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In summary: So it's not just a vibration of the atoms, it's a vibration of the whole system. Right. So when we talk about many states corresponding to a certain energy, we're not just talking about the number of different vibrational modes that the system can have, we're also talking about the number of allowed quanta of energy that can be assigned to each of those modes.In summary, Reif talks about the energy states of a system as corresponding to vibrations of the atoms within the system. He provides an example of a three particle system and explains that there are only 4 states that the system can be in.

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It might be a bit easier to understand a much simpler system, say only three particle, each one can have energy of 1,2,3,4,5,6 J (like dice).

If the particles are classical, there are the following states with the total energy of 7 J (using the notation (x,y,z) where x, y, z, are, respectively, energies of the first, second and third particle):

(1,1,5), (1,5,1), (5,1,1), (2,2,3), (2,3,2), (3,2,2), (1,3,3), (3,1,3), (3,3,1), (1,2,4), (1,4,2), (2,1,4), (2,4,1), (4,1,2), (4,2,1)

Therefore, we have 15 possible states of the three particle system and each of them gives us a total energy of 7 J.

This is the case when the particles are distinguishable. If these are idential quantum particles, they are not distinguishable. That means that the state with two particles of energy 1 J and the third with energy of 5 J is the same as the state with the first particle of energy 5 J and the remaining 2 of energy of 1 J ( you can't tell which is the first and which is the third particle!).

So, in quantum statistics, there are only the following states:

(1,1,5), (2,2,3), (1,3,3), (1,2,4), only 4 states.

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- Three degrees of freedom from translational velocity of its centre of mass,
- Two degrees of freedom from rotational frequency about axes perpendicular to the molecule's primary axis,
- One degree of freedom from vibrations between the two atoms.

Each degree of freedom can accept certain quanta of energy; the number of states is the total number of ways you can assign these quanta to individual components.

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I see, thanks for confirming that.

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