How does the wave-particle duality of electrons relate to the shape and energy levels of atomic orbitals?

Electrons behave like waves, so when they are confined to an atom they form standing‑wave patterns that satisfy the Schrödinger equation. The allowed standing waves are only those whose wavelengths fit exactly around the nucleus, which gives discrete energy levels and specific shapes for the orbitals. For example, the lowest‑energy “s” orbital is a spherical standing wave with no nodes, while the next “p” orbital has a dumbbell shape with one node, matching the electron’s wave pattern. Because the wave must fit an integer number of half‑wavelengths, only certain energies are possible, explaining why electrons occupy distinct energy shells. Thus, the wave‑particle duality directly determines both the geometry and the quantized energy of atomic orbitals.