/ y, whe e y=បky/M. Assuming ha he size o he wa e packe ␴ is e y small compa ed o L and aking in o accoun ha he ansi ime is much smalle han T, we can conside ha he magne ic ield s a s a T=0 and inishes a T= ␶ . So we ocus on sol ing he wo-dimensional ime-dependen p oblem, which co e- sponds o calcula ing he ime e olu ion be ween he ime T=0 and T= ␶ in a Hamil onian H=PX 2+PZ 2 2M− ␮ B ជ ·I ជ ,共4兲 conside ing ha he ini ial wa e unc ion is 具XZ;m兩⌽共T=0兲;m0典=Nexp 冉 −X2+Z2 2 ␴ 2 冊 ␦ 共m,m0兲.共5兲 I is con enien o make use o dimensionless a iables. So we de ine x=X/ ␴ ,z=Z/ ␴ , =T/ ␶ , and h=H ␶ /ប. Then, he equa ion o mo ion becomes h兩⌽共 兲;m0典=id d 兩⌽共 兲;m0典.共6兲 The dimensionless Hamil onian can be w i en as h=h0+ , wi h h0=A 2共px 2+pz 2兲, =−S关Iz共z+z0兲−Ixx兴,共7兲 whe e px=−id/dx,pz=−id/dz, and he dimensionless pa am- e e s A,S,z0a e A=ប ␶ M ␴ 2,S= ␮ B1 ␶ ␴ ប,z0=B0 ␴ B1.共8兲 The adiaba ici y pa ame e Ais he a io o he in e ac ion ime ␶ o he na u al ime o expansion o he Gaussian packe . The sepa a ion pa ame e Sis he a io o he mo- men um change induced by he magne ic ield g adien di- ided by he momen um wid h o he Gaussian packe . The inhomogenei y pa ame e z0de e mines he ela i e change o he magne ic ield in he ange o he Gaussian. No e ha in he posi ion 共x=0,z=−z0兲, he magne ic ield anishes. No e ha he p oduc AS= ␮ B1 ␶ 2/M ␴ is independen o ប. This magni ude is ela ed o he de ia ion o he beam in he magne . Fo a gi en ajec o y, which is de e mined by a ixed alue o he p oduc AS, he classical limi is eached as S→⬁and A→0. No e ha his co esponds o making ប→0 in Eqs. 共8兲. Validi y o he semiclassical desc ip ions We will now discuss he alidi y o he semiclassical and ex book desc ip ions o he S e n-Ge lach expe imen . I should be no iced ha , in gene al, a beam o pa icles is no gi en by a pu e quan um mechanical s a e, bu a he by a POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-2 mix u e o small quan um wa e packe s. Fo de ini eness, we conside ha ini ially one has a dis ibu ion o pa icles de- sc ibed as a Gaussian mix u e, o ange ␴ m, o small Gauss- ian wa e packe s o ange ␴ . The beam p o ile will hen be cha ac e ized by a Gaussian o ange ␴ =冑 ␴ m 2+ ␴ 2. The con- di ions equi ed, in o de o jus i y he semiclassical desc ip- ion done in 关7兴a e he ollowing. 共a兲The inhomogenei y o he magne ic ield o e he quan um size o he wa e packe should be small: ␴ B1ⰆB0. This implies ha z0Ⰷ1. 共b兲The momen um change should be la ge compa ed o he quan um sp ead o he beam momen um: ␮ B1 ␶ Ⰷប/ ␴ . This implies ha SⰇ1. No e ha hese condi ions a e e y well sa is ied in eal- is ic si ua ions o S e n-Ge lach expe imen s. Howe e , he alidi y o he ex book desc ip ion equi es also he a mo e s ingen condi ion ␴ B1ⰆB0, which equi es a e y s ong ield B0o , al e na i ely, a e y hin beam. The pu pose o his wo k is o in es iga e he ull quan- um solu ion o his p oblem o alues o he pa ame e s z0 and Swhich a e no necessa ily e y la ge, so ha he semi- classical and ex book desc ip ions become dubious. Ne e - heless, in o de o ha e a e e ence o compa e he quan um calcula ion, we ecall he expec ed esul s in he ex book desc ip ion. The ajec o y o he cen e o he wa e packe inside he magne is gi en by he exp ession zm共 兲= 1/2共SA兲m 2,共9兲 which depends on he spin p ojec ion m. No e ha , a e he in e ac ion 共 =1兲, he posi ions o he cen e o he wa e packe s o each spin p ojec ion a e gi en by zm共1兲 =SAm/2 and hei eloci ies a e z ˙m共1兲=SAm/2. I , a e he in e ac ion, he beam e ol es eely du ing a ime d, hen he posi ions o he cen e o he wa e packe s a e expec ed o be gi en by zm共 d兲=共1/2 + d兲共SA兲m.共10兲 As a ypical alue o he d i ime dwe will conside he ime necessa y o each he posi ion zm=−z0, o he spin p ojec ion m=−1/2, d=2z0/共SA兲− 1/2. 