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I have a question regarding the 2 Fermions in Two-level system and the possible states and resulting partition function.
As far as I know, Fermions are forbidden from occupying same quantum states by Pauli exclusion principle. Thus the possible states should look like, $$ \underline{\hspace{1cm}} \quad \underline{\uparrow \hspace{0.5cm}} \quad \underline{\downarrow \hspace{0.5cm}} \quad \underline{\uparrow \hspace{0.5cm}} \quad \underline{\downarrow \hspace{0.5cm}} \quad \underline{\uparrow\downarrow\hspace{0.6cm}}\\ \underline{\uparrow \downarrow \hspace{0.6cm}} \quad \underline{\downarrow \hspace{0.5cm}} \quad \underline{\downarrow \hspace{0.5cm}} \quad \underline{\uparrow \hspace{0.5cm}} \quad \underline{\uparrow \hspace{0.5cm}} \quad \underline{\hspace{1cm}} $$

But from the solutions, I saw it should look like below. $$ \underline{\hspace{1cm}} \quad \underline{\uparrow \hspace{0.5cm}} \quad \underline{\downarrow \hspace{0.5cm}} \quad \underline{\uparrow\downarrow\hspace{0.6cm}}\\ \underline{\uparrow \downarrow \hspace{0.6cm}} \quad \underline{\downarrow \hspace{0.5cm}} \quad \underline{\uparrow \hspace{0.5cm}} \quad \underline{\hspace{1cm}} $$

Shouldn't all the spin configuration be possible for the fermions?

jkhadka
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2 Answers2

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I do not know which "solutions" you are talking about, thus I cannot judge why such solutions would only allow 4 states for this system, but in all generality, you are right, and six states are allowed. The spins can be parallel if the fermions are lying on different energy levels.

Dominique Geffroy
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Yes. In general there must be the six states you mentioned. However, one common feature among the 4 states given in the “solution” is that they have total spin of 0. The other two states have a total spin of 1.

In case the system is constrained to have a total spin of 1, then only those 4 states will be allowed.