Q.Explain briefly the time independent Schrödinger wave equation.

Since classical mechanics needs simultaneous, precise knowledge of both position and momentum - exactly what Heisenberg's principle rules out for a microscopic particle - a fundamentally different framework, quantum mechanics, is needed to describe the electron. Erwin Schrödinger expressed the electron's wave behaviour as a differential equation. Its time-independent form is

where $\hat{H}$ is the Hamiltonian operator (combining kinetic and potential energy), $\psi$ is the wavefunction (a function of position, $\psi(x,y,z)$), and $E$ is the total energy of the system. Written out explicitly for a particle of mass $m$ in a potential $V$, this becomes the partial differential equation

The crucial mathematical property of this equation is that well-behaved solutions exist only for certain special, discrete values of $E$ - so energy quantisation falls directly out of the mathematics, rather than being assumed the way Bohr had to assume it. These permitted energies are called eigenvalues, and their corresponding wavefunctions $\psi$ are what we call atomic orbitals. The equation is exactly solvable for one-electron systems (H, $\text{He}^+$, etc.) but becomes far too complex to solve exactly once more than one electron - and therefore electron-electron repulsion - is present. …