About magnetic momentum in quantum mechanics

[KFC]

Hi all,
I have been reading lots of materials regarding the classical and quantum mechanics. The first subject I read is Bohr model, in which it is assumed the electron is in circular motion around the nucleus on the so-called orbital. I think it is semi-classical. With this assumption, we consider the moving electron is a minute current ($I$) moving on a circular plane with radius $r$ and area $\pi r^2$. Applying the definition of angular momentum $|L|=mvr$ and other relations about circular motion, it derives the magnetic momentum as
$\vec{\mu_B} = -\frac{e}{2m}\vec{L}$
All derivation above makes use of the classical concept (e.g. assuming the electron is really moving on an orbit as perfect circular motion), but actually, there is no precise orbit in which electron is moving. I wonder if it is correct by simply apply above expression with $\vec{L}$ replaced with quantized angular momentum.

 

[Charles Link]

Hi all,
I have been reading lots of materials regarding the classical and quantum mechanics. The first subject I read is Bohr model, in which it is assumed the electron is in circular motion around the nucleus on the so-called orbital. I think it is semi-classical. With this assumption, we consider the moving electron is a minute current ($I$) moving on a circular plane with radius $r$ and area $\pi r^2$. Applying the definition of angular momentum $|L|=mvr$ and other relations about circular motion, it derives the magnetic momentum as
$\vec{\mu_B} = -\frac{e}{2m}\vec{L}$
All derivation above makes use of the classical concept (e.g. assuming the electron is really moving on an orbit as perfect circular motion), but actually, there is no precise orbit in which electron is moving. I wonder if it is correct by simply apply above expression with $\vec{L}$ replaced with quantized angular momentum.

The answer is yes. If you google the Zeeman effect, you will see that this is how the calculation is performed. One of the more common uses of a magnetic moment is to see the effects that occur when it is in a magnetic field. In a first order perturbation treatment of the magnetic moment in a magnetic field, the atomic orbital electron wave function is assumed to be unaffected by the magnetic field, but an energy change occurs given by $\Delta E=-\mu \cdot B$.