13.5 Stern-Ger­lach Ap­pa­ra­tus

A con­stant mag­netic field will ex­ert a torque, but no net force on a mag­netic di­pole like an elec­tron; if you think of the di­pole as a mag­netic north pole and south pole close to­gether, the mag­netic forces on north pole and south pole will be op­po­site and pro­duce no net force on the di­pole. How­ever, if the mag­netic field strength varies with lo­ca­tion, the two forces will be dif­fer­ent and a net force will re­sult.

The Stern-Ger­lach ap­pa­ra­tus ex­ploits this process by send­ing a beam of atoms through a mag­netic field with spa­tial vari­a­tion, caus­ing the atoms to de­flect up­wards or down­wards de­pend­ing on their mag­netic di­pole strength. The mag­netic di­pole strengths of the atoms will be pro­por­tional to the rel­e­vant elec­tron an­gu­lar mo­menta, (the nu­cleus can be ig­nored be­cause of the large mass in its gy­ro­mag­netic ra­tio), and that will be quan­tized. So the in­com­ing beam will split into dis­tinct beams cor­re­spond­ing to the quan­tized val­ues of the elec­tron an­gu­lar mo­men­tum.

The ex­per­i­ment was a great step for­ward in the de­vel­op­ment of quan­tum me­chan­ics, be­cause there is re­ally no way that clas­si­cal me­chan­ics can ex­plain the split­ting into sep­a­rate beams; clas­si­cal me­chan­ics just has to pre­dict a smeared-out beam. An­gu­lar mo­men­tum in clas­si­cal me­chan­ics can have any value, not just the val­ues $m\hbar$ of quan­tum me­chan­ics. More­over, by cap­tur­ing one of the split beams, you have a source of par­ti­cles all in the same state with­out un­cer­tainty, to use for other ex­per­i­ments or prac­ti­cal ap­pli­ca­tions such as masers.

Stern and Ger­lach used a beam of sil­ver atoms in their ex­per­i­ment, and the sep­a­rated beams de­posited this sil­ver on a plate. Ini­tially, Ger­lach had dif­fi­culty see­ing any de­posited sil­ver on those plates be­cause the layer was ex­tremely thin. But for­tu­nately for quan­tum me­chan­ics, Stern was puff­ing his usual cheap cig­ars when he had a look, and the large amount of sul­phur in the smoke was enough to turn some of the sil­ver into jet-black sil­ver sul­fide, mak­ing it show clearly.

An irony is that that Stern and Ger­lach as­sumed that that they had ver­i­fied Bohr's or­bital mo­men­tum. But ac­tu­ally, they had dis­cov­ered spin. The net mag­netic mo­ment of sil­ver’s in­ner elec­trons is zero, and the lone va­lence elec­tron is in a 5s or­bit with zero or­bital an­gu­lar mo­men­tum. It was the spin of the va­lence elec­tron that caused the split­ting. While spin has half the strength of or­bital an­gu­lar mo­men­tum, its mag­netic mo­ment is about the same due to its $g$-​fac­tor be­ing two rather than one.

To use the Stern Ger­lach pro­ce­dure with charged par­ti­cles such as lone elec­trons, a trans­verse elec­tric field must be pro­vided to coun­ter­act the large Lorentz force that the mag­net im­parts on the mov­ing elec­trons.