Energy Operator

We can try and understand the Energy Operator through the example of the Infinite square well. The basis vector, which is also the solution to the Schrodinger equation for this is given by:

$${\phi }_n(x)=\sqrt{\frac{2}{L}}sin\left(\frac{n\pi x}{L}\right)$$

Where the Schrodinger equation is then

$$\left[-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)\right]\phi (x)=E\phi (x)$$

The content within brackets, $\left[-\frac{{\hslash }^2}{2m}\frac{d^2}{d{x}^2}+V(x)\right]$, constitutes the Energy Operator, also denoted by $\hat{H}$.

See also: standard Energy Eigenvalue Equation: