
A mathematical framework that relates complex quantum problems to simpler ones via supersymmetry transformations.Supersymmetric quantum mechanics is a mathematical framework that reveals a hidden symmetry in certain quantum systems, allowing a complex problem to be transformed into a simpler one. It was originally inspired by particle physics' supersymmetry, which pairs bosons and fermions, but in quantum mechanics it provides a powerful tool for exactly solving Schrödinger equations for a wide class of potentials. The key insight is that for many potentials, there exists a partner potential with the same energy spectrum, except for the ground state. By iterating this process, one can generate a chain of solvable potentials and obtain exact wavefunctions and energy levels.
Supersymmetric quantum mechanics was developed in 1981 by Edward Witten, who introduced it as a toy model to study supersymmetry breaking in quantum field theory, and it was later extended by Gendenshtein and others to solve exactly solvable potentials.
The framework introduces two operators, A and A†, which act as ladder operators that connect the eigenstates of two partner Hamiltonians, H1 and H2. If H1 has a known ground state, then H2 shares all its excited states with H1, but lacks the ground state. This relationship allows one to solve H2 by starting from H1, and vice versa. The method also reveals a hidden algebraic structure, often related to shape invariance, which provides a systematic way to find exact solutions for potentials like the harmonic oscillator, Coulomb potential, and many others.
It is used in theoretical physics to solve quantum mechanical problems exactly, and in quantum chemistry and condensed matter physics to approximate spectra of complex systems, as well as in the study of integrable models.
The shape invariance condition, a special case of supersymmetric quantum mechanics, allows one to solve the hydrogen atom without ever writing down the full wavefunction—just by using algebraic ladder operators.
