Past exam of the quantum condensed matter course of the University of Cambridge / 2015 / Paper 1 / 1 / b / Solution

Let the atomic spacing be $a$ and define the nearest-neighbour matrix elements $t_{\alpha }=⟨j,p_{\alpha }|H|j+1,p_{\alpha }⟩$, using the same translated orbital convention on every site. ThenFor the opposite sigma and pi hopping signs in a p-orbital chain, consider $p_z$ first: neighbouring lobes above the chain have the same sign, so the in-phase Bloch sum is bonding. Thus $t_{\pi }<0$ andFor $p_x$, the lobes facing one another have opposite signs for equal site coefficients. Bonding therefore needs alternating coefficients, giving $t_{\sigma }>0$ andEnd-on $\sigma$ overlap is normally stronger than side-on $\pi$ overlap, so the $p_x$ band is wider, with bandwidths $4|t_{\sigma }|$ and $4|t_{\pi }|$. The curve orientations require a stated hopping-sign convention; a Hamiltonian written with an explicit minus sign in front of the hopping uses correspondingly opposite parameter signs.^[ref]