共11兲 Thus we would expec ha , a e a d i ime d, pa icles wi h spin p ojec ion m=1/2 should appea a ound z=z0,x =0, and pa icles wi h spin p ojec ion m=−1/2 should ap- pea a ound z=−z0,x=0. III. NUMERICAL CALCULATIONS We conside he sca e ing o a spin-1/2 pa icle. We ex- pand he wa e unc ion in o wo componen s, which ha e de ini e spin p ojec ions along he zaxis, 具xz;m= 1/2兩⌽共 兲;m0典= ␣ 共x,z, 兲ei Sz0/2, 具xz;m= − 1/2兩⌽共 兲;m0典= ␤ 共x,z, 兲e−i Sz0/2,共12兲 and he Sch ödinge equa ion o he 共x,z兲plane can be w i - en as 冤 A 2共px 2+pz 2兲−S 2zS 2x S 2xA 2共px 2+pz 2兲+S 2z 冥 冋 ␣ 共x,z, 兲 ␤ 共x,z, 兲 册 =id d 冋 ␣ 共x,z, 兲 ␤ 共x,z, 兲 册 ,共13兲 whe e ␣ 共x,z, 兲and ␤ 共x,z, 兲a e he componen s o he spino in he basis o he eigens a es o Iz. The nume ical solu ion o his equa ion has al eady been pe o med by Ga - away and S enholm 关9兴. Howe e , hey conside ed he case in which z0was la ge, so hei nume ical esul co esponded o he ex book in e p e a ion. A simila p oblem has been add essed by F anca e al. 关10兴, bu hey made use o he adiaba ic app oxima ion, neglec ing he kine ic ene gy du - ing he in e ac ion ime. To ollow ou app oach we mus i s w i e bo h compo- nen s o he spino as linea combina ions o ha monic oscil- la o unc ions, so ha ␣ 共x,z, 兲=兺 nm anm共 兲 ␾ n共x兲 ␾ m共z兲, FIG. 1. P obabili y dis ibu ion o an unpola ized wa e packe a e going h ough an inhomogeneous magne ic ield. No e he ocusing e ec o he lowe componen , which co esponds p e- dominan ly o m=−1/2. The uppe igu e co esponds o A=0.5, S=4. The lowe igu e is o A=0.1, S=20, which is close o he classical limi . QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-3 ␤ 共x,z, 兲=兺 nm bnm共 兲 ␾ n共x兲 ␾ m共z兲,共14兲 whe e ␾ n共x兲and ␾ m共z兲a e he ha monic oscilla o eigen- s a es o o de nand min he xand zdi ec ions, espec i ely. To calcula e he ime-dependen coe icien s anm共 兲and bnm共 兲o he expansion, i is na u al o ew i e Eq. 共13兲in e ms o he well-known c ea ion and des uc ion ope a o s ax=1 冑2共x+ipx兲,ax †=1 冑2共x−ipx兲, az=1 冑2共z+ipz兲,az †=1 冑2共z−ipz兲.共15兲 Thus, subs i u ing he ope a o s 共15兲in o Eq. 共13兲, we ob ain he desi ed sys em o o dina y coupled di e en ial equa ions o he coe icien s o he expansion o ␣ 共x,z, 兲and ␤ 共x,z, 兲: a ˙nm =iA 4关an+2,m冑共n+1兲共n+2兲+an−2,m冑n共n−1兲+an,m+2冑共m+1兲共m+2兲+an,m−2冑m共m−1兲−2anm共n+m+1兲兴 +iS 2冑2关an,m+1冑m+1+an,m−1冑m−共bn+1,m冑n+1+bn−1,m冑n兲e−iSz0 兴, b ˙nm =iA 4关bn+2,m冑共n+1兲共n+2兲+bn−2,m冑n共n−1兲+bn,m+2冑共m+1兲共m+2兲+bn,m−2冑m共m−1兲−2bnm共n+m+1兲兴 +iS 2冑2关−bn,m+1冑m+1−bn,m−1冑m−共an+1,m冑n+1+an−1,m冑n兲eiSz0 兴,共16兲 whe e he o e do s ands o di e en ia ion wi h espec o he dimensionless pa ame e . This sys em is sol ed using a ou h-o de Runge-Ku a me hod. The numbe o ha monic oscilla o basis unc ions needed in he calcula ion was ypi- cally o he o de o 40 in each coo dina e. We ha e pe o med calcula ions using ypical alues o A=0.5, S=4, and z0=4. This co esponds o a case in which he magne ic ield anishes a a dis ance o 4 ␴ . The ime o he in e ac ion is such ha he wid h o he beam would inc ease by a ac o o 冑1+A2. The magne ic ield g adien is such ha each componen o he magne ic ield will acqui e a momen um o Sប/2 ␴ , in opposi e di ec ions. As a compa i- son, we ha e also conside ed calcula ions wi h A=0.1, S =20, and z0=4, which p oduce he same de ia ion o he beam, bu a e close o he classical limi . A e he in e ac ion, we conside a d i ime d, gi en by Eq. 共11兲, du ing which he sys em e ol es in he ee Hamil- onian, so ha he cen e o he m=±1/2wa e packe would each he poin z=±z0, acco ding o he ex book desc ip ion. In Fig. 1 we ep esen he p obabili y dis ibu ion o a wa e packe , co esponding ini ially o an unpola ized beam. This is gi en by P0共x,z兲=1 2兺 mm0 兩具x,z;m兩⌽共 兲;m0典兩2.