Final answer:
The normalized spherical harmonics for a spin-zero particle are D. uniformly distributed in space because the particle has an s-orbital with zero angular momentum, indicated by the quantum number l = 0, and s-orbitals are spherically symmetric with no directional preference.
Step-by-step explanation:
The normalized spherical harmonics associated with a spin-zero particle are uniformly distributed in space. The quantum number l for a spin-zero particle is 0, indicating zero angular momentum, which corresponds to an s-orbital. A characteristic of s-orbitals is their spherical symmetry, which means they have a uniform electron probability distribution in all directions around the nucleus. As we refer to the radial probability density function P(r), it shows the likelihood of finding the electron at any point within a spherical shell of radius r, leading to a uniform distribution.
The spherical harmonics for a particle with a principal quantum number n and an angular momentum quantum number l = 0 (indicating an s-orbital) do not show pronounced distributions in certain directions, unlike higher angular momenta orbitals such as p, d, or f orbitals. Therefore, the electron probability clouds of s-orbitals are spherical and do not exhibit directional preference, leading to a uniform spatial distribution.