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.