word_count: 200

The Schrodinger equation Hpsi = Epsi replaces Bohr's model. Solutions: wave functions psi(r,theta,phi) = R(r)Y(theta,phi). |psi|^2 = probability density. Orbital = 3D region enclosing ~90% probability. Radial distribution P(r) = 4pir^2R^2(r) shows probability at distance r. For 1s: maximum P(r) at r = a_0 (Bohr radius). Quantum numbers n, l, m_l emerge naturally from solving Schrodinger equation; m_s is added from relativistic quantum mechanics. Multi-electron atoms: orbital energies depend on both n and l. Effective nuclear charge Z_eff = Z - sigma (Slater's rules). Shielding: inner electrons reduce Z_eff for outer electrons. Penetration: s > p > d > f. Greater penetration means less shielding and lower energy. This explains why 4s fills before 3d: 4s has better penetration in neutral atoms. In cations, 3d drops below 4s because increased Z_eff favours d electrons. The quantum model successfully explains multi-electron atoms, periodic trends, chemical bonding, and spectral fine structure.