Particle in an Infinite Vee Potential

This Demonstration considers solutions of the Schrödinger equation for a particle in a one-dimensional "infinite vee" potential: , setting for simplicity. I believe this is the first published account of this problem. The solutions of the differential equation that approach zero as are Airy functions , as can be found using DSolve in Mathematica. The allowed values of are found by requiring continuity of at . The even solutions require , which leads to , with , , , … being the first, second, third, … zeros of the Airy prime function: . The odd solutions have nodes , which leads to , with , , , … being the first, second, third, … zeros of the Airy function: . The ground state is given by .

For user-selected and , the eigenfunctions , with normalization constants , are plotted as blue curves. The axis for each function coincides with the corresponding eigenvalue , the first ten values of which are shown on the scale at the right. The vertical scales are adjusted for optimal appearance.

A checkbox lets you compare the vee-potential eigenstates with the corresponding ones of the harmonic oscillator, drawn in red. The ground states of the two systems are chosen to coincide: . The harmonic oscillator is more confining, so its eigenvalues are more widely spaced. For higher values of , the oscillator functions might move off scale.