Electron in magnetic field Consider an electron moving in the plane with a perpendicular constant magnetic field. It is well-known I think that the direction of motion can be changed by the magnetic field but not the absolute value of the velocity.
However, in relativistic quantum mechanics, the electron is not just a charge, but also has a magnetic moment. I wonder: How does the magnetic moment of the electron interact with the external magnetic field and how does this effect the speed of the particle?
 A: Only the charge matters for the effect of an uniform magnetic field on its velocity. We can think of a limit experience where there is a magnetic dipole and no charge.
If we put a electric neutral magnet over a floating device on water, the effect of the (uniform) magnetic field from the Earth is only rotate it to align to the field. There is no attraction force to South or North.
A: As the charge has a magnetic moment, it will interact with the magnetic field. When the charge enters the uniform magnetic field, the direction of its velocity changes, while its inherent magnetic field (due to spin) starts to rotate. But the rotation itself doesn't change. This rotation (spin) is the same inside the magnetic field as before entering the field. So it doesn't influence the speed of the particle (the energy associated with rotation stays the same).
A: If the magnetic field is uniform it may exert a torque on the magnetic dipole.  This may result in a rotation of the dipole. If the magnetic field is not uniform there will be a net force on the dipole. See for example,  the Stern-Gerlach experiment, which showed the quantization of the magnetic moment.
A: 
It is well-known I think that the direction of motion can be changed by the magnetic field but not the absolute value of the velocity.

Have you also heard about the emission of photons during the deflection of electrons? This emission is directed almost tangentially to the deflection of the electron and thereby the electron radiates its kinetic energy in the form of photons. Therefore, the electron does not move on a circular path but on a spiral and comes to a standstill in its center after the exhaustion of its kinetic energy.

However, in relativistic quantum mechanics, the electron is not just a charge, but also has a magnetic moment.

The magnetic moment of the electron is a constant and an intrinsic property, thus independent of external circumstances or a theory. Quantum mechanics deals with the quantization of this magnetic moment under the influence of magnetic fields and is applied to the electron distribution in atoms.

I wonder: How does the magnetic moment of the electron interact with the external magnetic field and how does this effect the speed of the particle?

To understand this, the photon emission mentioned above and the magnetic moment of the electron must be considered together. Not for nothing there is an association between the spin of the electron via the gyroscopic effect. A rotation of the spin respectively of its magnetic dipole (under the influence of the external magnetic field) causes the deflection of the electron. It is not improbable that a photon is ejected thereby. Its recoil in turn disturbs the alignment of the spin. This continues in cycles until the kinetic energy of the electron is completely exhausted.