共17兲 The ocusing e ec can be clea ly seen by compa ing he shape o he dis ibu ions o he uppe and lowe compo- nen s, which co espond p edominan ly o m=1/2 and m =−1/2, espec i ely. The e ec o he ocusing is inc eased as Adec eases and Sinc eases. So we ha e con i med ha he ocusing e ec ha was p edic ed in he semiclassical calcula ion in 关8兴is a genuine esul ha appea s in he quan- um mechanical calcula ion, al hough i is di used i he adiaba ici y pa ame e Ahas a sizable alue. I should be no iced ha his ocusing e ec was also ound in he calcu- la ions p esen ed in 关9兴. In con as o he ex book desc ip ion, e en i he ini ial beam has a de ini e spin p ojec ion along he zaxis, a e he sca e ing p ocess his spin p ojec ion can change. We ha e e alua ed he p obabili y ha he pa icles change hei spin p ojec ion along he zaxis. I should be no iced ha he p obabili y o going om spin up o spin down is no exac ly he same as ha o going om spin down o spin up. Fo he e e ence case 共A=0.5, S=4, z0=4兲, we ob ain ha p共1/2, −1/2兲=0.0166 and p共−1/2,1/2兲=0.0198. The spin- lip phenomenon also appea s in he semiclassi- cal desc ip ion, because no all he pa icles ha compose he beam see he magne ic ield along he zaxis. The semiclas- sical spin- lip p obabili y is p共1/2,−1/2兲=p共−1/2,1/2兲 =0.0156, which depends only on he alue o z0. This is in good quali a i e ag eemen wi h he quan um calcula ions. In Fig. 2 we ep esen he spa ial dis ibu ion o he spin- lip p obabili y. No e ha he spin- lip p obabili y anishes o pa icles coming ou along he zaxis. The spa ial dis ibu ion o he spin- lip p obabili y is in quali a i e ag eemen wi h he semiclassical calcula ion, which becomes mo e accu a e as one makes he limi A→0, S→⬁, wi h AS cons an . The esul s o ou calcula ions can be summa ized as ol- lows: When a beam o pa icles, desc ibed by a Gaussian POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-4 wa e unc ion and wi h a gi en spin p ojec ion along he z axis, goes h ough an inhomogeneous magne ic ield, mos o he pa icles sca e as expec ed in he ex book desc ip ion. Howe e , a sizable ac ion o hem, which depends on z0 共abou 2% o z0=4兲, su e a change o he spin p ojec ion 共spin lip兲. F om hese pa icles ha su e spin lip, abou hal sca e in he same di ec ion as he majo i y o he pa - icles and he o he hal sca e in he opposi e di ec ion. We can conclude ha he spin- lip e ec desc ibed in he semi- classical desc ip ion, which was no p esen in he ex book desc ip ion o S e n-Ge lach expe imen s, is suppo ed by he ull quan um mechanical calcula ions. Also, we con i m ha he S e n-Ge lach expe imen , when conside ed as a measu emen appa a us o he spin p ojec ion, is no an ideal measu emen 共because he e is spin lip兲and i is no ully eliable 共because he e is no an exac co ela ion be ween he ini ial spin p ojec ion and he inal posi ion o he pa - icle兲. Howe e , he e a e quali a i e ea u es o he ull quan- um mechanical esul , such as he di e ence be ween up- down and down-up spin- lip p obabili ies, ha a e no p esen in he semiclassical desc ip ion and equi e u he in es iga ion. IV. APPROXIMATE TREATMENTS Ha ing sol ed nume ically he p oblem, we will conside se e al app oxima e ea men s o imp o e ou unde s and- ing o he phenomena unde conside a ion. The s a ing poin is he exac e olu ion ope a o and he ee e olu ion ope a- o U共 兲= exp关−i共h0+ 兲 兴,U0共 兲= exp共−ih0 兲.共18兲 I should be no iced ha h0and do no commu e. Thus, a p io i he e is no a single basis o spin s a es whe e he e olu ion ope a o is diagonal. Ne e heless, i can be a gued ha he in e ac ion domina es o e he ee Hamil onian h0. Tha would indica e ha he eigens a es o , which a e s a es wi h de ini e spin p ojec ion along he magne ic ield 共and hence cohe en in e nal s a es兲, should play an impo - an ole in he app oxima e solu ion o his p oblem. In his sec ion we de i e se e al app oxima e exp essions which make use o expansions o he exac e olu ion ope a o in e ms o and h0and i s commu a o s. We can use he coo dina es ␳ =冑共z+z0兲2+x2, ␤ = a c an x 共z+z0兲,共19兲 and e e he spin componen s o he di ec ion o he mag- ne ic ield a each posi ion: IB=Izcos共 ␤ 兲−Ixsin共 ␤ 兲,IT=Izsin共 ␤ 兲+Ixcos共 ␤ 兲. 