The particle in a periodic potential is an important topic in an undergraduate quantum mechanics
curriculum and a stepping stone on the way to more advanced topics, such as courses on interacting
electrons in crystalline solids, and graduate-level research in solid-state and condensed matter
physics. The interacting many-body phenomena are usually described in terms of the second quantized
lattice Hamiltonians which treat single-particle physics on the level of tight-binding approximation
and add interactions on top of it. The aim of this paper is to show how the tight-binding tunneling
amplitude can be related to the strength of the periodic potential for the case of a cosine
potential used in the burgeoning field of ultracold atoms. We show how to approach the problem of
computing the tunneling amplitude of a deep lattice using the JWKB
(Jeffreys–Wentzel–Kramers–Brillouin, also known as semiclassical) approximation. We also point out
that care should be taken when applying the…