Why isn't translational energy quantized? I was reading over this article on spectroscopy/molecular energy levels: https://en.wikiversity.org/wiki/Spectroscopy/Molecular_energy_levels. It says that translational energy is not quantized. Is this always true, or is it only true for a free particle?
 A: The potential energy is typically a function of the relative coordinates, v.g. $\vert x_1-x_2\vert$ (assuming 2 coordinates for simplicity).  Thus by going to the center of mass $X$ and relative coordinates $x$ the Hamiltonian separates
$$
H=-\frac{\hbar^2}{2(m_1+m_2)}\frac{\partial^2}{\partial X^2}
-\frac{\hbar^2}{2\mu}\frac{\partial^2}{\partial x^2} + V(x)
$$
with $\mu$ the reduced mass, so that the solution to the center of mass (or translational) degree of freedom is just the free-particle solution
\begin{align}
-\frac{\hbar^2}{2(m_1+m_2)}\frac{\partial^2}{\partial X^2}\psi_{CM}(x)&=E_{CM}\psi_{CM}(x)\\
\psi_{CM}(X)&=e^{iPX/\hbar}\, ,\qquad \frac{\hbar^2 P^2}{2(m_1+m_2)}=E_{CM}
\end{align}
which, like a free particle, has no restriction on $E$ and is thus not quantized.
A: The translational energy depends on the momentum, which is is a continuous parameter (as the Fourier transform of position, which is also continuous). For a particle restricted to live on a finite lattice or a box with periodic boundary conditions, the allowed momentum values will be quantized, but for a macroscopic system the allowed momenta will be incredibly close together and effectively continuous.
