# Eigenstates for Pöschl-Teller Potentials

Eigenstates for Pöschl-Teller Potentials

It has been long known that the Schrödinger equation for a class of potentials of the form (x)=-x, usually referred to as Pöschl–Teller potentials, is exactly solvable. The eigenvalue problem

V

λ

λ(λ+1)

2

2

sech

-(x)-xψ(x)=Eψ(x)

1

2

''

ψ

λ(λ+1)

2

2

sech

(in units with has physically significant solutions for , for both bound and continuum states. For , we find the solution , , which follows simply from the derivative relation sechx=sechx-2x. More generally, the Schrödinger equation has the bound state solutions

ℏ=m=1)

λ=1,2,3,…

λ=1

ψ(x)=sechx

E=-1/2

∂

x,x

3

sech

ψ

λ,μ

μ

P

λ

E

λ,μ

2

μ

2

λ=1,2,3,…

μ=λ,λ-1,…,1

where the are associated Legendre polynomials.

μ

P

λ

The Schrödinger equation has, in addition, continuum positive-energy eigenstates with =/2. The trivial case gives a free particle (x)=. The first two nontrivial solutions are (x)=1+tanhx and (x)=(1++3iktanhx-3x). These represent waves traveling left to right. A remarkable property of Pöschl-Teller potentials is that they are "reflectionless", meaning that waves are 100% transmitted through the barrier with no reflected waves.

E

λ,k

2

k

λ=0

+

ψ

0,k

ikx

e

+

ψ

1,k

i

k

ikx

e

+

ψ

2,k

-1

(1+)

2

k

2

k

2

tanh

ikx

e