Quantum effects$V(x)$ must show an appreciable relative variation over a distance of the order of a wavelength.

Square Potentials → discontinuities of the first kind.

Schrödinger equation$\psi''+(\epsilon - U)\psi = 0$

$\epsilon > U_i$ → linear combination of imaginary exponentials:

$Ae^{ik_ix}+ Be^{ik_ix}$

$\epsilon < U_i$ → linear combination of real exponentials:

$Fe^{\kappa_ix}+ Ge^{\kappa_ix}$

The parameters of these combinations are fixed by the condition of continuity of the wavefunction and of its first derivative at the points of discontinuity of the potential.

Eingenfunction → solution bounded everywhere (at both $x=-\infty$ and $x=+\infty$).

Energy $\epsilon$ lower than the potential over the entire interval ($-\infty$; $+\infty$) →No solutions