I was talking to my friend the other day about the origin of inertia and it pondered both of our brains. I would like to know if the concept of inertia exists at the quantum level? I realize that quantized particles are probabilistic and cannot be understood at a classic level. So, does this mean inertia is applicable to only complicated systems like an atom and molecules, etc? There is such a huge gap in connection between classical and quantum systems it destroys my brain..
The concept of inertia can be applied to quantum particles, although a localized quantum particle will never have a sharply defined momentum.
The core concept of inertia (in the sense, that bodies at rest stay at rest and bodies in motion remain in motion) is cleanly expressed by the Galilei invariance of a theory. And it is easy to show, that the Schrödinger equation is Galilei invariant, thus showing, that quantum particles have inertia in the usual sense.
Along another line, one can argue via the Ehrenfest theorem, that: $$ m \partial_t^2 \langle x \rangle = -\langle \nabla V(x) \rangle$$ Therfore, the Newton axioms hold for the center of mass, if $V(x)$ scales on a scale much smaller than the wave packet (or $V$ is the harmonic oscillator potential, then $-\langle \nabla V(x) \rangle = - \langle kx \rangle = -k \langle x \rangle$). ($F = ma$ implies that the law on inertia holds).