The terminology "spin up" and "spin down" used to describe the spin of a quantum particle is not directly related to the concept of clockwise or anticlockwise rotation. Instead, it refers to the two possible states or orientations of the particle's spin along a particular axis.
In quantum mechanics, the spin of a particle is an intrinsic property that doesn't have a classical analog. It is a quantum mechanical angular momentum that characterizes the particle's intrinsic rotation, but not in the sense of a physical rotation in three-dimensional space.
When discussing the spin of a particle, we typically choose a reference axis, often denoted as the z-axis. Along this axis, the spin can be measured as either "up" or "down." These terms are simply labels for the two possible eigenstates of the spin operator along the chosen axis.
The choice of "up" and "down" for spin states is arbitrary and not related to any physical rotation. They are simply convenient labels that have been adopted to describe the eigenstates of the spin operator. The spin states could have been labeled differently, such as "left" and "right" or "clockwise" and "anticlockwise," but the convention of "up" and "down" has become widely used in the field of quantum mechanics.