Operators

The Schrödinger equation comes directly out of our understanding of wave packets. To get from wave packets to a differential equation, we use the new concept of (linear) operators. We determine the momentum and energy operators by requiring that, when an operator for some variable $v$ acts on our simple wavefunction, we get $v$ times the same wave function.

$$\bgroup\color{black}p_x^{(op)}={\hbar\over i}{\partial\over\partial x}\egroup$$

$$\bgroup\color{black}p_x^{(op)}e^{i(\vec{p}\cdot\vec{x}-Et)/\h... ...ec{x}-Et)/\hbar} =p_xe^{i(\vec{p}\cdot\vec{x}-Et)/\hbar}\egroup$$

$$\bgroup\color{black}E^{(op)}=i\hbar{\partial\over\partial t}\egroup$$

$$\bgroup\color{black}E^{(op)}e^{i(\vec{p}\cdot\vec{x}-Et)/\hba... ...\vec{x}-Et)/\hbar} =Ee^{i(\vec{p}\cdot\vec{x}-Et)/\hbar}\egroup$$