Why do the \(3 s, 3 p,\) and \(3 d\) orbitals have the same energy in a hydrogen atom but different energies in a many-electron atom?

Hydrogen is known as a one-electron atom which means it consists of one electron orbiting a single proton. In this case, the energy levels of the orbitals only depends on the principal quantum number, n. Therefore, all orbitals having the same value of n have the same energy. In hydrogen atom, orbitals such as 3s, 3p and 3d all have the same principal quantum number, 3. Thus, in hydrogen atom, these orbitals possess the same energy.

In many-electron atoms, the energy of electrons does not solely depend on the principal quantum number. It is also influenced by electron-electron interactions and differences in penetration and shielding. According to the Pauli's exclusion principle, each electron in an atom is described by a unique set of quantum numbers. The principal quantum number defines the energy level of the electron and the possible values of the other three quantum numbers divide this energy level into sublevels. The shape of the orbital is determined by the azimuthal quantum number (l), which also affects the energy of the atom. For a many-electron atom the energies of 3s, 3p and 3d orbitals are different from each other due to different electron-electron interactions and differences in penetration and shielding. s orbitals are generally lower in energy than p orbitals, and p orbitals are lower in energy than d orbitals, because s orbitals have a greater ability to penetrate to the nucleus of the atom.