Does an electron with a measured Spin up $|1/2, 1/2>$ at $t= 0$ stay in the "Spin-up" state for all $t>0$? Does a free particle e.p. an electron with a measured Spin up |1/2, 1/2> at t= 0 stay in the "Spin-up" state for all t>0? 
Or in other words, is there a time-evolution which effects the Spin of a free particle (no magnetic field)? 
 A: Yes it maintains its spin orientation. 
Presumably there was at some earlier point an initial region with magnetic field so one could establish an “up” direction, but once it exits this region and propagates freely, it will maintain its original orientation in the sense that, when entering at some later point a second Stern-Gerlach apparatus with field gradient along the same direction as in the original region, the particle would still be deflected up.
A: We have to be careful with what means spin up and down. First at all


*

*the spin of the electron is an intrinsic property and

*one by one correlated with its magnetic dipole moment.


The positron has a spin of the same value but opposite sign and pointing in the opposite direction in relation to its magnetic dipole moment. The manifestation of these properties is the deflection of these partciles then moving in an external magnetic field. They get deflected in opposite directions.
For the dislocation of electrons in the atomic shells Hund formulated the earlier observed phenomenon, that two electrons can’t have all for quantum numbers identical. In the last quantum number (the spin quantum number) they have to be different. The terminology spin up and down is a bit misleading. In reality it was observed that the direction of their spins point always in opposite directions.
If you remember that spin and magnetic dipole moment are correlated you could answer your question by yourself. As long as an external magnetic field is not involved your free electron is moving in the same manner, maybe with the spin pointing constant in one direction or processing around an axis or rotation in a plane.
