5.25) 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 = - ( e / m ) S\\\\.\: \: \: \: \: \: \mu _l = - ( e / 2 m )l

Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.

Answers (1)

We know,

\mu_{l} = -(\frac{e}{2m})l 

\therefore \mu_{l} = -(\frac{e}{2m})l is in expected from classical physics.

Now, Magnetic moment associated with the orbital motion of the electron is:

\mu_{l} = Current x Area covered by orbit = I x A

(\frac{e}{T})\pi r^2

And, l = angular momentum = mvr

= m(\frac{2\pi r}{T}) r

(m is the mass of the electron having charge (-e), r is the radius of the orbit of by the electron around the nucleus and T is the time period.)

Dividing these two equations:

 \frac{\mu_{l}}{l} = -\frac{e}{T}\pi r^2\times \frac{T}{m\times2\pi r^2} = -\frac{e}{2m}

 \mu_{l} = (-\frac{e}{2m})l , which is the same result predicted by quantum theory.

The negative sign implies that \mu_{l} and l are anti-parallel.