# Superpotential
A **superpotential** is a function W(x) that appears in [supersymmetric quantum mechanics](https://www.edgepedia.org/supersymmetric-quantum-mechanics) (SUSYQM) as the building block from which a pair of related Schrödinger potentials, the supersymmetric partner potentials, are constructed. It is the central object of the one-dimensional supersymmetric factorization method: once W(x) is known, the partner potentials, their spectra, and the algebraic structure of the theory follow without solving the [Schrödinger equation](https://www.edgepedia.org/schrodinger-equation) directly.[1](https://www.ijpam.eu/contents/2014-97-1/10/index.html)
| Key fact | Detail |
|---|---|
| Definition | W(x) enters first-order operators A and A† that factorize the Hamiltonian; the partner potentials are V±(x) = W²(x) ± W′(x)[1](https://www.ijpam.eu/contents/2014-97-1/10/index.html) |
| Riccati equation | W satisfies a Riccati differential equation, and for a zero-energy ground state it is given by W(x) = −ψ₀′(x)/ψ₀(x) in terms of the nodeless ground-state wavefunction[1](https://www.ijpam.eu/contents/2014-97-1/10/index.html) • [2](https://www.mdpi.com/2073-8994/13/1/44) |
| Shape invariance | If the partners obey W²(x, a₁) + W′(x, a₁) = W²(x, a₂) − W′(x, a₂) + R(a₁), the spectra and eigenfunctions follow algebraically, as for the harmonic oscillator[3](https://www.mdpi.com/2073-8994/4/3/452) |
| Completeness | The known list of ℏ-independent (conventional) shape-invariant superpotentials is complete; it includes the harmonic oscillator, Morse, Coulomb, Eckart, Rosen–Morse, Pöschl–Teller, Scarf, and 3D oscillator superpotentials[3](https://www.mdpi.com/2073-8994/4/3/452) |
| Extended class | ℏ-dependent "extended" superpotentials, discovered by C. Quesne, obey shape invariance only with explicit ℏ dependence and are isospectral with conventional ones[3](https://www.mdpi.com/2073-8994/4/3/452) |
| SUSY breaking | The zero-mode candidates for the two partner Hamiltonians are proportional to e^(−∫W(x) dx) and e^(+∫W(x) dx), and whether either is normalizable decides whether a zero-energy ground state exists; for W(x) = x² − a² neither e^(−W) nor e^(+W) is square-integrable, so supersymmetry is broken and E₀ > 0[4](https://qft.org/supersymmetry-duality/susy-quantum-mechanics-cohomology-index/spectral-pairing-ground-states/) • [15](https://www.tpi.uni-jena.de/~wipf/lectures/susy/susyhead.pdf) |
| Approximation | A supersymmetry-inspired WKB (SWKB) approximation is exact for a class of shape-invariant potentials[5](https://arxiv.org/abs/hep-th/9405029) |
## Factorization and partner potentials
In SUSYQM a broad class of one-dimensional Schrödinger Hamiltonians is factorized into first-order differential operators. Writing the factorization in terms of the superpotential, the two partner potentials are
\[ V_{\pm}(x) = W^{2}(x) \pm W'(x), \]
where the prime denotes differentiation with respect to x. The two Hamiltonians built on V₋ and V₊ are called supersymmetric partner Hamiltonians, and their spectra are intertwined: apart from a possible unpaired ground state, the levels of one partner match those of the other.[1](https://www.ijpam.eu/contents/2014-97-1/10/index.html) • [6](https://arxiv.org/abs/0910.0192)
The superpotential is not an independent input in general. For a zero-energy ground state, the Schrödinger equation reduces to a first-order equation whose solution gives W in terms of the ground-state wavefunction, W(x) = −ψ₀′(x)/ψ₀(x), and substituting back turns the Schrödinger equation into the Riccati equation for W.[2](https://www.mdpi.com/2073-8994/13/1/44) Conversely, starting from any nodeless ground-state wavefunction one can generate a superpotential and hence a pair of partner systems; this is the basis of using SUSYQM as a tool for generating new potentials with known spectra from an initial solvable one.[6](https://arxiv.org/abs/0910.0192) A generalized construction extends the definition so that the factorization energy equals an arbitrary spectral level rather than only the ground state, connecting potentials whose spectra need not coincide.[7](https://arxiv.org/abs/hep-th/0607125)
## Shape invariance
**Shape invariance** is the condition that makes the superpotential practically useful. If the partner potentials built from W(x, a₁) have the same shape, differing only by a change of parameter a₁ → a₂ and an additive constant R(a₁), the system is called shape invariant. In operator form the condition reads
\[ W^{2}(x, a_{1}) + W'(x, a_{1}) = W^{2}(x, a_{2}) - W'(x, a_{2}) + R(a_{1}). \]
