Question

three identical spin -1/2 fermions are to be distributed in two non degenerate distinct energy levels the number of ways this can be done is pls give solution ans is 2

Answer 1

Let's denote the energy levels as $E_1$ and $E_2$. According to the Pauli exclusion principle, two or more fermions can not occupy the same state. This, there are two options here (I do not know which one is implied by the task).

1. Particles can not change their spin.

Then, iff we place one particle to $E_1$, the other two should be placed to $E_2$. But since these tow have the same spin -1/2 this would violate the Pauli exclusion principle. The same argument can be applied to the state $E_2$. Thus, the number of ways to distribute these particles is 0.

2. Particles can change their spin.

Then there are 4 possible options:

1. Two particles at $E_1$ with spins 1/2 and -1/2. One particle at $E_2$ with spin 1/2.

2. Two particles at $E_1$ with spins 1/2 and -1/2. One particle at $E_2$ with spin -1/2.

3. Two particles at $E_2$ with spins 1/2 and -1/2. One particle at $E_1$ with spin 1/2.

4. Two particles at $E_2$ with spins 1/2 and -1/2. One particle at $E_1$ with spin -1/2.

Answer. 0 or 4 depending on whether the particles can change their spins or not.