I am learning quantum chemistry. To have a comprehensive understanding of the Slater determinant, I studied the classical problem of two indistinguishable particles in a 1D box with infinite barriers.
At the beginning, it was very clear. But now, I am confused. When dealing with electrons (fermions), the wave function must be anti-symmetric. Two electrons could not be at the same place, .... except that the spin is not taken in account. Some explanation latter, two electrons could be at the same place if their spins are anti parallel. So the spatial part of the wave function is symmetric and the spin part is anti symmetric. The whole wave function is therefore anti symmetric. It is the singlet state.
And what about, if the triplet state is studied ? I understand that the spin part of the wave function is symmetric and thus, the spatial part must be antisymmetric. But there are three states in a triplet state. So what is the total wave function ?
And to go further, if there are three electrons in the infinite well, what are the wave functions for all state (ground and excited) ? How can I add three spins and what is the symmetry of the resulting spin part of the wave function ? ...
To sum up my question, where can I find for example, a complete and detailed demonstration of two and three fermions in a 1D box that takes in account the spatial and the spin parts of the wave function. By complete and detailed, I mean without justification "by hand-waving arguments" as above (i.e. if the spin part is symmetric, the spatial part is anti symmetric and conversely). For ground state, I understand and it justifies for example Hund's rules. But I am not able to derive equations for excited states. So, this is why I am confused and I realize that I haven't understood the root of the problem.
Thanks for answer
