Particle in an Infinite Vee Potential

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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 .

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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.

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Contributed by: S. M. Blinder (November 2010)
Open content licensed under CC BY-NC-SA


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