Mathematically, the time-independent Schrödinger equation is an example of an eigenvalue problem whose eigenvectors ψ are called 'wavefunctions' or 'quantum states' and whose eigenvalues E correspond to energy levels.
It should also be noted that the fact that the time independent equation takes the form of a simple eigenvalue problem (thus being more amenable to mathematical analysis) that makes it so useful. The conditions that must be met to enable this separation of variables arise very frequently in chemistry and physics. The second one is an eigenvalue equation, written commonly as Hψ = E ψ. Actually there are two different equations known as Schrödinger equation: The first is sometimes called the time-dependent, while the second is called the time independent equation, though in reality, it is simply an equation derived from the first using a mathematical technique known as separation of variables. Austrian physicist Erwin Schrödinger first proposed the equation in early 1926. The Schrödinger equation is one of the fundamental equations of quantum mechanics and describes the spatial and temporal behavior of quantum-mechanical systems.
