# Particle in an Infinite Vee Potential

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. 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 =0.808614.

-ψ''(x)+axψ(x)=Eψ(x)

1

2

ℏ=m=1

x±∞

ψ(x)=Ai(a|x|-E)

1/3

2

-2/3

a

E

ψ'(x)

x=0

n=0,2,4,…

ψ'(0)=0

E

n

-1/3

2

2/3

a

β

n

β

0

β

2

β

4

Ai'(-)=0

β

n

n=1,3,5,…

ψ(0)=0

E

n

-1/3

2

2/3

a

α

n

α

1

α

3

α

5

Ai(-)=0

α

n

E

0

2/3

a

For user-selected and , the eigenfunctions (x)=Ai(a|x|-), 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.

n

a

ψ

n

N

n

1/3

2

-2/3

a

E

n

N

n

-1/2

2dx

∞

∫

0

2

Ai(ax-)

1/3

2

-2/3

a

E

n

x

E

n

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.

E

0

1

2

ω

0

n