Here are the MOs (more accurately, the breakdown of the MOs) for conjugated hexatriene:
Now, when first asked to draw the MOs myself, I drew this one for $\psi_3$:
It still has two nodes, and is still symmetric. I see nothing wrong with it, except that it doesn't match the "correct" MOs.
Similarly, for $\psi^*_4$, I can draw
Now, as I understand it, MOs are formed from various combinations of atomic orbitals, subject to a few rules regarding symmetry. From what I know, the two MOs I've given above should be valid MOs for hexatriene, or at least components of an MO of hexatriene.
Why aren't these two MOs valid? Also, is there a methodical way to obtain the sequence of MOs in the general case(of a conjugated polyene)?