### D.73 Or­bital mo­tion in a mag­netic field

This note de­rives the en­ergy of a charged par­ti­cle in an ex­ter­nal mag­netic field. The field is as­sumed con­stant.

Ac­cord­ing to chap­ter 13.1, the Hamil­ton­ian is

where and are the mass and charge of the par­ti­cle and the vec­tor po­ten­tial is re­lated to the mag­netic field by . The po­ten­tial en­ergy is of no par­tic­u­lar in­ter­est in this note. The first term is, and it can be mul­ti­plied out as:

The mid­dle two terms in the right hand side are the changes in the Hamil­ton­ian due to the mag­netic field; they will be de­noted as:

Now to sim­plify the analy­sis, align the -​axis with so that . Then an ap­pro­pri­ate vec­tor po­ten­tial is

The vec­tor po­ten­tial is not unique, but a check shows that in­deed for the one above. Also, the canon­i­cal mo­men­tum is

There­fore, in the term above,

the lat­ter equal­ity be­ing true be­cause of the de­f­i­n­i­tion of an­gu­lar mo­men­tum as . Be­cause the -​axis was aligned with , , so, fi­nally,

Sim­i­larly, in the part of the Hamil­ton­ian, sub­sti­tu­tion of the ex­pres­sion for pro­duces

or writ­ing it so that it is in­de­pen­dent of how the -​axis is aligned,