Inverse-Square Potential in Quantum Mechanics
The inverse-square potential in quantum mechanics
Intro
The inverse-square potential in quantum mechanics is connected to phenomena like electron capture by neutral polar molecules, the Efimov effect in a system of three identical bosons, the transition between asymptotically-free and conformal phases in QCD-like theories as a function of the ratio of the number of quark flavors to colors, and the AdS/CFT correspondence. The attractive \[ 1/x^2 \] potential is interesting because it naturally calls upon the framework of the Renormalization Group. This approach will be seen to mirror the modern treatment of quantum effective field theories whereby one demans that long-distance observables remain insensitive to the adjustment of fine details at short distances.
Pure Inverse-Square Potential
Consider the potential \[ V(x) = \begin{cases} \infty & \text{if } x \leq 0 \\ \frac{\alpha}{x^2} & \text{if } x > 0 \end{cases} \] for \[ -1/4 < \alpha < 0\]. Work in units with \[ \hbar = 1\] so energy is the reciprocal of time, \[ \hbar^2/2m = 1 \] so that energy is also the reciprocal of length-squared. Thus time and length-squared have equivalent dimensions. \[\alpha\] is dimensionless.
Ref
Taken directly from: https://arxiv.org/pdf/1707.04388