Different forms of the parameter change define classes of shape invariance, the most commonly discussed being translational, where a₂ = a₁ + δ for a constant δ.[3](https://www.mdpi.com/2073-8994/4/3/452)
For a shape-invariant system the eigenvalues and eigenfunctions can be determined analytically by an algebraic procedure comparable to the ladder-operator solution of the harmonic oscillator: the first excited energy of one partner equals the ground energy of the other shifted by R(a₁), and iterating the argument gives the full spectrum. Every shape-invariant system also carries an underlying potential algebra, a symmetry algebra of the potential, that guarantees its solvability.[3](https://www.mdpi.com/2073-8994/4/3/452) • [8](https://doi.org/10.1103/PhysRevA.58.4287) The Cooper, Khare, and Sukhatme review (Fred Cooper of Los Alamos National Laboratory, Avinash Khare of the [Institute of Physics](https://www.edgepedia.org/institute-of-physics), Bhubaneswar, and Uday Sukhatme of the [University at Buffalo](https://www.edgepedia.org/university-at-buffalo)) reports that the familiar analytically solvable potentials all have the property of shape invariance, which is why SUSYQM gives a much deeper understanding of their solvability.[5](https://arxiv.org/abs/hep-th/9405029)
## Classification of superpotentials
The additive (translational) shape-invariance condition is a difference-differential equation for W. Bougie, Gangopadhyaya, Mallow, and Rasinariu ([Loyola University Chicago](https://www.edgepedia.org/loyola-university-chicago) and [Columbia College Chicago](https://www.edgepedia.org/columbia-college-chicago)) showed that this equation is equivalent to an infinite set of nonlinear partial differential equations, of which only two need to be solved, and derived a new proof that the known list of ℏ-independent superpotentials is complete.[3](https://www.mdpi.com/2073-8994/4/3/452) Solving the equations case by case reproduces the standard families: the harmonic oscillator and 3D harmonic oscillator, the [Morse potential](https://www.edgepedia.org/morse-potential) (whose superpotential decreases with the parameter, signaling a finite number of bound states), the Coulomb, Eckart, Rosen–Morse I and II, general Pöschl–Teller, and Scarf I and II superpotentials.[3](https://www.mdpi.com/2073-8994/4/3/452)
A separate class of ℏ-dependent extended superpotentials, found by C. Quesne, satisfies the shape-invariance condition only when W is allowed to depend explicitly on ℏ. Each of these new potentials is isospectral with a conventional one, and later work added an infinite number of potentials to this class; they remain objects of active research.[3](https://www.mdpi.com/2073-8994/4/3/452) Beyond the standard list, new classes continue to be introduced, such as superpotentials of the form W(x) = g ε(x) x²ⁿ, whose n = 0 case reproduces supersymmetrically related Dirac delta well and barrier potentials.[9](https://arxiv.org/abs/1111.1198)
## Supersymmetry breaking
Whether a superpotential yields an unpaired zero-energy ground state is decided by normalizability. For W(x) = ωx the function e^(−W) is square-integrable and gives a normalizable zero mode, so supersymmetry is unbroken and the positive spectrum consists of paired oscillator levels E = nω. For W(x) = x² − a², by contrast, W tends to +∞ as x → +∞ and to −∞ as x → −∞, so the zero-mode candidates, proportional to e^(−∫W dx) and to e^(+∫W dx), decide by their normalizability whether a zero-energy ground state exists; the partner potentials still grow as x⁴/2 and have a discrete spectrum, and the absence of a zero mode forces the ground-state energy to be positive: supersymmetry is broken and every level is paired.[15](https://www.tpi.uni-jena.de/~wipf/lectures/susy/susyhead.pdf) The two zeros of w(x) produce semiclassical wells, but tunneling couples their approximate states and removes the would-be zero-energy degeneracy.[4](https://qft.org/supersymmetry-duality/susy-quantum-mechanics-cohomology-index/spectral-pairing-ground-states/) This normalizability criterion is the one-dimensional mechanism used in [Edward Witten](https://www.edgepedia.org/edward-witten)'s 1981 work that introduced supersymmetric quantum mechanics.[4](https://qft.org/supersymmetry-duality/susy-quantum-mechanics-cohomology-index/spectral-pairing-ground-states/)
