# Mass, Spin, Internal Energy and 1-Particle States in Galilean Quantum Mechanics

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'?

• Why do you think "internal energy" and "potential energy" are different things in this case? – ACuriousMind Aug 10 '15 at 20:26
• Hello ACuriousMind. I think internal energy is the energy of the internal structure of that particle. Am I right? But in QM, the potential is due to external force, which has nothing to do with internal energy in thermal dynamics. – Xiaoyi Jing Aug 10 '15 at 20:29
• I gave an answer at physicsoverflow.org/34237 – Arnold Neumaier Nov 12 '15 at 12:52