Supersymmetry in Quantum Mechanics
The problem that usually arises in the study of a quantum system is the integration of the eigenvalue equation of the Hamiltonian, namely the search for eigenstates and the corresponding eigenvalues of energy. The eigenstates, in the representation of the position, are described by eigenfunctions with the same position variable. The relations work modulo a normalization constant. The number of nodes of a function allows to determine its parity: an odd number of nodes characterizes as an antisymmetric function, while a even number of nodes is a symmetric one. The state of a system can be characterized as bosonic or fermionic depending on whether the wave function is even or odd, respectively. The two operators are nilpotent, when applied more than once to an eigenstate, it generates a null ket, regardless of the starting state.