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Supersymmetric quantum mechanics based on higher excited states II: a few examples of isospectral partner potentials
chao-dyn · Submitted 14 Mar 1997
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The generalization is technically almost straightforward but physically quite nontrivial since it yields an infinity of new classes of susy-partner potentials, whose spectra are exactly identical except for the lowest $n+1$ states, if the superpotential is defined in terms of the $(n+1)$-st eigenfunction, with $n=0$ reserved for the ground state.
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We apply the generalized formalism and the techniques of the supersymmetric (susy) quantum mechanics to the cases where the superpotential is generated/defined by higher excited eigenstates (Robnik 1997, paper I). The generalization is technically almost straightforward but physically quite nontrivial since it yields an infinity of new classes of susy-partner potentials, whose spectra are exactly identical except for the lowest $n+1$ states, if the superpotential is defined in terms of the $(n+1)$-st eigenfunction, with $n=0$ reserved for the ground state. First we show that there are practically no possibilities for shape invariant potentials based on higher excited states. Then we calculate the isospectral partner potentials for the following 1-dim potentials (after separation of variables where appropriate): (i) 3-dim (spherically symmetric) harmonic oscillator, (ii) 3-dim (isotropic) Kepler problem, (iii) Morse potential, (iv) P\"oschl-Teller type I potential, and (v) the 1-dim box potential. In all cases except in (v) we get new classes of solvable potentials. In (v) the partner potential to the box potential is a special case of P\"oschl-Teller type I potential. potentials. In this paper we present results of a straightforward further application of this formalism to a few most important exactly solvable 1-dim potentials, namely (i) spherically symmetric 3-dim harmonic oscillator, (ii) 3-dim isotropic (spherically symmetric) Kepler potential, (iii) Morse potential, (iv) P\"oschl-Teller type I potential, and (v) 1-dim box potential. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We apply the generalized formalism and the techniques of the supersymmetric (susy) quantum mechanics to the cases where the superpotential is generated/defined by higher excited eigenstates (Robnik 1997, paper I). | contribution 0.65 | 0.86 | 0.05 | asserted 0.99 |
| s1 | The generalization is technically almost straightforward but physically quite nontrivial since it yields an infinity of new classes of susy-partner potentials, whose spectra are exactly identical except for the lowest $n+1$ states, if the superpotential is defined in terms of the $(n+1)$-st eigenfunction, with $n=0$ reserved for the ground state. | result 0.55 | 0.71 | 0.52 | asserted 1.00 |
| s2 | First we show that there are practically no possibilities for shape invariant potentials based on higher excited states. | result 0.95 | 0.85 | 0.15 | asserted 0.99 |
| s3 | Then we calculate the isospectral partner potentials for the following 1-dim potentials (after separation of variables where appropriate): (i) 3-dim (spherically symmetric) harmonic oscillator, (ii) 3-dim (isotropic) Kepler problem, (iii) Morse potential, (iv) P\"oschl-Teller type I potential, and (v) the 1-dim box potential. | contribution 0.86 | 0.92 | 0.08 | asserted 0.98 |
| s4 | In all cases except in (v) we get new classes of solvable potentials. | result 1.00 | 0.92 | 0.23 | asserted 1.00 |
| s5 | In (v) the partner potential to the box potential is a special case of P\"oschl-Teller type I potential. potentials. | result 0.99 | 0.91 | 0.09 | asserted 1.00 |
| s6 | In this paper we present results of a straightforward further application of this formalism to a few most important exactly solvable 1-dim potentials, namely (i) spherically symmetric 3-dim harmonic oscillator, (ii) 3-dim isotropic (spherically symmetric) Kepler potential, (iii) Morse potential, (iv) P\"oschl-Teller type I potential, and (v) 1-dim box potential. | contribution 0.98 | 0.88 | 0.09 | asserted 0.99 |
| s7 | PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq | background 0.87 | 0.10 | 0.02 | unclear 0.95 |
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Original abstract
We apply the generalized formalism and the techniques of the supersymmetric (susy) quantum mechanics to the cases where the superpotential is generated/defined by higher excited eigenstates (Robnik 1997, paper I). The generalization is technically almost straightforward but physically quite nontrivial since it yields an infinity of new classes of susy-partner potentials, whose spectra are exactly identical except for the lowest $n+1$ states, if the superpotential is defined in terms of the $(n+1)$-st eigenfunction, with $n=0$ reserved for the ground state. First we show that there are practically no possibilities for shape invariant potentials based on higher excited states. Then we calculate the isospectral partner potentials for the following 1-dim potentials (after separation of variables where appropriate): (i) 3-dim (spherically symmetric) harmonic oscillator, (ii) 3-dim (isotropic) Kepler problem, (iii) Morse potential, (iv) P\"oschl-Teller type I potential, and (v) the 1-dim box potential. In all cases except in (v) we get new classes of solvable potentials. In (v) the partner potential to the box potential is a special case of P\"oschl-Teller type I potential. potentials. In this paper we present results of a straightforward further application of this formalism to a few most important exactly solvable 1-dim potentials, namely (i) spherically symmetric 3-dim harmonic oscillator, (ii) 3-dim isotropic (spherically symmetric) Kepler potential, (iii) Morse potential, (iv) P\"oschl-Teller type I potential, and (v) 1-dim box potential. PACS numbers: 03.65.-w, 03.65.Ge, 03.65.Sq
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