( i hbar del/ del t ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi
This equation is the backbone of quantum mechanics. If we consider a free particle, the potential energy of the particle is zero. Then this equation explains how a free particle evolves with time. The solution to this equation under a suitable potential always gives the time evolution of a quantum system under that potential. If we replace ( i hbar del/ del t ), by the energy operator E, the above equation takes a form:
( E ) psi = (- hbar^2/(2m) * laplacian^2) psi + V psi
This is time-independent (stationary) Scrodinger equation. The solution to this equation gives the stationary state of a system depending upon the nature of the potential. For example, if we take the potential to be parabolic, V = (1/2)m (omega)^2 (x)^2, in the time-independent schrodinger equation, we will get the stationary states of this potential, which is popularly known as the stationary states of a harmonic oscillator.