Quantum Surfing

Here we report a project in which time-dependent supersymmetry has been employed to derive a new potential and eigen- functions that satisfy the Schr ̈odinger equation. The supersymmetry method is outlined and we apply it to a wavefunction obeying the free-particle Schr ̈odinger equation. This leads to an exactly soluble model in which a quantum particle is seen to “surf” on a time-dependent potential. The model can be solved and understood within both classical and quantum mechanics and the relationship between the two approaches is discussed. The mathematics of this formalism is accessible to a final year British undergraduate making supersymmetry derived Hamiltonians suitable as a final year theoretical physics research project.