### D.82 Clas­si­cal spin-or­bit de­riva­tion

This note de­rives the spin-or­bit Hamil­ton­ian from a more in­tu­itive, clas­si­cal point of view than the Dirac equa­tion math­e­mat­ics.

Pic­ture the mag­netic elec­tron as con­tain­ing a pair of pos­i­tive and neg­a­tive mag­netic monopoles of a large strength . The very small dis­tance from neg­a­tive to pos­i­tive pole is de­noted by and the prod­uct is the mag­netic di­pole strength, which is fi­nite.

Next imag­ine this elec­tron smeared out in some or­bit en­cir­cling the nu­cleus with a speed . The two poles will then be smeared out into two par­al­lel mag­netic cur­rents that are very close to­gether. The two cur­rents have op­po­site di­rec­tions be­cause the ve­loc­ity of the poles is the same while their charges are op­po­site. These mag­netic cur­rents will be en­cir­cled by elec­tric field lines just like the elec­tric cur­rents in fig­ure 13.15 were en­cir­cled by mag­netic field lines.

Now as­sume that seen from up very close, a seg­ment of these cur­rents will seem al­most straight and two-di­men­sion­al, so that two-di­men­sion­al analy­sis can be used. Take a lo­cal co­or­di­nate sys­tem such that the -​axis is aligned with the neg­a­tive mag­netic cur­rent and in the di­rec­tion of pos­i­tive ve­loc­ity. Ro­tate the -​plane around the -​axis so that the pos­i­tive cur­rent is to the right of the neg­a­tive one. The pic­ture is then just like fig­ure 13.15, ex­cept that the cur­rents are mag­netic and the field lines elec­tric. In this co­or­di­nate sys­tem, the vec­tor from neg­a­tive to pos­i­tive pole takes the form .

The mag­netic cur­rent strength is de­fined as , where is the mov­ing mag­netic charge per unit length of the cur­rent. So, ac­cord­ing to ta­ble 13.2 the neg­a­tive cur­rent along the -​axis gen­er­ates a two-di­men­sion­al elec­tric field whose po­ten­tial is

To get the field of the pos­i­tive cur­rent a dis­tance to the right of it, shift and change sign:

If these two po­ten­tials are added, the dif­fer­ence be­tween the two arc­tan func­tions can be ap­prox­i­mated as times the de­riv­a­tive of the un­shifted arc­tan. That can be seen from ei­ther re­call­ing the very de­f­i­n­i­tion of the par­tial de­riv­a­tive, or from ex­pand­ing the sec­ond arc­tan in a Tay­lor se­ries in . The bot­tom line is that the monopoles of the mov­ing elec­tron gen­er­ate a net elec­tric field with a po­ten­tial

Now com­pare that with the elec­tric field gen­er­ated by a cou­ple of op­po­site elec­tric line charges like in fig­ure 13.12, a neg­a­tive one along the -​axis and a pos­i­tive one above it at a po­si­tion . The elec­tric di­pole mo­ment per unit length of such a pair of line charges is by de­f­i­n­i­tion , where is the elec­tric charge per unit length. Ac­cord­ing to ta­ble 13.1, a sin­gle elec­tric charge along the -​axis cre­ates an elec­tric field whose po­ten­tial is

For an elec­tric di­pole con­sist­ing of a neg­a­tive line charge along the -​axis and a pos­i­tive one above it at , the field is then

and the dif­fer­ence be­tween the two log­a­rithms can be ap­prox­i­mated as times the -​de­riv­a­tive of the un­shifted one. That gives

Com­par­ing this with the po­ten­tial of the monopoles, it is seen that the mag­netic cur­rents cre­ate an elec­tric di­pole in the -​di­rec­tion whose strength is . And since in this co­or­di­nate sys­tem the mag­netic di­pole mo­ment is and the ve­loc­ity , it fol­lows that the gen­er­ated elec­tric di­pole strength is

Since both di­pole mo­ments are per unit length, the same re­la­tion ap­plies be­tween the ac­tual mag­netic di­pole strength of the elec­tron and the elec­tric di­pole strength gen­er­ated by its mo­tion. The primes can be omit­ted.

Now the en­ergy of the elec­tric di­pole is where is the elec­tric field of the nu­cleus, ac­cord­ing to ta­ble 13.1. So the en­ergy is:

and the or­der of the triple prod­uct of vec­tors can be changed and then the an­gu­lar mo­men­tum can be sub­sti­tuted:

To get the cor­rect spin-or­bit in­ter­ac­tion, the mag­netic di­pole mo­ment used in this ex­pres­sion must be the clas­si­cal one, . The ad­di­tional fac­tor 2 for the en­ergy of the elec­tron in a mag­netic field does not ap­ply here. There does not seem to be a re­ally good rea­son to give for that, ex­cept for say­ing that the same Dirac equa­tion that says that the ad­di­tional -​fac­tor is there in the mag­netic in­ter­ac­tion also says it is not in the spin-or­bit in­ter­ac­tion. The ex­pres­sion for the en­ergy be­comes

Get­ting rid of us­ing , of us­ing , and of us­ing , the claimed ex­pres­sion for the spin-or­bit en­ergy is found.