The chapter introduces the concept of Supersymmetry (SUSY) in Quantum Mechanics, a symmetry between fermions and bosons. It begins by explaining the basic formalism of SUSYQM, where two operators \( A \) and \( A^\dagger \) are used to construct two Hamiltonians \( H_1 \) and \( H_2 \). The relationship between the eigenvalues, eigenfunctions, and scattering matrices of these Hamiltonians is derived, showing that if \( E \) is an eigenvalue of \( H_1 \), it is also an eigenvalue of \( H_2 \) and vice versa. The chapter also discusses the reflection and transmission coefficients for SUSY partner potentials, and how to construct new solvable potentials using SUSY.
The chapter then delves into the concept of shape invariance, which is an integrability condition that allows the operator method for the harmonic oscillator to be generalized to a wider class of solvable potentials. Shape-invariant potentials (SIPs) are potentials that differ only in the parameters that appear in them, and the chapter provides a detailed derivation of the energy eigenvalues and eigenfunctions for SIPs using the shape invariance condition.
Finally, the chapter discusses the supersymmetric WKB (SWKB) approximation, which is an extension of the semiclassical approach inspired by SUSY. It is shown that for many problems, the SWKB method provides better accuracy than the standard WKB method, and it is proven that the lowest order SWKB approximation gives exact energy eigenvalues for all SIPs with translation, including most analytically solvable potentials in standard textbooks on quantum mechanics.The chapter introduces the concept of Supersymmetry (SUSY) in Quantum Mechanics, a symmetry between fermions and bosons. It begins by explaining the basic formalism of SUSYQM, where two operators \( A \) and \( A^\dagger \) are used to construct two Hamiltonians \( H_1 \) and \( H_2 \). The relationship between the eigenvalues, eigenfunctions, and scattering matrices of these Hamiltonians is derived, showing that if \( E \) is an eigenvalue of \( H_1 \), it is also an eigenvalue of \( H_2 \) and vice versa. The chapter also discusses the reflection and transmission coefficients for SUSY partner potentials, and how to construct new solvable potentials using SUSY.
The chapter then delves into the concept of shape invariance, which is an integrability condition that allows the operator method for the harmonic oscillator to be generalized to a wider class of solvable potentials. Shape-invariant potentials (SIPs) are potentials that differ only in the parameters that appear in them, and the chapter provides a detailed derivation of the energy eigenvalues and eigenfunctions for SIPs using the shape invariance condition.
Finally, the chapter discusses the supersymmetric WKB (SWKB) approximation, which is an extension of the semiclassical approach inspired by SUSY. It is shown that for many problems, the SWKB method provides better accuracy than the standard WKB method, and it is proven that the lowest order SWKB approximation gives exact energy eigenvalues for all SIPs with translation, including most analytically solvable potentials in standard textbooks on quantum mechanics.