I have been reading an article discussing the unitary representation of Galilean group and non-relativistic quantum mechanics. The link to the article is given below.

http://arxiv.org/abs/1107.2442

From equation 2.17a~2.17b, we see three Casimir operators. The authors claimed that the 1-particle states are therefore labeled by three numbers: mass, spin and internal energy.

Unfortunately, I do not understand why $\hat{H}-\frac{1}{2\hat{M}}\hat{P}^{2}$ should be interpreted as internal energy.

As far as I could remember from my bachelor QM course, the 1-particle states have nothing related with thermal dynamics or internal energy.

From the above paper, it seems that the energy (or Hamiltonian) is the internal energy plus kinetic energy. But from my bachelor QM, the Hamiltonian should be kinetic energy plus potential.

Can anyone help understand the 'internal energy'?