Franz Schwabl Quantum Mechanics 4th Edit · Problem 3.20

Angular Momentum

Physics — Quantum Mechanics · well-with-delta-barriers-transmission

Problem

Calculate the transmission coefficient S(E)S(E) for scattering from the potential V(x)=−V0 Θ(a−∣x∣)+λ (δ(x−a)+δ(x+a))  ,V(x) = -V_0\,\Theta(a-|x|) + \lambda\,(\delta(x-a)+\delta(x+a))\;, where λ≥0\lambda \geq 0 and Θ(x)\Theta(x) is the step function. (The δ\delta-peaks simulate Coulomb barriers in realistic problems.) Investigate the change in the form of the resonances in comparison with the case λ=0\lambda = 0 and discuss ∣λ/V0∣→∞|\lambda/V_0| \to \infty.
Intuitive explanation

This problem combines a finite square well with delta-function barriers at its edges—a model where you must match wavefunctions across both smooth and singular potentials. The transmission coefficient emerges from boundary conditions at three locations: the two delta peaks (which impose discontinuous derivative jumps) and the well boundaries. The resonances in S(E)S(E) reveal when quasi-bound states nearly form; adding λ\lambda shifts and broadens these peaks by modifying the effective barrier height. As λ/V0\lambda/V_0 grows large, the delta barriers dominate, suppressing transmission and reshaping resonance structure—watch how the well's role diminishes.

Intuitive explanation

This problem combines a finite square well with delta-function barriers at its edges—a model where you must match wavefunctions across both smooth and singular potentials. The transmission coefficient emerges from boundary conditions at three locations: the two delta peaks (which impose discontinuous derivative jumps) and the well boundaries. The resonances in S(E)S(E) reveal when quasi-bound states nearly form; adding λ\lambda shifts and broadens these peaks by modifying the effective barrier height. As λ/V0\lambda/V_0 grows large, the delta barriers dominate, suppressing transmission and reshaping resonance structure—watch how the well's role diminishes.

Related theory

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