| Rosen–Morse potential | |
|---|---|
| Name | Rosen–Morse potential |
| Introduced | 1932 |
| Author | Nathan Rosen; Philip M. Morse |
| Field | Quantum mechanics |
| Equations | Schrödinger equation |
Rosen–Morse potential
The Rosen–Morse potential is a one-dimensional model potential used in quantum mechanics to describe bound and scattering states with analytic solutions. It is significant for its exactly solvable nature, serving as a pedagogical and practical tool in studies of the Schrödinger equation, supersymmetric quantum mechanics, and molecular and nuclear model systems. The potential interpolates between well-known solvable potentials and illustrates techniques such as factorization and algebraic methods.
The Rosen–Morse potential, introduced by Nathan Rosen and Philip M. Morse in 1932, is defined on the real line and combines features of asymmetric wells and hyperbolic tails. In physical contexts it models diatomic molecular vibrations, quasi-one-dimensional motion in heterostructures, and potential wells with nontrivial asymptotic behaviour relevant to scattering theory. Because it yields closed-form eigenvalues and eigenfunctions, the Rosen–Morse potential is a standard example in textbooks on exactly solvable models and is used to benchmark numerical methods developed at institutions such as Los Alamos National Laboratory and CERN.
The potential typically appears in two closely related forms: the Rosen–Morse I (with a hyperbolic cotangent term) and Rosen–Morse II (with a hyperbolic tangent term). A common form is V(x)= -A sech^2(αx) + B tanh(αx), where A and B are real parameters controlling depth and asymmetry, and α sets the length scale. Parameters map to physical quantities such as well depth, force constant, and asymptotic energy offsets used in molecular models derived from the Born–Oppenheimer approximation and effective one-dimensional reductions. The hyperbolic functions connect the potential to the Pöschl–Teller potential and to solvable potentials studied by E. Schrödinger and Paul Dirac.
The time-independent Schrödinger equation for the Rosen–Morse potential admits exact bound-state solutions in terms of special functions. Eigenfunctions can be expressed using associated Legendre functions, Jacobi polynomials, or hypergeometric functions such as the Gauss hypergeometric function {}_2F_1. Energy eigenvalues follow quantization conditions determined by the potential parameters and are often presented in closed analytic form, facilitating comparison with experimental spectra and with algebraic models like the quantum harmonic oscillator and the Morse potential. The solvability is closely connected to the existence of ladder operators and shape-invariance under parameter shifts, concepts developed in the context of factorization method (quantum mechanics).
For energies above the continuum threshold, the Rosen–Morse potential supports scattering states with analytic reflection and transmission amplitudes. Closed-form expressions for S-matrix elements can be derived using analytic continuation of bound-state solutions, matching asymptotic plane waves at infinity. These results are important in one-dimensional scattering theory and are applied in tunnelling analyses and resonant transmission studies for quantum wells and quantum dots fabricated in semiconductor devices such as those studied by Bell Labs and IBM Research. The potential’s asymmetric term (B tanh) produces different left/right reflection coefficients, making it useful in studies of parity breaking and nonreciprocal transport.
The Rosen–Morse potential reduces to several well-known solvable models in particular parameter limits. With B=0 it becomes the symmetric Pöschl–Teller potential; in a suitable limit of small α and scaled A it connects to the Morse potential used in molecular spectroscopy and to the harmonic oscillator near its minimum. Other limits reproduce reflectionless potentials studied in the inverse scattering method and by the Korteweg–de Vries equation community. These connections place the Rosen–Morse potential within the broader family of exactly solvable potentials analyzed by researchers such as L. Infeld and T.E. Hull.
Applications span molecular vibration modeling, effective potentials in nuclear cluster models, and engineered potentials in cold-atom experiments employing optical lattices, as performed in laboratories like the Max Planck Institute for Quantum Optics. The analytic eigenfunctions facilitate calculation of transition matrix elements, selection rules, and spectroscopic line strengths relevant to molecular spectroscopy and to computational chemistry codes developed at universities including Harvard University and Massachusetts Institute of Technology. In solid-state physics, the Rosen–Morse form models asymmetric heterojunction barriers and informs design of tunnelling devices explored in condensed matter physics research.
The Rosen–Morse potential is amenable to algebraic solution techniques including the factorization method (quantum mechanics), ladder operators, and the framework of supersymmetric quantum mechanics (SUSY QM). It exhibits shape invariance, a property identified by L. Gendenshteĭn that guarantees exact solvability via an operator algebra. SUSY QM connects partner potentials and generates isospectral families; for Rosen–Morse, supersymmetric transformations produce hierarchies of solvable potentials and facilitate derivation of scattering data. Algebraic approaches link this potential to studies in mathematical physics by groups around institutions such as the Steklov Institute of Mathematics and to classic works like those by E. Witten on supersymmetry.
Category:Quantum mechanics potentials Category:Exactly solvable models in quantum mechanics