# How does electron spin change instantaneously without violating inertia principle?

The inertia in one of the main properties of matter. That is why all process in macro world do not happen instantaneously.

What I do not understand is how we should apply this general idea of inertia to quantum world? Usually people say that electron spin change instantaneously from one direction two another. But this violates the principle of inertia.

Your intuition about inertia is essentially correct. The spin state of an electron does not change instantly. If the electron is in the spin-up state then the z-component of its angular momentum is $\frac{1}{2} \hbar$. If it is in the spin-down state then its angular momentum is $-\frac{1}{2} \hbar$. Classically, angular momentum cannot change instantly, just as linear momentum cannot. More quantitatively, the rate of change of angular momentum is equal to the torque applied. Analogously, in quantum mechanics, one can apply a "torque" on the electron spin degree of freedom by placing the electron in a magnetic field. This will cause the spin to precess into a different orientation. To double the precession speed one must double the strength of the applied magnetic field.