共20兲 In e ms o hese a iables, he ini ial s a e can be exp essed as 具 ␳ ␤ ;m兩⌽共 =0兲;m0典 =Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2 冊 ␦ 共m,m0兲 共21兲 and h0and ake he exp essions h0=A 2共p ␳ 2+ ␳ −2p ␤ 2兲, =−S ␳ IB,共22兲 whe e p ␳ and p ␤ a e he momen a associa ed wi h ␳ and ␤ . The ele an commu a o s a e he ollowing: 关h0, 兴=iAS共p ␳ IB−兵p ␤ ,IT其/2 ␳ 兲,共23兲 关关h0, 兴, 兴=−AS2共IB 2+IT 2−兵p ␤ ,Iy其/2兲.共24兲 No e ha 关关h0, 兴,h0兴=0 and 关关关h0, 兴, 兴,h0兴=0. Fo spin- 1/2 pa icles, I=1/2, IB 2=IT 2=1/4. A. Adiaba ic app oxima ion The simples app oxima ion o he e olu ion ope a o consis s in neglec ing comple ely he e ec o h0. This leads o he adiaba ic app oxima ion, gi en by U共 兲⯝exp共−i 兲= exp共i S ␳ IB兲.共25兲 No e ha his exp ession conse es he p ojec ion o he spin along he di ec ion o he magne ic ield. Thus, i is con e- nien o expand he ini ial spin s a e in o s a es 兩n典which ul ill IB兩n典=n兩n典. This can be done conside ing he o a ion o an angle ␤ a ound he yaxis which akes he z axis o he di ec ion o he magne ic ield. Thus, he adiaba ic exp es- sion o he wa e unc ion a e he in e ac ion becomes 具 ␳ ␤ ;m兩⌽共 兲;m0典=Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2 冊 ⫻兺 ndnm 1/2共 ␤ 兲exp共in ␳ S 兲dnm0 1/2 共 ␤ 兲. 共26兲 No e ha his exp ession is equi alen o Eq. 共3.3兲in 关10兴, FIG. 2. Con ou plo o he p obabili y dis ibu ion o he spin- lip componen 共spin up o spin down兲o he wa e unc ion. The maximum is 3.3⫻10−4. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-5 whe e hey expanded he wa e unc ion in componen s ha had de ini e spin p ojec ions along he local magne ic ield. This exp ession con ains he quali a i e ea u es desc ibed in he nume ical calcula ion. The e is a spin- lip p obabili y, as m⫽m0. The ocusing e ec appea s when his adiaba ic wa e unc ion unde goes a ee e olu ion du ing a ime d a e he in e ac ion. Howe e , du ing he in e ac ion ime, he p obabili y dis ibu ion is ozen. B. Pseudoadiaba ic app oxima ion The nex app oxima ion consis s in neglec ing he com- mu a o 关h0, 兴. This leads o he pseudoadiaba ic app oxi- ma ion, gi en by U共 兲⯝exp共−i 兲exp共−i h0兲= exp共i S ␳ IB兲U0共 兲.共27兲 This exp ession also conse es he p ojec ion o he spin along he di ec ion o he magne ic ield, bu s a ing om a wa e unc ion ha has e ol ed eely du ing he in e ac ion ime . The wa e unc ion has an analy ic exp ession gi en by 具 ␳ ␤ ;m兩⌽共 兲;m0典=Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2共1+iA 兲 冊 ⫻兺 ndnm 1/2共 ␤ 兲exp共in ␳ S 兲dnm0 1/2 共 ␤ 兲. 共28兲 The di e ence o his exp ession wi h he adiaba ic one lies in he ac ha he Gaussian wa e packe ge s wide du ing he in e ac ion ime, by a ac o 冑1+A2, which is he widening o he ee wa e packe du ing he in e ac ion ime. C. Cohe en -s a e app oxima ion We conside he expansion o he e olu ion ope a o up o he hi d o de commu a o . The ollowing ela ions can be de i ed: U共 兲⯝exp 冉 共−i 兲3 6关关关h0, 兴, 兴兴 冊 exp共−i 兲 ⫻exp 冉 共−i 兲2 2关h0, 兴 冊 U0共 兲.共29兲 This exp ession is he basis o an analy ic ea men o he wa e unc ion. Fo ha pu pose, we no e ha he dominan e ms in he e olu ion ope a o a e hose which conse e he spin p ojec ion along he di ec ion o he magne ic ield. The s ongly oscilla ing ac o exp共−i 兲 ends o cancel he e ms ha do no conse e IB. We e ain in he expansion only hose e ms which commu e wi h IB. This leads o he exp es- sion U共 兲⯝exp共−i 3AS2/12兲exp共i S ␳ IB兲exp共−i 2ASp ␳ IB兲U0共 兲. 共30兲 The ope a o exp共−i 2ASp ␳ IB兲, when ac ing on eigens a es o IB, gene a es a displacemen in ␳ , which is gi en by ␳ = ␳ i + 2ASIB. This leads o an analy ic exp ession o he wa e unc ion, gi en by 具 ␳ ␤ ;m兩⌽共 兲,m0典 = exp共iAS2 3/12兲N兺 ndnm 1/2共 ␤ 兲exp共in ␳ S 兲冑 ␳ n ␳ ⫻exp 冉 − ␳ n 2−2 ␳ nz0cos ␤ −z0 2 2共1+iA 兲 冊 dnm0 1/2 共 ␤ 兲,共31兲 whe e ␳ n= ␳ −nAS 2/2. This wa e unc ion conse es he spin p ojec ion along he di ec ion o he magne ic ield. Thus, he s a es wi h a de ini e spin p ojec ion along he magne ic ield in each posi ion co espond o he cohe en in e nal s a es in oduced in Re . 