In mathematical physics, a Pöschl–Teller potential is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions. It is named after the physicists Gertrud Pöschl and Edward Teller who developed it in 1933.
Definition In its symmetric form is explicitly given by
V ( x ) = − λ ( λ + 1 ) 2 s e c h 2 ( x ) {\displaystyle V(x)=-{\frac {\lambda (\lambda +1)}{2}}\mathrm {sech} ^{2}(x)}
and the solutions of the time-independent Schrödinger equation
− 1 2 ψ ″ ( x ) + V ( x ) ψ ( x ) = E ψ ( x ) {\displaystyle -{\frac {1}{2}}\psi ''(x)+V(x)\psi (x)=E\psi (x)}
with this potential can be found by virtue of the substitution u = t a n h ( x ) {\displaystyle u=\mathrm {tanh(x)} } , which yields
[ ( 1 − u 2 ) ψ ′ ( u ) ] ′ + λ ( λ + 1 ) ψ ( u ) + 2 E 1 − u 2 ψ ( u ) = 0 {\displaystyle \left[(1-u^{2})\psi '(u)\right]'+\lambda (\lambda +1)\psi (u)+{\frac {2E}{1-u^{2}}}\psi (u)=0} . Thus the solutions ψ ( u ) {\displaystyle \psi (u)} are just the Legendre functions P λ μ ( tanh ( x ) ) {\displaystyle P_{\lambda }^{\mu }(\tanh(x))} with E = − μ 2 2 {\displaystyle E=-{\frac {\mu ^{2}}{2}}} , and λ = 1 , 2 , 3 ⋯ {\displaystyle \lambda =1,2,3\cdots } , μ = 1 , 2 , ⋯ , λ − 1 , λ {\displaystyle \mu =1,2,\cdots ,\lambda -1,\lambda } . Moreover, eigenvalues and scattering data can be explicitly computed. In the special case of integer λ {\displaystyle \lambda } , the potential is reflectionless and such potentials also arise as the N-soliton solutions of the Korteweg–De Vries equation. The more general form of the potential is given by
V ( x ) = − λ ( λ + 1 ) 2 s e c h 2 ( x ) − ν ( ν + 1 ) 2 c s c h 2 ( x ) . {\displaystyle V(x)=-{\frac {\lambda (\lambda +1)}{2}}\mathrm {sech} ^{2}(x)-{\frac {\nu (\nu +1)}{2}}\mathrm {csch} ^{2}(x).}
Related potentials The symmetric form of the Poschl–Teller potential is a special case of the Epstein profile, also called Eckart potential or hyperbolic Rosen–Morse potential.
V ( x ) = − λ ( λ + 1 ) 2 s e c h 2 ( x ) − g tanh x . {\displaystyle V(x)=-{\frac {\lambda (\lambda +1)}{2}}\mathrm {sech} ^{2}(x)-g\tanh x.}
Notes
References
External links Eigenstates for Pöschl-Teller Potentials