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Schrodinger equation

The Schrodinger equation in quantum mechanics, is a mathematical equation that describes the change over time of a physical system in which quantum effects,  such as wave-particle duality, are significant.
The equation is a mathematical formulation for studying quantum mechanical systems. It is used to find the allowed energy levels of the quantum mechanical system (atoms, or transistors). The associated wave function gives the probability of finding the particle at a certain position.
The Schrodinger equation exist in different forms:
I.  Energy form of Schrodinger equation:
$$\frac{E^{2}}{2m} = E - V(r)$$
II. The time independent Schrodinger equation:
$$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) + V(r) \psi(r) = E \psi(r)$$
III.  Time dependent Schrodinger equation:
$$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = i\hbar \frac{d\Psi(r,t)}{dt}$$
IV.  Auxiliary time dependent Schrodinger equation:
$$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = -i\hbar \frac{d\Psi(r,t)}{dt}$$
Check the links below for the derivation of the four forms of the Schrodinger equation:

Derivation of the energy form of Schrodinger equation.

Derivation of the time independent Schrodinger equation.

Derivation of the time dependent Schrodinger equation

Derivation of the auxiliary time dependent Schrodinger equation

Also look at:

Schrodinger equation as a law in physics
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