Explain the Schrödinger wave equation and also explain $\psi$ and $|\psi|^{2}$ for a one-electron system.

(N/A) The Schrödinger wave equation describes the behavior of electrons in atoms as waves. When solved for a hydrogen atom:
$(i)$ The solution provides the possible energy levels the electron can occupy and the corresponding wave function$(s)$ $(\psi)$ associated with each energy level.
$(ii)$ These quantized energy states and wave functions are characterized by three quantum numbers: principal $(n)$,azimuthal $(l)$,and magnetic $(m_{l})$.
$(iii)$ The wave function contains all the information about the electron in a given energy state.
Definition of $\psi$: The wave function is a mathematical function whose value depends on the coordinates $(x, y, z)$ of the electron in the atom. It does not have direct physical significance.
One-electron system: Wave functions for hydrogen or hydrogen-like species (e.g.,$He^{+}$,$Li^{2+}$) containing only one electron are known as atomic orbitals.
Meaning of $|\psi|^{2}$: The value $|\psi|^{2}$ represents the probability density of finding an electron at a specific point within an atom. It is always a positive quantity.
For one-electron species,the energy of the orbitals depends solely on the principal quantum number '$n$'.