# The Energy of Spin-Orbit Coupling

Can you show that the energy of interaction is proportional to the scalar product s∙l.? See attachment for symbols.

The energy of a magnetic moment mu in a magnetic field B is equal to their scalar product (see attachment). If the magnetic field arises from the orbital angular momentum of the electron, it is proportional to the l: if the magnetic moment mu is that of the electron spin, then it is proportional to s. It then follows that the energy of the interaction is proportion to the scalar product s-l.

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#### Solution Preview

See the attachments.

The energy stored in the interaction between a magnetic moment and external magnetic field B is:

(1.1)

We want to show that in quantum mechanics this is proportional to the dot product between the intrinsic angular momentum (spin) operator of the electron and the electron's angular momentum operator

The electron can be modeled as a charged sphere rotating about an axis, say

Say the electron has a constant charge density, radius R and rotates at angular speed

a ring of radius r carrying current i has a magnetic moment of

(1.2)

And the direction is normal to the plane of the ring

A spherical ring located at distance r from ...

#### Solution Summary

The solution shows that the classical energy expression U= - m . B is proportional to the dot product of the spin and the angular momentum operators