<prompt> Fikiria chembe isiyo na spin inayowakilishwa na utendaji kazi wa wimbipsiKxy2 z e-alpha rambapo rsqrtx2y2z2, na K na alpha ni viambajengo halisi.Ikiwa z-kijenzi cha kasi ya angular, L_z, kilipimwa, kuna uwezekano gani kwamba matokeo yangekuwa L_zhbar ?Unaweza kupata misemo ifuatayo kwa maumbo machache ya kwanza ya duara kuwa muhimu:anzailiyopangwaY_00sqrtfrac14 pi, quad Y_1pm 1mp sqrtfrac38 pi dhambi theta epm i phi, Y_10sqrtfrac34 pi cos theta, quad Y_2pm 1mp sqrtfrac15 8 pi sin theta cos theta epm i phi .mwishoiliyopangwa  <response>: Kitendaji cha wimbi kinaweza kuandikwa upya katika viwianishi vya duara kamapsiK rcos phi sin thetasin phi sin theta2 cos theta e-alpha rsehemu yake ya angular kuwapsitheta, phiKprimecos phi sin thetasin phi sin theta2 cos theta,ambapo Kprime ni hali ya kuhalalisha mara kwa mara hivyoKprime 2 int_thetapi d theta int_02 pi sin thetacos phi sin thetasin phi dhambi theta2 cos theta2 d phi1 maandishi . Tangucos phifrac12kushotoei phie-i phikulia, quad sin phifrac12 ikushotoei phi-e-i phikuliatunaanzailiyopangwapsitheta, phi Kprimeleftfrac12leftei phie-i phikulia sin thetafrac12kushotoei phi-e-i phikulia sin thetacos 2 thetakulia, Kprimeleft-frac121-i sqrtfrac8 pi3 Y_11frac 121i sqrtfrac8 pi3 Y_1-12 sqrtfrac4 pi3 Y_10kulia .mwishoiliyopangwaHali ya kuhalalisha na uhalisi wa Y_lm kisha toaKprime 2leftfrac12 cdot frac8 pi3frac12 cdot frac8 pi3 4 cdot frac4 pi3right1auKprimesqrtfrac18 pi,na hivyoanzailiyopangwapsitheta, phi sqrtfrac18 pileft-frac121-i sqrtfrac8 pi3 Y_11kulia.kushoto.frac121i sqrtfrac8 pi3 Y_1-12 sqrtfrac4 pi3 Y_10kulia .mwishoiliyopangwaJumla ya kasi ya angular ya chembe nisqrtleftlanglemathbfL2rightranglesqrtll1 hbarsqrt2 hbar .kama kitendakazi cha wimbi kinalingana na l1.Kipengele cha z-cha mwendo wa angular nianzailiyopangwaleftlanglepsileftL_zkulia psirightrangle Kprime 2leftfrac12 cdot frac8 pi3 cdot hbarleftY_1 1kulia2frac12 cdot frac8 pi3-hbarkushotoY_1-1kulia 2kulia.kushoto.4 cdot frac4 pi30kushotoY_10kulia2kulia frac18 pileftfrac12 cdot frac8 pi3hbarfrac12 cdot frac8 pi3-hbarright0 .mwishoiliyopangwaUwezekano wa kupata L_zhbar nianzailiyopangwaP leftleftlangle L_zhbar mid psitheta, phirightrangleright2 frac18 pi cdot frac12 cdot frac8 pi3frac16 .mwishoiliyopangwa 