### D.71 Elec­tro­mag­netic com­mu­ta­tors

The pur­pose of this note is to iden­tify the two com­mu­ta­tors of chap­ter 13.1; the one that pro­duces the ve­loc­ity (or rather, the rate of change in ex­pec­ta­tion po­si­tion), and the one that pro­duces the force (or rather the rate of change in ex­pec­ta­tion lin­ear mo­men­tum). All ba­sic prop­er­ties of com­mu­ta­tors used in the de­riva­tions be­low are de­scribed in chap­ter 4.5.4.

The Hamil­ton­ian is

when the dot prod­uct is writ­ten out in in­dex no­ta­tion.

The rate of change in the ex­pec­ta­tion value of a po­si­tion vec­tor com­po­nent is ac­cord­ing to chap­ter 7.2 given by

so you need the com­mu­ta­tor

Now the term can be dropped, since func­tions of po­si­tion com­mute with each other. On the re­main­der, use the fact that each of the two fac­tors comes out at its own side of the com­mu­ta­tor, to give

and then again, since the vec­tor po­ten­tial is just a func­tion of po­si­tion too, the can be dropped from the com­mu­ta­tors. What is left is zero un­less is the same as , since dif­fer­ent com­po­nents of po­si­tion and mo­men­tum com­mute, and when , it is mi­nus the canon­i­cal com­mu­ta­tor, (mi­nus since the or­der of and is in­verted), and the canon­i­cal com­mu­ta­tor has value , so

Plug­ging this in the time de­riv­a­tive of the ex­pec­ta­tion value of po­si­tion, you get

so the nor­mal mo­men­tum is in­deed given by the op­er­a­tor .

On to the other com­mu­ta­tor! The -​th com­po­nent of New­ton’s sec­ond law in ex­pec­ta­tion form,

re­quires the com­mu­ta­tor

The eas­i­est is the term , since both and are func­tions of po­si­tion and com­mute. And the com­mu­ta­tor with is the gen­er­al­ized fun­da­men­tal op­er­a­tor of chap­ter 4.5.4,

and plug­ging that into New­ton’s equa­tion, you can ver­ify that the elec­tric field term of the Lorentz law has al­ready been ob­tained.

In what is left of the de­sired com­mu­ta­tor, again take each fac­tor to its own side of the com­mu­ta­tor:

Work out the sim­pler com­mu­ta­tor ap­pear­ing here first:

the first equal­ity be­cause mo­men­tum op­er­a­tors and func­tions com­mute, and the sec­ond equal­ity is again the gen­er­al­ized fun­da­men­tal com­mu­ta­tor.

Note that by as­sump­tion the de­riv­a­tives of are con­stants, so the side of that this re­sult ap­pears is not rel­e­vant and what is left of the Hamil­ton­ian be­comes

Now let be the in­dex fol­low­ing in the se­quence and the one pre­ced­ing it (or the sec­ond fol­low­ing). Then the sum above will have a term where , but that term is seen to be zero, a term where , and a term where . The to­tal is then:

and that is

and the ex­pres­sion in brack­ets is the -​th com­po­nent of and pro­duces the term in New­ton’s equa­tion pro­vided that .