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Kronig-Penney model

For
$$
V(x)=-\frac{\hbar^2\lambda}{m}\sum_{n\in\mathbb Z}\delta(x-na),
$$
the <wavefunction> is continuous across every lattice point and obeys
$$
\psi'(na^+)-\psi'(na^-)=-2\lambda\psi(na).
$$
At negative energy $E=-\hbar^2\mu^2/(2m)$, an <allowed energy band> satisfies
$$
\boxed{\lambda\tanh\frac{\mu a}{2}<\mu<\lambda\coth\frac{\mu a}{2}.}
$$
At positive energy $E=\hbar^2k^2/(2m)$, it satisfies
$$
\boxed{\left|\cos(ka)-\frac\lambda k\sin(ka)\right|<1.}
$$

Attractive delta-comb Kronig-Penney model

Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 34A / Solution

Attractive delta-comb Kronig-Penney model

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Attractive delta-comb Kronig-Penney model

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