关7兴. So we call his app oxi- ma ion he cohe en -s a e app oxima ion. No e ha in his app oxima ion he wa e unc ion no only ge s wide du ing he in e ac ing egion, bu he componen s wi h di e en al- ues o IBsepa a e. D. Symme ized app oxima ion We can app oxima e he e olu ion ope a o by he ollow- ing exp ession, which is co ec up o commu a o s o ou h o de : U共 兲⯝U0共 /2兲exp兵−i −共−i 兲3关关h0, 兴, 兴/12其U0共 /2兲. 共32兲 Neglec ing he e ms ha do no commu e wi h IB,we ha e U共 兲⯝exp共i 3AS2/24兲U0共 /2兲exp共i S ␳ IB兲U0共 /2兲.共33兲 The wa e unc ion can be w i en as 兩⌽共 兲;m0典= exp共iAS2 3/24兲U0共 /2兲兩⌽⬘共 兲;m0典,共34兲 whe e 具 ␳ ␤ ;n兩⌽⬘共 兲;m0典=Nexp 冉 − ␳ 2−2 ␳ z0cos ␤ +z0 2 2共1+iA /2兲 冊 ⫻兺 ndnm 1/2共 ␤ 兲exp共−in ␳ S 兲dnm0 1/2 共 ␤ 兲, 共35兲 which, al hough i is no comple ely analy ic, i can be ap- plied o e alua e he expansion o he wa e unc ion in a ha monic oscilla o basis. This app oxima ion co esponds o spli he e ec o U0共 兲du ing he in e ac ion symme ically, aking hal o i be o e and hal o i a e he in e ac ion. No e ha he e also he e olu ion associa ed wi h he in e ac- ion conse es he spin p ojec ion along he magne ic ield. We call his he symme ized app oxima ion. E. Compa ison wi h he exac calcula ion We ha e pe o med calcula ions wi h all he app oxima- ions. We ind ha he quali a i e cha ac e is ics o he exac calcula ions discussed abo e, which a e he ocusing e ec in he componen which goes o nega i e z alues and he POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-6 p esence o spin- lip componen s, appea in all he calcula- ions. The quan i a i e di e ences be ween he di e en ap- p oaches a ise in he momen um dis ibu ion o he spin lip componen . This comes ou symme ic in he adiaba ic and pseudoadiaba ic app oxima ions 共same p obabili y dis ibu- ion o posi i e and nega i e momen a兲and no ully sym- me ic in he cohe en -s a e o symme ized app oxima ions, in close ag eemen wi h he exac calcula ions. To e alua e he quali y o hese app oxima ions, we ha e calcula ed he a e age o he o e lap be ween he exac and app oxima e calcula ions. This o e lap is de ined as O=1 2冏兺 m0 具⌽ex共 =1兲;m0兩⌽ap共 =1兲;m0典冏.共36兲 They a e displayed in Fig. 3, as a unc ion o he adiaba ici y pa ame e A, o a ixed alue o he p oduc AS=2, which de e mines he de ia ion o he cen e o he wa e packe in he magne ic ield, as shown in Eq. 共9兲. The quan i y 1−Ois abou 10% o a wide ange o alues o A. In pa icula , o A=0.5 and S=4, 1−0=0.088 o he adiaba ic calcula ion and 1−O=0.064 o he pseudoadiaba ic calcula ion. On he con a y, he symme ized and cohe en -s a e app oxima ions a e much be e , so ha 1−Ois abou 0.1%. In pa icula , o A=0.5 and S=4, 1−0=0.0015 o he cohe en -s a e and 1−0=0.0006 o he symme ized calcula ions. The eason o his be e ag eemen a ises om he ac ha he cohe en -s a e and symme ized calcula ions allow o he dis o ion in he wa e unc ion p oduced by he magne ic ield g adien , while o he adiaba ic and pseudoadiaba ic calcula ions he e ec o he ield con ibu es only o a phase. In all he calcula ions ha we ha e pe o med, he quali y o he app oxima ed calcula ions imp o es as one goes om he adiaba ic o he pseudoadiaba ic o he cohe en s a e and inally o he symme ized app oxima ions. Globally consid- e ed, he app oxima ions de e io a e as he p oduc SA ge s la ge , because hen he e is mo e dis o ion in oduced in he wa e unc ion due o he combined e ec o he in e ac ion and he ee Hamil onian. A e y in e es ing case is he limi A→0, S→⬁ o ixed alues o AS. Nai ely, one would expec ha he adiaba ic app oxima ion would be adequa e he e, as he ee Hamil- onian h0is negligible compa ed o . Howe