Eigenfunctions and Energies for Sloped-Bottom Square-Well Potential

potential x

0

0

potential slope s

1

quantum level n

1

ψ

n

&

(0)

ψ

n

Δψ

V(x)

ψ

n

(0)

ψ

n

Eigenenergies

E

n

and eigenfunctions

ψ

n

of the potential

V(x)=s|x-

x

0

|

for

x≤1/2

and

V(x)=∞

for

x>1/2

are obtained numerically. This is treated as a perturbation of the infinite square-well potential, with

s=0

,

V(x)=0

for

x≤1/2

and

V(x)=∞

for

x>1/2

. The

th

n

eigenfunction of the unperturbed problem is denoted by

(0)

ψ

n

. Note that, for increasing quantum number

n

, the effect of the perturbation diminishes.