Universality and quantum systems  In quantum mechanics, universality is the observation that there are properties for a large class of systems that are independent of the exact structural details of the system. The notion of universality is familiar in the study and application of statistical mechanics to various physical systems since its introduction in a very precise fashion by Leo Kadanoff. Although not quite the same, it is closely related to universality as applied to quantum systems. This concept links to the essence of renormalization and scaling in many problems. Renormalization is based on the notion that a measurement device of wavelength formula_1 is insensitive to details of structure at distances much smaller than formula_1. An important consequence of universality is that one can mimic the "real" short-structure distance of the measurement device and the system to be measured by "simple" short-distance structure. Even though it is seen that scaling, universality and renormalization are closely related, they are not to be used interchangeably. For low energy particles, the exact detail of the potential step is irrelevant. This means that the intermediate change in the potential between formula_3 and formula_4 is irrelevant for particles with extremely low energies. An important consequence of this is that theoretical calculations made for a Heaviside step function potential work perfectly well for potentials that have a finite spatial rate of change in potential: finite ∆formula_5 over ∆formula_6, which is the case with physical situations, for e.g. - quantum confinement systems of nano dots, etc. For a general potential barrier: The time-dependent Schrödinger's equation in one-dimension is: It is instructive to analyse this equation after expanding formula_9 in the plane wave basis formula_10. formula_11, then, is real for formula_12. For when formula_13, the relevant length scale in this problem is the range of the wavefunction under the barrier which is given by: formula_14 for low energy particles, i.e. - formula_15. When formula_16, the wave cannot resolve the detailed shape of the potential and one must have formula_17. The wavelength of a particle in the region of zero potential (essentially, a free particle) with energy formula_18 is given as: formula_19. For low-energy waves incident on this potential from the left: formula_20. Essentially, for low energy, or large wavelength, particles, the exact detail of the potential barrier is irrelevant.