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Supersymmetry and Quantum Mechanics

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Fred Cooper, Avinash Khare, Uday Sukhatme

High Energy Physics - Theory · Submitted 4 May 1994 · 265 references

Pith's verdict: The standard SUSY QM reference, and it holds up; the one soft spot that lands is the series-defined scaling potentials in Sec. 4.3.2, not the classification claim the reader flagged.

T0 · AI review

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arXiv IDhep-th/9405029 (version 2)
Pith IDPR72NIQZ, Pith's permanent record of this paper and its review
CitationAvinash Khare, Fred Cooper, Uday Sukhatme. "Supersymmetry and Quantum Mechanics." arXiv:hep-th/9405029 (1994). https://pith.science/paper/PR72NIQZ
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The paper

The paper claims

The review argues that shape invariance is the common algebraic reason a large family of quantum potentials is exactly solvable.

The reading

This review asks why a short list of quantum-mechanical potentials — oscillator, Coulomb, Morse, Pöschl-Teller and relatives — can be solved in closed form. Its answer is that their supersymmetric partner potentials are the same shape up to a parameter shift, a condition called shape invariance. Once that condition holds, the spectrum is a sum of remainders, the eigenfunctions are built by repeated raising operators, and a supersymmetry-inspired WKB rule gives the exact energies for the translation-type cases. The review extends the same logic to newly found scaling-type potentials, to isospectral families and KdV solitons, and to Dirac and Pauli problems; if the picture is right, exact solvability becomes an algebraic property rather than a case-by-case accident.

What carries the argument

Shape invariance is the load-bearing mechanism: the two SUSY partner potentials $V_1=W^2-W'$ and $V_2=W^2+W'$ are required to have the same spatial form after a parameter transformation, $V_2(x;a_1)=V_1(x;a_2)+R(a_1)$. That condition turns the Riccati/factorization structure into a ladder: $H_s$ has ground energy $�sum_{k=1}^{s-1}R(a_k)$, so the original spectrum is the partial sums of $R$, while $A^�dagger(x;a_k)$ maps one rung to the next. The same condition is what makes the SWKB quantization condition exact in the translation case and what organizes the new scaling and multi-step potentials.

Extended reading notes

Core claim

The central claim is that shape invariance, $V_2(x;a_1)=V_1(x;a_2)+R(a_1)$, is the organizing integrability condition behind the standard analytically solvable potentials. Starting from a factorized Hamiltonian $H_1=A^�dagger A$ with superpotential $W(x)$, the SUSY partner is $H_2=AA^�dagger$; when the partner potentials differ only by a parameter map and an $x$-independent remainder, the whole bound-state spectrum follows as $E_n=�sum_{k=1}^n R(a_k)$ with $E_0=0$, and the eigenfunctions follow from repeated application of $A^�dagger$. The review presents the translation case $a_2=a_1+�alpha$ as covering the familiar textbook potentials, adds scaling and multi-step variants, and argues that the lowest-order SWKB quantization condition is exact for the translation-type shape-invariant class. It also uses the same machinery to generate isospectral potentials and multi-soliton KdV solutions and to exhibit supersymmetry in Pauli and Dirac problems.

Load-bearing premise

The load-bearing premise is that the translation-type shape-invariance classification built from the factorized superpotential ansatz in Eq. (130) captures the standard analytically solvable potentials, even though the authors state they have no rigorous proof that the ansatz exhausts the class.

Editorial extensions

If this is right

  • For any potential whose SUSY partner is shape invariant with a translation parameter map, the bound-state energies can be written down as partial sums of the remainder without solving a differential equation.
  • The eigenfunctions of the same class are generated algebraically from the ground state by repeated application of the operators $A^�dagger(x;a_k)$.
  • The lowest-order supersymmetric WKB quantization condition reproduces the exact spectrum for translation-type shape-invariant potentials, unlike ordinary WKB.
  • The scaling and multi-step shape-invariance constructions produce new reflectionless potentials with infinite bound-state spectra, including q-deformed relatives of the Rosen-Morse and oscillator potentials.
  • The same SUSY bookkeeping yields n-parameter isospectral families and explicit multi-soliton solutions of the KdV hierarchy.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If shape invariance is the right organizing principle, it gives a practical test for exact solvability: compute the SUSY partner and check whether it closes under a parameter map before attempting a direct integration.
  • A useful next test would be to look for a closed-form expression for one of the series-defined scaling superpotentials; success would turn that class from a formal construction into a directly usable potential family.
  • The isospectral families can be read as a control knob: one can change a potential's shape and scattering data while holding its bound-state energies fixed, which is a natural setting for testing whether tunneling rates depend on barrier shape beyond the energy splitting.
  • Singular superpotentials, which the review uses to produce negative-energy states and bound states in the continuum, suggest that the same SUSY bookkeeping could classify which singular potentials admit normalizable states embedded in the continuum.
Original abstract

