4: One-Dimensional Potentials
( \newcommand{\kernel}{\mathrm{null}\,}\)
In this chapter, we shall investigate the interaction of a non-relativistic particle of mass m and energy E with various one-dimensional potentials, V(x). Because we are searching for stationary solutions with unique energies, we can write the wavefunction in the form (see Section [sstat]) ψ(x,t)=ψ(x)e−iEt/ℏ,
where ψ(x) satisfies the time-independent Schrödinger equation: d2ψdx2=2mℏ2[V(x)−E]ψ.
In general, the solution, ψ(x), to the previous equation must be finite, otherwise the probability density |ψ|2 would become infinite (which is unphysical). Likewise, the solution must be continuous, otherwise the probability current ([eprobc]) would become infinite (which is also unphysical).
