Problem 2.52 Consider the potential V (x) : ka? -sech²(ax), m 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 Vo (x) = A sech(ax), and find its energy. Normalize vo, and sketch its graph. (c) Show that the function ´ik Yk (x) = A a tanh(ax) ik + a (where k = /2m E/h, as usual) solves the Schrödinger equation for any (positive) energy E. Since tanh z → -1 as z → -0, Vk (x) ~ Aekx, for large negative xr. 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 Yk (x) at large positive x? What are R and T, for this potential? Comment: This is a famous example of a reflectionless potential–every incident particle, regardless its energy, passes right through.

Problem 2.52 Consider the potential V (x) : ka? -sech²(ax), m 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 Vo (x) = A sech(ax), and find its energy. Normalize vo, and sketch its graph. (c) Show that the function ´ik Yk (x) = A a tanh(ax) ik + a (where k = /2m E/h, as usual) solves the Schrödinger equation for any (positive) energy E. Since tanh z → -1 as z → -0, Vk (x) ~ Aekx, for large negative xr. 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 Yk (x) at large positive x? What are R and T, for this potential? Comment: This is a famous example of a reflectionless potential–every incident particle, regardless its energy, passes right through.