Question

The particle-in-a-box (n), the particle on a ring (ml) and the particle on a sphere (l + ml) successively increased what aspect of the restriction of motion

Answer 1

The particle in a box is restricted to move in one dimensional space. Hence, its energy e-values are a function of just the principal quantum number (n). For a particle on a ring, the movement happens in 2D space and so the energy e-values are degenerate in nx and ny. This gives rise to the the next quantum number, that is the angular momentum quantum number (l). For a particle moving on a sphere, the movement happens in 3D. This leads to a threefold degeneracy of the energy e-values and the magnetic quantum number ml is introduced.

Therefore, the three cases are in increasing order of the complexity of the motion in terms of spatial dimensions to be considered. This means the spatially resitrictions in the particle is maximum for the first case and none for the last case.