General Magnetic Dipole Moment For an Electron in an Atom

[rtareen]

TL;DR Summary

The book (Young and Freedman Univeristy Physics E14) uses the same symbol $\mu_z$ for the magnetic dipole moment associated with orbital angular momentum and the moment associated with spin. Are these the same?

On the first attached page $\mu_z$ is associated with orbital angular momentum (Eq. 41.34). On the following pages (Eq. 41.38) it is associated with spin angular momentum? Are these both part of the same thing? I tried to read further but the book does not address this. In example 41.6 it implies that both contribute to the total as the orbital angular momentum is zero. What is the equation for the total magnetic moment?

 

[Haborix]

This smells like a chemistry book (despite it being a physics book)... They should really define a new quantity and not reuse the same symbol. In general, there are two contributions to the energy, one for intrinsic angular momentum and one for orbital angular momentum.

 

[rtareen]

This smells like a chemistry book (despite it being a physics book)... They should really define a new quantity and not reuse the same symbol. In general, there are two contributions to the energy, one for intrinsic angular momentum and one for orbital angular momentum.

Thank you, but do these two momentums combine into one single, total, angular momentum? (I'm two semesters removed from mechanics so I forgot).

 

[rtareen]

Also, do they both contribute to a single magnetic moment, or are these exclusive quantities?

 

[PeterDonis]

do these two momentums combine into one single, total, angular momentum?

All of these things are operators. There are three "angular momentum" operators; the usual nomenclature and symbols are $L$ for orbital angular momentum, $S$ for spin angular momentum, and $J$ for total angular momentum. Usually textbooks will tell you that $J = L + S$; but it should be noted that it is not always possible to split $J$ up into $L$ and $S$ in any invariant way, so in some cases $J$ is the only really meaningful operator.

do they both contribute to a single magnetic moment, or are these exclusive quantities?

"Magnetic moment" is really just another word for "angular momentum" (or strictly speaking, "angular momentum of something that has magnetic properties"), usually with different units to further confuse people. Or one can think of "magnetic moment" as the result of measuring angular momentum about some axis.

 

[rtareen]

All of these things are operators. There are three "angular momentum" operators; the usual nomenclature and symbols are $L$ for orbital angular momentum, $S$ for spin angular momentum, and $J$ for total angular momentum. Usually textbooks will tell you that $J = L + S$; but it should be noted that it is not always possible to split $J$ up into $L$ and $S$ in any invariant way, so in some cases $J$ is the only really meaningful operator.
"Magnetic moment" is really just another word for "angular momentum" (or strictly speaking, "angular momentum of something that has magnetic properties"), usually with different units to further confuse people. Or one can think of "magnetic moment" as the result of measuring angular momentum about some axis.

So if $\vec{J} = \vec{L} + \vec{S}$ then is it as simple as $\mu_{zj} = \mu_{zl} + \mu_{zs} = -e/2m( L_z+2.002S_z)$? I just added both the moments together. Would this be the total moment in the z-direction?

 

[PeterDonis]

So if $\vec{J} = \vec{L} + \vec{S}$ then is it as simple as $\mu_{zj} = \mu_{zl} + \mu_{zs} = -e/2m( L_z+2.002S_z)$?

No, because your first equation is a vector equation and your second is not. In the simple case where we are measuring all operators about the same axis, in this case $z$, yes, you can just add the moments since they are measurement results and the measurement results will add in that way. But if you are not measuring all the operators about the same axis, that simple relationship will no longer hold.

 

[rtareen]

Thank you

No, because your first equation is a vector equation and your second is not. In the simple case where we are measuring all operators about the same axis, in this case $z$, yes, you can just add the moments since they are measurement results and the measurement results will add in that way. But if you are not measuring all the operators about the same axis, that simple relationship will no longer hold.

Thank you PeterDonis! You are always great help. Please never leave PhysicsForums!

 

[PeterDonis]

Thank you PeterDonis! You are always great help. Please never leave PhysicsForums!

You're welcome! Thanks for the kudos.

 

[vanhees71]

"Magnetic moment" is really just another word for "angular momentum" (or strictly speaking, "angular momentum of something that has magnetic properties"), usually with different units to further confuse people. Or one can think of "magnetic moment" as the result of measuring angular momentum about some axis.

Magnetic moment is physically not the same as angular momentum but the quantities are related by
$$\hat{\vec{\mu}}=\mu_{\text{B}}(g_L \hat{\vec{L}} + g_s \hat{\vec{s}}).$$
Since we deal with electrons you have $g_L \simeq -1$ and $g_s \simeq -2$. For details, you may start at

https://en.wikipedia.org/wiki/Landé_g-factor

 

[PeterDonis]

Magnetic moment is physically not the same as angular momentum

Ah, yes, that's right, the gyromagnetic ratio is different for orbital and spin angular momentum.