e , his is no he case. As shown in Fig. 3, he adiaba ic and pseudoadi- aba ic app oxima ions a e a he poo , gi ing alues o 1−Oo abou a ew pe cen . The cohe en -s a e and symme- ized app oxima ions a e e y good o A=0.015, bu hen hey become wo se o smalle alues o A. Nume ical cal- cula ions a e e y di icul when Sis la ge, because a la ge oscilla o basis is needed. An analy ic solu ion o his limi - ing case would be desi able. The in e es o his limi case 共A→0, AS cons an 兲is no only o mal. In nuclea physics he e a e cases in which weakly bound nuclei in e ac s ongly wi h a ge s du ing a e y sho ime, so ha he quan um s a e is signi ican ly dis o ed. The alidi y o he adiaba ic app oxima ion in hese si ua ions is open o deba e 关12兴. No e ha in he de ini ion o he o e lap we allow o an o e all phase di e ence be ween he exac and app oxima e wa e unc ions. This o e all phase di e ence does no a ec any obse able. We ind ha he bes app oxima e calcula- ions 共cohe en s a e and symme ized兲only ep oduce accu- a ely he phase o he exac wa e unc ion when bo h Aand Sa e small. We hink ha his is ela ed o he e ec o highe -o de e ms in he commu a o se ies o he e olu ion ope a o , which seem o a ec only a global phase in he wa e unc ion. So we see om hese app oxima ions ha a c ucial ea- u e o hem is he ac ha he mos ele an e ms in he e olu ion ope a o conse e he spin p ojec ion along he lo- cal di ec ion o he magne ic ield. This is he basis o he semiclassical calcula ion pe o med in 关7兴, in which he s a es wi h de ini e spin p ojec ions along he local magne ic ield we e aken as cohe en in e nal s a es, and hence hei mo ion could be desc ibed in e ms o ajec o ies. Despi e he ac ha he app oxima ions discussed he e, especially he cohe en -s a e and symme ized app oxima- ions, a e e y accu a e, hey do no desc ibe an impo an e ec o he exac e olu ion ope a o . In all he app oaches desc ibed he e, he sca e ing ampli udes o gi en spin p o- jec ions along he yaxis 共 he beam axis兲a e equal, up o a phase ac o , o he ampli udes in which he spin p ojec ions a e e e sed. This is a esul o he ac ha only e ms which commu e wi h IBa e allowed in he expansion o he e olu- ion ope a o . V. REEXAMINING THE STERN-GERLACH EXPERIMENTS In he ex book desc ip ion o he S e n-Ge lach expe i- men , he de lec ion o he beam gi es in o ma ion o he spin p ojec ion along he zaxis, which is he one ha poin s along he magne ic ield a he cen e o he beam. The de- lec ion o he beam is no sensi i e o he spin componen s along o he di ec ions. I , o a spin-1/2 pa icle, he ini ial FIG. 3. O e laps o he app oxima e wa e unc ions wi h he exac one, as a unc ion o he adiaba ici y pa ame e , o SA=2. The alue 1−O=0 co espond o pe ec ag eemen . The solid line is he adiaba ic app oxima ion, he dashed line is he pseudoadi- aba ic app oxima ion, he do ed line is he cohe en -s a e app oxi- ma ion, and he do -dashed line is he symme ized app oxima ion. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-7 spin poin s along he xaxis, mx=+1/2, he ex book desc ip- ion would indica e ha he pa e n o sca e ed pa icles would be comple ely equi alen o ha one p oduced by a mix u e o 50% mz=+1/2 and 50% mz=−1/2 pa icles. The same would be ue o mx=−1/2. So a S e n-Ge lach expe i- men is no expec ed o gi e any asymme y be ween di e - en spin p ojec ions pe pendicula o he zaxis. To in es iga e his ques ion, we de ine he asymme y o a gi en axis as he di e ence in he p obabili ies o inding he sca e ed pa icles in a gi en posi ion in he 共z,x兲plane o he wo spin p ojec ions. Thus, we ha e Az共x,z兲=兺 mm0m0 ⬘ 具x,z;m兩⌽共 兲;m0典具x,z;m兩⌽共 兲;m0 ⬘典*具m0兩 ␴ z兩m0 ⬘典, 共37兲 Ax共x,z兲=兺 mm0m0 ⬘ 具x,z;m兩⌽共 兲;m0典 ⫻具x,z;m兩⌽共 兲;m0 ⬘典*具m0兩 ␴ x兩m0 ⬘典,共38兲 Ay共x,z兲=兺 mm0m0 ⬘ 具x,z;m兩⌽共 兲;m0典 ⫻具x,z;m兩⌽共 兲;m0 ⬘典*具m0兩 ␴ y兩m0 ⬘典.