Discussion of length scales.

A "length scale", or characteristic length, is an important thing that has a significant role in physics. A number of people have asked about it. What does it mean?  Let's first summarize
a few examples. For the harmonic oscillator we noticed that electron wave-functions could be written in terms of a quantity with units of length, a, where \(a^4 = \hbar^2/(mk)\). For a delta-function potential, there is a length scale, \(a = \hbar^2/(m\alpha)\), that appears naturally in the electron ground-state wave-function. Later in the quarter we will see that there is a length scale for the hydrogen atom, \(a=\hbar^2/(m \frac{e^2} {4\pi\epsilon_o})\). This later one is often called the Bohr radius, but it is not really a radius or anything with a simple classically-based meaning, it is an emergent quantum length scale (like the first two examples).
      Notice that they all have some things in common. \(\hbar\) appears in the numerator in all cases. \(\hbar\) makes the quantum length scale non-zero. m appears always in the denominator. A light mass makes the quantum length scale larger. That is why electrons extend far away from the nucleus; nucleons are much heavier; the electron is light and that is critical to the electron's behavior and to the phenomenology of the world. Finally, the strength of the interaction potential, k, \(\alpha\) and \(e^2\), always appears in the denominator. This gives us an idea of where the length scale comes from. It comes from a dynamic tension between potential energy considerations, which favor an electron localized at the minimum of the potential, and kinetic energy considerations, which reflect the considerable cost of confinement. A confinement associated kinetic energy is a quantum thing with no classical analogue, so there are no classical length scales that are truly analogous to these quantum length scales which emerge in a natural way from the Schrodinger wave equation.