Question

The magnetic moment vectors ${\mu }_s$ and ${\mu }_l$ 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:
${\mu }_s=-\left(e/m\right)S$,
${\mu }_l=-\left(e/2m\right)l$
Which of these relations is in accordance with the result expected classically ? Outline the derivation of the classical result.

Answer 1

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

Magnetic mom, ${\overrightarrow{\mu }}_1=-\left(\frac{e}{2m}\right)\overrightarrow{l}$

It follows from the definitions of ${\mu }_l$ and $l$.

${\mu }_l-iA=(-\frac{e}{T})\pi r^2$ ...(i)

Angular momentum, $l=mvr=m\left(\frac{2\pi r}{T}\right)r$ ...(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), $\frac{{\mu }_l}{l}=\frac{-e}{T}\pi r^2\times \frac{T}{m2\pi r^2}=\frac{-e}{2m}$

$\therefore {\overrightarrow{\mu }}_l=\left(\frac{-e}{2m}\right)\overrightarrow{l}$

Clearly ${\overrightarrow{\mu }}_l$ and $\overrightarrow{l}$ will be antiparallel (both being normal to the plane of the orbit)

In contrast, $\frac{{\mu }_s}{S}=\frac{e}{m}$. It is obtained on the basis of quantum mechanics.