Schrödinger Equation

The Schrödinger equation is one of the most fundamental equations of quantum physics. It cannot be derived from a more fundamental concept, but it is possible to make the correctness plausible by using the equation of energy conservation. Assuming a particle moving in a potential $V(x,t)$, total energy is given as the sum of the kinetic and potential energy: $$E = E_\mathrm{kin} + V$$ The kinetic energy can then be replaced by the momentum according to the equation: $$E_\mathrm{kin} = \frac{p^2}{2m}$$ In the next step, we can apply the following replacements for the momentum: $$p\rightarrow i\hbar \vec{\nabla}$$ and for the energy: $$E\rightarrow i\hbar \frac{\partial}{\partial t}$$ The resulting equation would make only sense if both sides are applied with a wave function $\psi(x,t)$ which leads to the general form of the time-dependent Schrödinger equation: $$\boxed{\left(-\frac{\hbar}{2m}\vec{\nabla}^2 + V\right)\psi(x,t) = i\hbar\frac{\partial}{\partial t}\psi(x,t)}$$ In case of a stationary potential, this equation simplifies to the time-independent Schrödinger equation: $$\boxed{\left(-\frac{\hbar}{2m}\vec{\nabla}^2 + V\right)\psi(x) = E\psi(x)}$$