共39兲 No e ha , in he s anda d desc ip ion o he S e n-Ge lach expe imen , he spin p ojec ion along he zaxis is conse ed, and hus he asymme ies Axand Ayshould anish a all poin s. This is no he case. As shown in Fig. 4共b兲, he e is a di e ence in he pa e n o pa icles sca e ed depending on he spin p ojec ion along he xaxis. This e ec is ound o depend on he inhomogenei y o he magne ic ield, which is de e mined by z0=B0/B1 ␴ .I z0is la ge, he inhomogenei y o he magne ic ield explo ed by he beam is small and so is Ax. This asymme y can be calcula ed, wi h a ious deg ees o accu acy, making use o he app oxima e ea men s dis- cussed he e. I can also be calcula ed wi h he semiclassical ea men o 关7兴. The o igin o his asymme y can be unde - s ood by a guing ha he mo ion in an inhomogeneous mag- ne ic ield conse es he spin p ojec ion along he local mag- ne ic ield, which has a di e en di ec ion o he di e en pa s o he wa e unc ion. This links wi h he concep o cohe en in e nal s a es, which we e in oduced in Re . 关8兴. The calcula ions in Fig. 4共a兲show also ha he e is an asymme y Aywhich means ha he e is a dependence o he spin p ojec ion along he yaxis. This is a dynamical e ec , which does no appea in he semiclassical desc ip ion. In ac , in he analy ic app oxima ions p esen ed he e, he alue o Ay anishes a e he in e ac ion. Only a e allowing o some ime o ee e olu ion do non anishing alues o Ay de elop. The o igin o his asymme y a ises om he e m AS2p ␤ Iywhich appea s in he double commu a o 关关h0, 兴, 兴. The e ec o his e m can be unde s ood because p ␤ is he gene a o o o a ions in he 共x,z兲plane, a ound he poin x =0, z=−z0, whe e he ield anishes. The e ec o his e m in he expansion o he e olu ion ope a o would gene a e a o a ion in he wa e unc ion a ound he poin whe e he ield anishes, which will be opposi e o he di e en spin p o- jec ions along he zaxis. Indeed, his e ec compe es wi h he in e ac ion =S ␳ IB, which ends o p ese e he spin p o- jec ion along he di ec ion o he ield. The esul o his compe i ion is ha he magni ude o he asymme y depends on he a io AS/z0. No e ha he asymme y Ayis associa ed o he dynamically gene a ed e m AS2p ␤ Iy. This e m de- pends on he spin p ojec ion Iybu is independen o Yo PY. So he mo ion in he Ydi ec ion is una ec ed by he dynam- ics, and hence i is gi en by he ee e olu ion o he Y componen o he ini ial wa e packe . FIG. 4. Asymme ies o pa icles pola ized along he y共a兲,x 共b兲, and z共c兲di ec ions. No e ha he maximum asymme y occu s o pa icles pola ized along he zaxis, bu ha he e a e impo an asymme ies o pa icles pola ized along he xand yaxes. POTEL e al. PHYSICAL REVIEW A 71, 052106 共2005兲 052106-8 The ac ha all he asymme ies a e non anishing and also ha hey ha e di e en beha io as a unc ion o 共x,z兲 leads o an exci ing possibili y. Conside ha we ha e a beam o pa icles, so ha we do no know hei pola iza ion s a e. We can make he beam go h ough an inhomogeneous ield, as desc ibed he e, and de ec he pa e n o sca e ed pa icles. Le he pola iza ion s a e be desc ibed ini ially as a densi y ma ix ␳ =1/2共I+px ␴ x+py ␴ y+pz ␴ z兲, whe e p ជ is a ec o which measu es he deg ee and di ec ion o he beam pola iza ion. Then, he densi y o pa icles de ec ed in he 共x,z兲plane will be p opo ional o P共x,z兲=P0共x,z兲+1 2关pxAx共x,z兲+pyAy共x,z兲+pzAz共x,z兲兴. 共40兲 This allows us o ob ain all he componen s o he pola iza- ion ec o om he pa e n o sca e ed pa icles, when a su icien numbe o pa icles a e de ec ed. No e ha , in con- as o exp ession 共40兲, he ex book desc ip ion o he S e n-Ge lach expe imen would be consis en wi h a p ob- abili y densi y gi en by P共x,z兲=P0共x,z兲+1 2pzAz共x,z兲, Az共x,z兲=2P0共x,z兲,z⬎0, Az共x,z兲=−2P0共x,z兲,z⬍0. 