## Applications
The superpotential formalism extends well beyond textbook one-dimensional problems. A supersymmetry-inspired [WKB approximation](https://www.edgepedia.org/wkb-approximation) built from W is exact for a class of shape-invariant potentials, and SUSY ideas give improved results for the tunneling rate in double-well potentials and for large-N expansions.[5](https://arxiv.org/abs/hep-th/9405029) The formalism has been applied to periodic potentials, considerably enlarging the class of analytically solvable one-dimensional periodic systems,[10](https://arxiv.org/abs/quant-ph/0402206) and to particles with position-dependent effective mass, for which shape invariance has been generalized and harmonic-oscillator-like and Morse-like spectra obtained algebraically.[11](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.60.4318) In kink-bearing field theories on the line, the translational zero mode of the kink is determined by the superpotential, and the Pöschl–Teller hierarchy of reflectionless potentials is built from a sequence of superpotentials related by shape invariance.[12](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.036020) Numerical work has treated superpotentials with no known analytical solution, such as W(x) = x³, whose partner potentials are V₋(x) = x⁶ − 3x² and V₊(x) = x⁶ + 3x².[13](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.35.1255) In mathematics, supersymmetric quantum mechanics underlies Witten's proof of the Morse inequalities and the physicist's proof of the [Atiyah–Singer index theorem](https://www.edgepedia.org/atiyah-singer-index-theorem).[12](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.036020)
The same word names a distinct object in relativistic supersymmetric field theory: there the superpotential is a holomorphic function of superfields, such as the cubic Yukawa-plus-Higgs superpotential of the minimal supersymmetric standard model, that determines the scalar interaction terms of the Lagrangian.[14](https://journals.aps.org/prd/abstract/10.1103/vkmb-4jnn)
## References
1. [R. Thibes, E.F. Batista, G.D. Barbosa (2014). Briefly Revisiting the Quantum Mechanics Supersymmetry Algebra and Group. International Journal of Pure and Applied Mathematics 97, 99–109.](https://www.ijpam.eu/contents/2014-97-1/10/index.html)
2. [The Generalized OTOC from Supersymmetric Quantum Mechanics, Symmetry 13, 44 (2021).](https://www.mdpi.com/2073-8994/13/1/44)
3. [J. Bougie, A. Gangopadhyaya, J. Mallow, C. Rasinariu (2012). Supersymmetric Quantum Mechanics and Solvable Models. Symmetry 4, 452–473.](https://www.mdpi.com/2073-8994/4/3/452)
4. [Spectral Pairing, Ground States, and Supersymmetry Breaking, QFT.org.](https://qft.org/supersymmetry-duality/susy-quantum-mechanics-cohomology-index/spectral-pairing-ground-states/)
5. [F. Cooper, A. Khare, U. Sukhatme (1994). Supersymmetry and Quantum Mechanics. arXiv:hep-th/9405029.](https://arxiv.org/abs/hep-th/9405029)
6. [D.J. Fernandez C. (2009). Supersymmetric Quantum Mechanics. arXiv:0910.0192.](https://arxiv.org/abs/0910.0192)
7. [S.P. Maydanyuk (2006). Search of a general form of superpotential in hierarchy with discrete energy spectrum. arXiv:hep-th/0607125.](https://arxiv.org/abs/hep-th/0607125)
8. [Translational shape invariance and the inherent potential algebra, Phys. Rev. A 58, 4287 (1998).](https://doi.org/10.1103/PhysRevA.58.4287)
9. [F. Marques, O. Negrini, A.J. da Silva (2011). A new simple class of superpotentials in SUSY Quantum Mechanics. arXiv:1111.1198.](https://arxiv.org/abs/1111.1198)
10. [A. Khare, U. Sukhatme (2004). Periodic Potentials and Supersymmetry. arXiv:quant-ph/0402206.](https://arxiv.org/abs/quant-ph/0402206)
11. [A.R. Plastino et al. (1999). Supersymmetric approach to quantum systems with position-dependent effective mass, Phys. Rev. A 60, 4318.](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.60.4318)
12. [Self-gravitating kinks in two-dimensional pseudo-Riemannian universes, Phys. Rev. D 101, 036020 (2020).](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.101.036020)
13. [L.J. Boya, M. Kmiecik, A. Bohm (1987). Calculations with supersymmetric potentials, Phys. Rev. D 35, 1255.](https://journals.aps.org/prd/abstract/10.1103/PhysRevD.35.1255)
14. [Vacuum moduli space of the minimal supersymmetric standard model, Phys. Rev. D (2025).](https://journals.aps.org/prd/abstract/10.1103/vkmb-4jnn)
15. [tpi.uni-jena.de](https://www.tpi.uni-jena.de/~wipf/lectures/susy/susyhead.pdf)
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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum formalism and states › Exactly solvable quantum systems › Supersymmetric quantum mechanics and shape invariance*
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