Abstract
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ.
Highlights
Supersymmetric quantum mechanics (SUSYQM) is a generalization of the factorization method commonly used for the harmonic oscillator
We will describe the general formalism of supersymmetric quantum mechanics (SUSYQM) and in Section 3 we introduce the shape invariance condition that makes a potential solvable
While supersymmetric quantum mechanics began as a simplified model to account for dynamical symmetry breaking, the application of this formalism to quantum mechanics has become an important field in its own right
Summary
Supersymmetric quantum mechanics (SUSYQM) is a generalization of the factorization method commonly used for the harmonic oscillator. For a system with a given shape invariant superpotential, the eigenvalues and eigenfunctions can be determined analytically. If we define a third operator J3 in terms of the operator i ∂φ , the shape invariance condition becomes one of the commutation relations of the potential algebra.
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