The magnetic moment vectors µs and µl associated with the intrinsic spin angular momentum S and orbital angular momentum l, respectively, of an electron are predicted by quantum theory (and verified experimentally to a high accuracy) to be given by: µs = –(e/m) S, µl = –(e/2m)l Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

Out of the two relations given, only one is in accordance with classical physics.

The magnetic momemtum vector  which as a result of orbital Angular Moment is given by ,

μ_l​=−e/2m l

It follows from the definitions ofμ_l​ and l.

μ_l​=iA=(−e/T)πr2     …(i)

Angular momentum, l=mvr=m(2πr/T​)r     …(ii)

where r is the radius of the circular orbit, which the electron of mass m and charge (−e) completes in time T. 

Divide (i) by (ii), μl/l​​=[(−e/T)πr2 ]/(m(2πr/T​)r )​=−e/2m​

∴μ​l​=(−e​/2m)l

Clearly μ​l​ and l will be antiparallel (both being normal to the plane of the orbit)

In contrast, μs​​/S=e​/m. It is obtained on the basis of quantum mechanics.