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Remembering Square Wells…

December 16, 2008
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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

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