Definition of magnetic moment in quantum mechanics

[Kashmir]

* The general formula for the magnetic moment of a charge configuration is defined as $\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r$* For an electron it's said that the correct equation relating it's spin and magnetic moment is is
$\vec{\mu} =g\frac{q}{2m}\vec{S}$

* It's said that the above equation cannot be justified classically and is a quantum mechanical phenomenon. What is the definition of magnetic moment used in the quantum mechanical equation $\vec{\mu} =g\frac{q}{2m}\vec{S}$

 

[anuttarasammyak]

Spin S has physical dimension of angular momentum. $\mu$ has dimension of dipole magnetic momentum as if a charge point is circulating along a ring but actually such an image does not hold.

 

[Spinnor]

You might like this paper, "What is Spin" by Hans C Ohanian, American Journal of Physics 54, 500 (1986).

Abstract,

According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave.

1. © 1986 American Association of Physics Teachers.

Available here,

https://www.physics.mcmaster.ca/PHYS3MM3/notes/whatisspin.pdf

From Google search, "hans ohanian what is spin pdf"

 

[Kashmir]

Spin S has physical dimension of angular momentum. $\mu$ has dimension of dipole magnetic momentum as if a charge point is circulating along a ring but actually such an image does not hold.

Yes it's true such an image isn't correct. I wanted to know how does one calculate the magnetic moment of a particle such as electron. We can't use $\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r$ so what is the definition?

 

[Kashmir]

You might like this paper, "What is Spin" by Hans C Ohanian, American Journal of Physics 54, 500 (1986).

Abstract,

According to the prevailing belief, the spin of the electron or of some other particle is a mysterious internal angular momentum for which no concrete physical picture is available, and for which there is no classical analog. However, on the basis of an old calculation by Belinfante [Physica 6, 887 (1939)], it can be shown that the spin may be regarded as an angular momentum generated by a circulating flow of energy in the wave field of the electron. Likewise, the magnetic moment may be regarded as generated by a circulating flow of charge in the wave field. This provides an intuitively appealing picture and establishes that neither the spin nor the magnetic moment are ‘‘internal’’—they are not associated with the internal structure of the electron, but rather with the structure of its wave field. Furthermore, a comparison between calculations of angular momentum in the Dirac and electromagnetic fields shows that the spin of the electron is entirely analogous to the angular momentum carried by a classical circularly polarized wave.

1. © 1986 American Association of Physics Teachers.

Available here,

https://www.physics.mcmaster.ca/PHYS3MM3/notes/whatisspin.pdf

From Google search, "hans ohanian what is spin pdf"

Thank you but it's above my level right now.

 

[Meir Achuz]

Yes it's true such an image isn't correct. I wanted to know how does one calculate the magnetic moment of a particle such as electron. We can't use $\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r$ so what is the definition?

The magnetic moment of an electron cannot be calculated classically.
It comes out of the Relativistic QM Dirac equation.
It is defined by one term in the Dirac equation.
The electron's charge and mass also cannot be derived classically.

 

[Kashmir]

The magnetic moment of an electron cannot be calculated classically.
It comes out of the Relativistic QM Dirac equation.
It is defined by one term in the Dirac equation.
The electron's charge and mass also cannot be derived classically.

Ohh! Got it! Thank you so much