= Solution
Let the atomic spacing be $a$ and define the nearest-neighbour matrix elements $t_\alpha=\langle j,p_\alpha|H|j+1,p_\alpha\rangle$, using the same translated orbital convention on every site. Then
$$
E_\alpha(k)=\epsilon_\alpha+2t_\alpha\cos(ka),\qquad-\pi/a\leq k\leq\pi/a.
$$
For 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 $t_\pi<0$ and
$$
\boxed{E_{p_z}=\epsilon_z-2|t_\pi|\cos(ka):\quad\text{minimum at }k=0.}
$$
For $p_x$, the lobes facing one another have opposite signs for equal site coefficients. Bonding therefore needs alternating coefficients, giving $t_\sigma>0$ and
$$
\boxed{E_{p_x}=\epsilon_x+2|t_\sigma|\cos(ka):\quad\text{minimum at }k=\pm\pi/a.}
$$
End-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.<Solid State Physics by David Tong>[Chapter 2: nearest-neighbour tight binding and Bloch sums]{ref}
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2015/paper-1-p-orbital-bands.png]
{title=End-on and side-on p-orbital hopping give opposite tight-binding band orientations}
{source=paper-1-p-orbital-bands.py}
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