Eigenstates for Pöschl-Teller Potentials

energy levels

bound states

continuum

λ

1

2

3

4

bound states only

μ

1

2

continuum only

k

1

It has been long known that the Schrödinger equation for a class of potentials of the form

V

λ

(x)=-

λ(λ+1)

2

sech

x

, usually referred to as Pöschl–Teller potentials, is exactly solvable. The eigenvalue problem

-

1

2

''

ψ

(x)-

λ(λ+1)

2

sech

xψ(x)=Eψ(x)

(in units with

ℏ=m=1)

has physically significant solutions for

λ=1,2,3,…

, for both bound and continuum states. For

λ=1

, we find the solution

ψ(x)=sechx

,

E=-1/2

, which follows simply from the derivative relation

∂

x,x

sechx=sechx-2

3

sech

x

. More generally, the Schrödinger equation has the bound state solutions

ψ

λ,μ

=

μ

P

λ

(tanhx)

,

E

λ,μ

=-

2

μ

2

,

λ=1,2,3,…

,

μ=λ,λ-1,…,1

,

where the

μ

P

λ

are associated Legendre polynomials.

The Schrödinger equation has, in addition, continuum positive-energy eigenstates with

E

λ,k

=

2

k

/2

. The trivial case

λ=0

gives a free particle

+

ψ

0,k

(x)=

ikx

e

. The first two nontrivial solutions are

+

ψ

1,k

(x)=1+

i

k

tanhx

ikx

e

and

+

ψ

2,k

(x)=

-1

(1+

2

k

)

(1+

2

k

+3iktanhx-3

2

tanh

x)

ikx

e

. 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.