8. Plot the radial probability densities of 2p, 3p, and 4p orbitals in a single plot altogether. Explain their characteristic features and compare those to each other.

8. For two angular momenta of quantum numbers j₁ and j2, there are (2j1 + 1) × (2j2 + 1) possible products |j1,m,)|j2,m,,) of eigenstates of the individual angular momenta. Count all the possible eigenstates lj,mj) of the total angular momentum, and show that there are exactly (2j1 + 1) × (2j2 + 1) such eigenstates.

2. A wave function is a linear combination of 1s, 2s, and 3s orbitals: y(r) = N(0.25 y,s + 0.50W2, +0.30W). Find the normalization constant N, knowing that 1s, 2s, and 3s orbitals are normalized.

3. For free particles in two dimensions, what is density of states (DOS) in low speed limit (=p²/2m), and in high speed limit (=pc)?