Schrodinger equation

well, the equation #

$$ i\hbar \frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V\psi $$

$H\psi=E\psi$ H: Hamiltonian operator E: Energy operator

Intuition behind the Energy Operator #

https://youtu.be/QeUMFo8sODk?t=329

$$ \begin{align} E=T+V=\frac{p^2}{2m}+V\\ General\\ form\\ of\\ wave\\ equation:\\ \psi=e^{i(kx-wt)}\\ \lambda = \frac{2\pi}{k}\\ De-broglie\\ wavelength: \lambda=\frac{h}{p}\\ \frac{2\pi}{k}=\frac{h}{p}\Rightarrow k=\frac{p}{\hbar}\\ \\ (1)\\ \\ \frac{\partial \psi}{\partial x}=ik\psi\\ \frac{\partial^2 \psi}{\partial x^2}=(ik)^2\psi =-k^2\psi \\ =-\frac{p^2}{\hbar^2}\psi \\ [\because (1)] \\ -\hbar^2 \frac{\partial^2 \psi}{\partial x^2}=p^2 \psi \\ \\ (2)\\ \\ E=T+V\Rightarrow E\psi = T\psi + V\psi\\ \frac{p^2}{2m}\psi + V\psi \\ E\psi=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + V\psi \end{align} $$

Rearranging the terms in this equation, we get the time independent schrodinger equation.

Intuition behind the hamiltonian operator #

https://youtu.be/QeUMFo8sODk?t=363

$$ \begin{align} \psi = e^{i(kx-wt)}\\ w=2\pi v\\ and\\ E=hv=hw/2\pi=\hbar w\\\ \frac{\partial \psi}{\partial t}=-iw\psi=-i\frac{E}{\hbar}\psi \\ \Leftrightarrow i\hbar \frac{\partial \psi}{\partial t}=E\psi \end{align} $$

Equating this to the energy operator expression from , we get the time dependent schrodinger equation.