2.2: Energy Function

Of course the kinetic energy is $\frac{1}{2} m v^2$, with $v=r$

$$E=\frac{1}{2} m v^2+V(r)$$

Actually, this form is not very convenient for quantum mechanics. We rather work with the so-called momentum variable $p=m v$. Then the energy functional takes the form

$$E=\frac{1}{2} \frac{p^2}{m}+V(r)$$

The energy expressed in terms of $p$ and $r$ is often called the (classical) Hamiltonian, and will be shown to have a clear quantum analog.