Consider the potential

where a is a positive constant, and “sech” stands for the hyperbolic secant.
(a) Graph this potential.
(b) Check that this potential has the ground state

and find its energy. Normalize Ψ₀, and sketch its graph.
(c) Show that the function

(where as usual) solves the Schrödinger equation for any(positive) energy E. Since tanh z → -1 as z → - ,

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)). What is the asymptotic form of Ψ_k (x) at large positive x? What are R and T, for this potential? This is a famous example of a reflectionless potential—every incident particle, regardless its energy, passes right through.

Transcribed Image Text:

V (x) = ħ²a² m -sech² (ax),