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