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The upper picture is my ideas which represent states by using the tensor product. but the lower picture, as you see, includes uppermost states. i don't know how to treat the uppermost states in lower picture. could you help me?

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I think what the exercise aims at is making the connection between the occupation-number representation to the tensor product representation.

The occupation-number representation only makes sense if the particles are indistinguishable, as it tells you "$n_g$ particles are in state g, $n_e$ particles are in state e, $n_u$ particles are in state u".

By contrast, the tensor product representation tells you which particle is in which state, for example $|u,u,g>$ means that the first and the second particle are in the uppermost state while the third particle is in the ground state. However, you have to take into account the indistinguishability of the quantum particles. Bosonic wavefunctions are symmetric under particle exchange. So "two particles in the uppermost state and one in the ground state", which is $|1,0,2>$ in the occupation-number representation would be $\frac{1}{\sqrt{3}}(|u,u,g>+|u,g,u>+|g,u,u>)$ in the tensor product representation.

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