% This is the formula for calculating the expectation value of energy in quantum mechanics:
 
\begin{aligned}
\langle E \rangle &= \langle \psi | \hat{H} | \psi \rangle \\
                  &= \int_{-\infty}^{\infty} \psi^*(x) \left( -\frac{\hbar^2}{2m} \frac{d^2}{dx^2} + V(x) \right) \psi(x)  dx \\
                  &= -\frac{\hbar^2}{2m} \int \psi^* \frac{d^2 \psi}{dx^2} dx + \int V(x) |\psi(x)|^2 dx \\
                  &= \frac{\hbar^2}{2m} \int \left| \frac{d \psi}{dx} \right|^2 dx + \int V(x) |\psi(x)|^2 dx \quad \text{(after integration by parts)} \\
                  &= \langle T \rangle + \langle V \rangle
\end{aligned}