WOLFRAM|DEMONSTRATIONS PROJECT

Particle in an Infinite Vee Potential

​
potential coefficient a
1.75
quantum number n
0
1
2
3
4
5
6
7
8
9
show harmonic oscillator
This Demonstration considers solutions of the Schrödinger equation for a particle in a one-dimensional "infinite vee" potential:
-
1
2
ψ''(x)+axψ(x)=Eψ(x)
, setting
ℏ=m=1
for simplicity. The solutions of the differential equation that approach zero as
x±∞
are Airy functions
ψ(x)=Ai
1/3
2
-2/3
a
(a|x|-E)
, as can be found using DSolve in Mathematica. The allowed values of
E
are found by requiring continuity of
ψ'(x)
at
x=0
. The even solutions
n=0,2,4,…
require
ψ'(0)=0
, which leads to
E
n
=
-1/3
2
2/3
a
β
n
, with
β
0
,
β
2
,
β
4
, … being the first, second, third, … zeros of the Airy prime function:
Ai'(-
β
n
)=0
. The odd solutions
n=1,3,5,…
have nodes
ψ(0)=0
, which leads to
E
n
=
-1/3
2
2/3
a
α
n
, with
α
1
,
α
3
,
α
5
, … being the first, second, third, … zeros of the Airy function:
Ai(-
α
n
)=0
. The ground state is given by
E
0
=0.808614
2/3
a
.
For user-selected
n
and
a
, the eigenfunctions
ψ
n
(x)=
N
n
Ai
1/3
2
-2/3
a
(a|x|-
E
n
)
, with normalization constants
N
n
=
-1/2
2
∞
∫
0
2
Ai
1/3
2
-2/3
a
(ax-
E
n
)
dx
, are plotted as blue curves. The
x
axis for each function coincides with the corresponding eigenvalue
E
n
, 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:
E
0
=
1
2
ω
0
. The harmonic oscillator is more confining, so its eigenvalues are more widely spaced. For higher values of
n
, the oscillator functions might move off scale.