Simply Lecture

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) + ...

Recall that classical energy is given by the equation: $$E_c = \frac{p^{2}}{2m} + V(r) + V_0$$ ........(1) Where, V( r ) = spatially dependent potential              \(V_0\) = constant potential energy Recall from the square of the magnitude of the momenergy 4-vector associated with quantum object is given by: $$E^{2} - p^{2} = m^{2}$$ ..........(2) Divide equ (2) by 2m $$\frac{E^{2}}{2m} - \frac{p^{2}}{2m} = \frac{m^{2}}{2m}$$ $$\frac{E^{2}}{2m} - \frac{p^{2}}{2m} = \frac{m}{2}$$ Making \(\frac{p^{2}}{2m}\) subject of the formula, we have: $$\frac{p^{2}}{2m} = \frac{E^{2}}{2m} - \frac{m}{2}$$ .........(3) Also let: $$V_0 = \frac{m}{2} + E_c - E$$ ............(4) Substitute equ (4) and equ (3) into equ (1), we have: $$E_c = \frac{E^{2}}{2m} - \frac{m}{2} + V(r) + \frac{m}{2} + E_c - E$$ Making \(\frac{E^{2}}{2m}\) subject of the formula, we have: $$\frac{E^{2}}{2m} = E_c - E_c + \frac{m}{2} - \frac{m}{2} - V(r) + E$$ $$\frac{E^{2}}{2m}...