The magnetic moment vectors μ_sand μ_lassociated 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:

μ_s= –(e/m) S,

μ_l = –(e/2m)l

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

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Solution

The magnetic moment associated with the orbital angular momentum is valid with the classical mechanics.

The magnetic moment associated with the orbital angular momentum is given as

$μ_{l} = - (\frac{e}{2 m}) l$

For current i and area of cross-section A, we have the relation:

Magnetic moment

$μ_{l} = i A = > μ_{l} = (\frac{- e}{T}) π r^{2}$ ...............................(1)

Where,

e= Charge of the electron

r= Radius of the circular orbit

T= Time taken to complete one rotation around the circular orbit of radius r

Orbital angular momentum, l= mvr

$l = m * \frac{2 π r}{T} * r$ ..................................(2)

Where,

m= Mass of the electron

v= Velocity of the electron

r= Radius of the circular orbit

Dividing equation (1) by equation (2), we get:

$\frac{μ_{l}}{l} = - (\frac{e}{2 m}) = > μ_{l} = - (\frac{e}{2 m}) l$

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