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large $N$ expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order $p$.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 8,293 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.25 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large $N$ expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order $p$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. background 1.00 0.05 0.07 asserted 1.00
s1 In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. background 0.99 0.07 0.11 asserted 1.00
s2 In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. contribution 0.98 0.50 0.11 asserted 0.75
s3 Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. background 0.40 0.20 0.12 asserted 1.00
s4 Familiar solvable potentials all have the property of shape invariance. background 0.64 0.13 0.06 asserted 1.00
s5 We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. contribution 0.93 0.91 0.11 asserted 1.00
s6 The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multi-soliton solutions of the KdV equation are constructed. contribution 0.82 0.76 0.06 asserted 0.98
s7 Approximation methods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. result 0.72 0.84 0.12 asserted 1.00
s8 Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large $N$ expansions. result 0.84 0.58 0.07 asserted 0.96
s9 We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantum mechanics. contribution 0.98 0.59 0.03 asserted 0.92
s10 Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order $p$. contribution 0.92 0.57 0.07 asserted 0.91

What kind of paper the classifier reports this is: review 1.00 · evidence from proof or derivation 0.56.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 2. 1: The energy levels of two supersymmetric partner
Figure 7. 1: Selected members of the family of potentials wit
Figure 9. for the two choices
Figure 12. 2: The Morse and its supersymmetric partner poten

Showing 4 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:hep-th/9405029 by the authors.

References
How later papers used it

220 later papers cite it

Evidence: substantial. 220 of the citing papers share no author with it, and 32 of those carry a confident label from Pith's citation classifier.

Pith read 27 of the 343 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Uses its method or tool 4 of 27 papers with this label
How the literature uses it

162 arXiv papers cite this in 259 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 144 sentences without a confident label

Builds on it 1 paper
  • The fermionic operators [equation] generate an $sl(1|1)$ superalgebra [ref] , [equation] where [equation] and $S_3$ is defined by [equation] Choosing $H$ as the Hamiltonian, we get a system that generalizes the usual $N=2$ supersymmetry of a spin-$1/2$ particle with gyromagnetic ratio $g=2$ in arbitrary magnetic field [ref] to the non-commutative case.

    Super-extended noncommutative Landau problem and conformal symmetry · $N=2$ supersymmetry: arbitrary magnetic field · 0.90

Uses its method or tool 14 papers
Compares with it 4 papers
  • Potentials: [equation] [equation] where $V_0$ is an irrelevant constant given by [equation] Solvable sectors: [equation] It is worth mentioning here that the potential $V^(0)+(q)$ coincides with the potential whose bound state wave functions are given in terms of exceptional $X_1$ Jacobi polynomial [ref] for $a=1$, $b_1=B+/2$, $z_0=-(2A-1)/(2B)$ whereas potential $V^(0)-(q)$ coincides with the Scarf I potential [ref] [after making an change $B→B+$] whose bound state wave functions are given in terms of classical Jacobi polynomials.

    Effect of Position-dependent Mass on Dynamical Breaking of Type B and Type X_2 N-fold Supersymmetry · Effects of PDM on Dynamical Symmetry Breaking of Type B $$-fold SUSY · 0.97

  • Such a representation reveals an obvious similarity of the hidden supersymmetry of the spinless Aharonov-Bohm system with supersymmetry of a usual form, associated with introduction of the spin degrees of freedom [ref] .

    Hidden superconformal symmetry of spinless Aharonov-Bohm system · Hidden N=2 supersymmetry in spinless AB system · 0.88

  • Note relative simplicity of the found potential in a comparison with variety of the reflectionless shape invariant potentials (with scaling of parameters; for example, see [ref] ).

    Double complex SUSY-transformations: deformations of real potentials and their spectral characteristics · Conclusions · 1.00

Cites it as background 78 papers

Discussion

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