Question
The magnetic moment vectors μs and μl associated with the intrinsic spin angular momentum S and orbital angular momentum I, 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) 1
Which of these relations is in accordance with the result expected classically? Outline the derivation of the classical result.
Solution
The relation is in accordance with the result expected from classical physics.
It can be derived as follows:
Magnetic moment associated with the orbital motion of the electron is
and, angular momentum of the orbiting electron is given by
Here, r is the radius of the circular orbit which the electron of mass m and charge (– e) completes in time T.
As charge of the electron is negative (– e) it is easily seen that magnetic moment, μ1 and angular momentum, l are antiparallel, both normal to the plane of the orbit.
Therefore, which is same result as predicted by quantum theory in contrast, is twice the classically expected value which is .
This latter result (verified experimentally) is an outstanding consequence of modem quantum theory.
It can be derived as follows:
Magnetic moment associated with the orbital motion of the electron is
and, angular momentum of the orbiting electron is given by
Here, r is the radius of the circular orbit which the electron of mass m and charge (– e) completes in time T.
As charge of the electron is negative (– e) it is easily seen that magnetic moment, μ1 and angular momentum, l are antiparallel, both normal to the plane of the orbit.
Therefore, which is same result as predicted by quantum theory in contrast, is twice the classically expected value which is .
This latter result (verified experimentally) is an outstanding consequence of modem quantum theory.
