Does a free electron have a moment of inertia tensor?
Hmm. Thats an interesting question.
Well since the electron is considered as a point particle, I dont think we can distinguish any "axis" of spinning in the classical sense.
But since there is the purely quantum mechanical effect of spin, maybe we can define some "moment of inertia" likeish tensor, which is also purely quantum mechanical...
The electron has spin angular momentum s=\hbar/2.
But that is QM spin which does not come from rotation of the electron.
That is, the electron has angular momentum, but no angular velocity.
Moment of inertia is defined by L=I\omega,
so moment of inertial cannot be defined for an electron.
Yes, in the classical sense of course it cannot be defined, but in the classical sense spin is also non sensical since the electron is a point particle.
But in the QM sense the electron has spin, hence in the QM there is a possibility of defining some QM "moment of inertia"...., but most probably there would be no use of defining it...
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