This Demonstration considers solutions of the Schrödinger equation for a particle in a one-dimensional "infinite vee" potential:

, setting

for simplicity. 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.