Pöschl–Teller Potential

One-Sentence Description

The hyperbolic Pöschl–Teller potential is a smooth attractive one-dimensional well whose bound states and scattering data can be obtained exactly.

Physical Setup

The common symmetric attractive convention is

$$V(x) = -\frac{\hbar^2}{2ma^2} \lambda(\lambda+1) \operatorname{sech}^2\left(\frac{x}{a}\right), \qquad \lambda>0.$$

This is an idealized potential well on the full real line. It is useful as a solvable alternative to square wells because the potential is smooth and its scattering problem is also analytically controlled.

Hilbert Space

The bound-state problem lives in

$$\mathcal H=L^2(\mathbb R).$$

The Hamiltonian is a one-dimensional Schrödinger operator with a smooth, short-range attractive potential. Scattering states are treated with generalized continuum normalization.

Hamiltonian

The Hamiltonian is

$$\hat H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} -\frac{\hbar^2}{2ma^2} \lambda(\lambda+1) \operatorname{sech}^2\left(\frac{x}{a}\right).$$

With the dimensionless coordinate $y=x/a$, the stationary equation reduces to a hypergeometric or associated-Legendre equation.

Parameters

Symbol	Meaning
$m$	particle mass
$a$	width scale of the well
$\lambda$	dimensionless depth parameter
$n$	bound-state index

Solvability

The model is exactly solvable. Bound-state wavefunctions can be written using associated Legendre functions or Jacobi-polynomial forms, depending on convention. The same potential family is also important in supersymmetric quantum mechanics because it is shape-invariant.

For integer $\lambda$, the attractive hyperbolic well is reflectionless: incoming continuum waves transmit without reflected flux, although they acquire a phase shift.

Spectrum and Eigenstates

For the convention above, the bound-state energies are

$$E_n = -\frac{\hbar^2}{2ma^2} (\lambda-n)^2, \qquad n=0,1,2,\ldots, \quad n<\lambda.$$

If $\lambda$ is a positive integer, this gives $\lambda$ normalizable bound states, with $n=0,\ldots,\lambda-1$. The possible zero-energy threshold state at $n=\lambda$ is not counted as a normalizable bound state on the full line.

The ground-state wavefunction is proportional to

$$\psi_0(x) \propto \operatorname{sech}^{\lambda}\left(\frac{x}{a}\right),$$

up to normalization.

Key Observables

• Bound-state energies and node count.
• Transmission and reflection amplitudes in the continuum.
• Phase shifts for short-range scattering.
• Shape-invariance ladder structure in supersymmetric treatments.

What It Teaches

The model teaches how smooth finite wells differ from hard-wall boxes, how exact bound-state spectra can coexist with continuum scattering, and how special-function solvability appears in one-dimensional Schrödinger operators.

It is also a useful benchmark for numerical solvers because the potential is smooth and localized while retaining exact answers.

Canonical Links

• Finite Square Well
• Sturm-Liouville Theory
• Ordinary Differential Equations
• Benchmark Problems

Variants

• trigonometric Pöschl–Teller potential on a finite interval;
• generalized hyperbolic forms with an additional inverse-sinh-squared term;
• supersymmetric partner potentials;
• reflectionless integer-$\lambda$ wells;
• relativistic or position-dependent-mass extensions in specialized literature.

Common Mistakes

• Mixing the hyperbolic and trigonometric Pöschl–Teller conventions.
• Counting a threshold solution as a square-integrable bound state without checking normalizability.
• Assuming every deformation of the potential remains exactly solvable.
• Dropping the scale factor $a$ when converting dimensionless formulas back to physical units.

Quick Checks

1. For integer $\lambda=3$, how many normalizable bound states does the attractive hyperbolic well have in the convention above?

Solution

The allowed values are $n=0,1,2$, so there are three normalizable bound states. The formula would give $E_3=0$, but that threshold state is not counted as a bound state on the full line.

2. What happens to the energy scale if the width $a$ is doubled at fixed $\lambda$?

Solution

The bound energies scale as $1/a^2$. Doubling $a$ makes the binding energies four times smaller in magnitude, while the dimensionless level pattern in $(\lambda-n)^2$ is unchanged.

References

• G. Pöschl and E. Teller, “Bemerkungen zur Quantenmechanik des anharmonischen Oszillators,” Zeitschrift für Physik 83, 143–151, 1933.
• S. Flügge, Practical Quantum Mechanics, Springer, 1999.
• F. Cooper, A. Khare, and U. Sukhatme, Supersymmetry in Quantum Mechanics, World Scientific, 2001.