Schrodinger Equation

Schrödinger Equation

Schrödinger proposed that, in order for a corresponding plane-wave $\psi (x,t)=e^{i(k\cdot x-\omega t)}$ to describe the motion of this free particle, then $\psi$ ought to satisfy an equation involving a first-order time derivative and a second-order spatial derivative. Taking the Fourier transform of $E$ and $p$ suggests replacing $E$ and $p$ in the Energy-momentum relation by the differential operators $\hat{E}=i\hslash {\partial }_t\text{ and }\hat{p}=-i\hslash ({\partial }_{x^1},{\partial }_{x^2},{\partial }_{x^3}).$ This results in SCHRÖDINGER’S EQUATION (for a free particle), $i{\partial }_t\psi =-\frac{\hslash }{2m}\Delta \psi .$