共41兲 This exp ession, when applicable, would allow one o ob ain in o ma ion only on he alue o pz. VI. SUMMARY AND CONCLUSIONS We ha e in es iga ed he mo ion o a pa icle wi h spin in an inhomogeneous magne ic ield using a quan um mechani- cal amewo k. Ou aim is o in es iga e in de ail he limi a- ions o he usual ex book app oach o S e n-Ge lach expe i- men s, which assumes ha he spin p ojec ion along he di ec ion o he magne ic ield is conse ed, while di e en spin componen s acqui e a momen um which depends on he g adien o he ield. We ind ha , consis en ly wi h a p e ious semiclassical analysis, he e is a sizable p obabili y o spin lip, which depends on he inhomogenei y o he ield. Besides, he e is a ocusing e ec in he componen ha de ia es owa ds he di ec ion in which he modulus o he ield dec eases. These cha ac e is ics a e e y obus and occu in dynamical si ua- ions which a e a om he semiclassical limi . Thus, we can conclude ha he S e n-Ge lach expe imen is no , e en in p inciple, an ideal expe imen , which would “p ojec ” he in e nal s a e in o he eigen alues o he mea- su emen ope a o . Mo eo e , he expe imen is no ully e- liable, as he posi ions o momen a o he pa icles do no gi e unequi ocal in o ma ion on he spin p ojec ion. The magni ude ha de e mines how close a S e n-Ge lach expe i- men is o an ideal eliable measu emen is z0= ␴ B0/B1. Only when he magne ic ield B0is e y la ge compa ed o i s g adien o when he size o he beam ␴ is e y small would he S e n-Ge lach expe imen app oxima e o an ideal eli- able measu emen . We ha e in es iga ed di e en app oxima e ea men s o he exac quan um mechanical p oblem. We ind ha , o a good app oxima ion, he in e ac ion occu s as i he spin p o- jec ion along he magne ic ield a each posi ion was con- se ed. This indica es ha , o each posi ion in he inhomo- geneous ield, he s a es wi h a gi en spin p ojec ion along he magne ic ield a e cohe en in e nal s a es. Then, p o- ided ha he quan um size o he wa e unc ion is small compa ed o he inhomogenei y o he magne ic ield, i is meaning ul o app oxima e he mo ion o hese s a es in e ms o classical ajec o ies. This jus i ies he ea men pe - o med in 关7兴. I is in e es ing o no e ha he adiaba ic app oxima ion is no accu a e, e en in he limi o small A共la ge mass o sho in e ac ion ime兲, i , a he same ime, he in e ac ion is la ge so ha i gene a es a ixed de lec ion angle. This obse a ion can be ele an o cases, such as in nuclea physics 关12兴,in which, al hough he collision imes a e sho o gua an ee he alidi y o he adiaba ic app oxima ion, he o ces a e so s ong o p oduce a ini e de lec ion. Ou calcula ions indica e ha he S e n-Ge lach expe i- men is no an ideal measu ing appa a us, in he sense o Re . 关5兴. Howe e , his does no mean ha one canno acqui e accu a e knowledge om he spin s a e o he p ojec ile by obse ing he s a is ical esul s o he expe imen . On he con a y, while an idealized S e n-Ge lach expe imen will no gi e any in o ma ion o he spin p ojec ion along he xo yaxis, he analysis o a ealis ic S e n-Ge lach expe imen , such as modeled in ou calcula ions, can gi e he alue o all he componen s o he densi y ma ix ha desc ibes he po- la iza ion o he beam. Ou analysis suppo s he idea ha he in e p e a ion o ealis ic expe imen s does no equi e he use o he educ- ion p inciple, as discussed by se e al au ho s in 关11兴. Thus, he in e ac ion be ween he spin and magne ic ield, which is desc ibed in a pu ely quan um mechanical amewo k, gen- e a es a co ela ion be ween he spin pola iza ion o he beam and he inal posi ion o he pa icles o he beam. A mea- su emen o a su icien ly la ge numbe o hese posi ions allows one o de e mine he componen s o he densi y ma- ix o he beam wi h su icien s a is ical accu acy. The e- duc ion p inciple is no equi ed in his a gumen . ACKNOWLEDGMENT This wo k has been pa ially suppo ed by he Spanish MCyT, P ojec s No. FPA2002-04181-C04-04 and BFM2002- 03315. QUANTUM MECHANICAL DESCRIPTION OF STERN-…PHYSICAL REVIEW A 71, 052106 共2005兲 052106-9