Introduction to Quantum Mechanics · Problem 2.52

Time-Independent Schrödinger Equation — David J. Griffiths

Physics — Quantum Mechanics · reflectionless_potential

Problem

Consider the potential V(x)=−ℏ2a2msech⁡2(ax),V(x) = -\frac{\hbar^2 a^2}{m}\operatorname{sech}^2(ax), where aa is a positive constant, and "sech" stands for the hyperbolic secant. (a) Graph this potential. (b) Check that this potential has the ground state ψ0(x)=Asech⁡(ax),\psi_0(x) = A\operatorname{sech}(ax), and find its energy. Normalize ψ0\psi_0, and sketch its graph. (c) Show that the function ψk(x)=A(ik−atanh⁡(ax)ik+a)eikx,\psi_k(x) = A\left(\frac{ik - a\tanh(ax)}{ik + a}\right)e^{ikx}, (where k≡2mE/ℏk \equiv \sqrt{2mE}/\hbar, as usual) solves the Schrödinger equation for any (positive) energy EE. Since tanh⁡z→−1\tanh z \to -1 as z→−∞z \to -\infty, ψk(x)≈Aeikx,for large negative x.\psi_k(x) \approx Ae^{ikx}, \quad \text{for large negative } x. This represents, then, a wave coming in from the left with no accompanying reflected wave (i.e. no term exp⁡(−ikx)\exp(-ikx)). What is the asymptotic form of ψk(x)\psi_k(x) at large positive xx? What are RR and TT, for this potential? Comment: This is a famous example of a reflectionless potential—every incident particle, regardless its energy, passes right through.

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