restoring force
A restoring force is one that always pushes a system back toward its resting point, and grows stronger the farther the system strays. A stretched spring pulls inward; a compressed one pushes outward; either way the force points home. When that force is proportional to the displacement — twice as far means twice the pull — the motion it produces is called harmonic.
This simple proportionality is the heart of the harmonic oscillator. It corresponds to a potential energy that rises as the square of the displacement, a smooth parabola with its lowest point at equilibrium. Because nearly every smooth energy valley looks like a parabola when you zoom in close, almost any object resting in a stable minimum feels an approximately linear restoring force for small nudges.
In quantum mechanics the restoring force does not act on a definite trajectory, since the particle has no single path. Instead it shapes the potential in the Schrödinger equation, and that potential dictates which energies and wavefunctions are allowed. The same parabolic pull that would make a classical mass oscillate gives the quantum system its evenly spaced ladder of energy levels.
Force proportional to displacement gives a parabolic potential — the signature of harmonic motion.
The word 'force' is a classical idea. In quantum mechanics what really matters is the potential energy V(x) it comes from; the parabolic V, not a force pushing a particle along a path, is what enters the Schrödinger equation.