Write down the Schrodinger wave equation for the hydrogen atom.

(N/A) The Schrodinger wave equation for a system like the hydrogen atom is given by the operator equation: $\hat{H}\Psi = E\Psi$,where $\hat{H}$ is the Hamiltonian operator,$\Psi$ is the wave function,and $E$ is the total energy of the system.
When this equation is solved for the 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 a set of three quantum numbers: principal quantum number $(n)$,azimuthal quantum number $(l)$,and magnetic quantum number $(m_{l})$.
$(iii)$ The wave function $(\Psi)$ 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 of the electron in the atom. It does not have direct physical meaning,but its square,$|\Psi|^{2}$,represents the probability density of finding an electron at a point.
One-electron system: The wave functions for hydrogen or hydrogen-like species $(He^{+}, Li^{2+}, \dots)$ are called atomic orbitals.
The quantum mechanical model successfully predicts all aspects of the hydrogen atom spectrum,including phenomena that could not be explained by the Bohr model.