Lecture 8 Q2.6 and 2.7 wave functions (notation and energy levels)

Physics Q&A
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Hi! I have two questions about exercises 2.6 and 2.7 from lecture 8:

  • I was wondering why the \phi_0 and \psi_0 are not left out of the wave function, because I thought we just fill in \phi_n=\phi_0e^{ikna} and \psi_n=\psi_0e^{ikna} in the wave function, with [\phi_0 \; \psi_0] = \frac{1}{\sqrt2}[1 \; \pm1]. Then \phi_0 and \psi_0 would not be in the wave function right?
  • Is there a general rule(s) for knowing which wave function has lowest energy? Especially for the wave functions in 2.7, I’m not sure how we know that the wave function which is even around the bond with largest hopping corresponds to the lowest energy level. Also, why are the energy levels of the k=\pi/a wavefunctions below the energy level of the k=0 wave function?

Thank you in advance!

The solutions:

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for the first part: see the solution of 6, which shows that the wavefunction |\Psi\rangle does indeed contain \phi_0 and \psi_0. These are the 2 components of the eigenvectors of the tight-binding matrix, and we find them to be [1 \quad \pm1] at k=0 and k=\pi/a

For the second question: the trick here is to not to remember a rule, although the node theorem does help here, but rather compare the systems to simpler systems of which you know the behaviour. For intance: try to recall how the difference in eigenenergies of the two-atom LCAO molecule depends on the strength of the hopping

To your last point: the eigenenergies are the eigenvalues of your tight-binding matrix, so once you know the eigenvectors you can quickly check which one has which energy

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