Past exam of the advanced Quantum Condensed Matter Physics course of the University of Cambridge / 2015 / Paper 1 / 3 / b / Solution

Expand the field operator in orthonormal localized orbitals, $\hat{\psi }(\mathbf{r})={\sum }_i{\varphi }_i(\mathbf{r})c_i$. For a one-electron continuum Hamiltonian $h_0={\hat{p}}^2/(2m^{\ast })+V(\mathbf{r})$, projection givesThe coefficient in the displayed lattice Hamiltonian is $-t_{12}$. Thus the projected hopping matrix element and sign convention requireOrthogonality states $\int {\varphi }_1^{\ast }{\varphi }_2=0$; it does not imply $\int {\varphi }_1^{\ast }V{\varphi }_2=0$ for a spatially varying potential. If the off-diagonal potential is assumed zero, for example when it is a constant, the hopping coefficient is $h_{12}=K_{12}$. This is the kinetic integral requested by the paper, but it equals $-t_{12}$ under the paper's own leading-minus convention. A convention writing the hopping term with a plus sign would instead call $h_{12}$ itself $t_{12}$.

TODO: the printed positive equality for $t_{12}$ requires a missing assumption about $V_{12}$ and a consistent hopping-sign convention. It does not follow from orthogonality alone. The general projected relation above is the result supported by the supplied information. Rephasing an orbital changes both $h_{12}$ and the coefficient of the corresponding operator together; it does not remove this algebraic sign mismatch in a fixed convention. The spectrum in the remaining two-site calculation depends on $|t{|}^2$, so the missing convention does not prevent solving those parts.