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

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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'?

This post imported from StackExchange Physics at 2015-11-12 12:46 (UTC), posted by SE-user Xiaoyi Jing
asked Aug 10, 2015 in Theoretical Physics by Xiaoyi Jing (10 points) [ no revision ]
retagged Nov 12, 2015
Why do you think "internal energy" and "potential energy" are different things in this case?

This post imported from StackExchange Physics at 2015-11-12 12:46 (UTC), posted by SE-user ACuriousMind
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.

This post imported from StackExchange Physics at 2015-11-12 12:46 (UTC), posted by SE-user Xiaoyi Jing

1 Answer

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Galilei invariance implies that there is no external potential energy. Thus any contribution to a Galilei invariant energy must come from the internal structure.

Any nonrelativistic scattering state in a Galilei invariant interacting nonrelativistic $N$-particle system can be interpreted in terms of a cluster decomposition. Each cluster represents a bound state with mass $m_i$ (the sum of the masses of its constituent particles), internal energy $E_i$ (an eigenenergy of the cluster Hamiltonian in the rest frame of the cluster, corresponding to the internal state of the cluster), and momentum $p_i$ (for the cluster's center of mass motion relative to the origin of the whole system), and contributes to the energy of the scattering state the additive energy term $E_i+p_i^2/2m_i$.

If there is an additional external potential, all cluster Hamiltonians get an extra term with the potential energy, and the above description gives only the zeroth order in perturbation theory. But this is enough to see that in general, the internal energy is an additional shift in the potential energy expressing the contribution of the internal state of the cluster to the total energy.

answered Nov 12, 2015 by Arnold Neumaier (12,355 points) [ no revision ]

Cannot we have an external potential that is Galilean invariant?

An external potential is space and possibly time dependent. Galilei invariance forces this to be a constant. This constant can be absorbed into the internal energy.

An internal potential $U(|\mathbf{r}_1-\mathbf{r}_2|)$ is invariant, but an external is not if it's not a constant. However, mechanics works in an external potential too ;-).

@VladimirKalitvianski: It is not a matter of working mechanics. A nonconstant external potential breaks the symmetry group. Thus the Casimir's the PO was talking about no longer apply.

@AnroldNeumaier: It means the CM contains everything, often implicitly, what some limiting cases make evident via funny things like Casimir's one, Noether one, etc. In particular, the case without external potential is the same as the case with it when the kinetic energy is much larger, roughly speaking. (Strictly speaking, one has to compare kinetic energy relative variations in an external field).

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