Inverse square potential

The canonical commutation relation of quantum mechanics, ${\displaystyle [{\hat {x}}_{i},{\hat {p}}_{i}]=i\hbar }$, has the property of being invariant in a scaling

${\displaystyle {\hat {p}}_{i}'={\hat {p}}_{i}/\lambda }$, and ${\displaystyle {\hat {x}}_{i}'=\lambda {\hat {x}}_{i}}$,

where ${\displaystyle \lambda }$ is some scaling factor. The momentum ${\displaystyle \mathbf {p} }$ and the position ${\displaystyle \mathbf {x} }$ are vectors, while the components ${\displaystyle p_{i}}$,${\displaystyle x_{i}}$ and the radius ${\displaystyle r}$ are scalars. In an inverse square potential system, if a wavefunction ${\displaystyle \psi (r)}$ is an eigenfunction of the Hamiltonian operator ${\displaystyle {\hat {H}}({\hat {\mathbf {p} }},{\hat {\mathbf {x} }})}$, it is also an eigenfunction of the operator ${\displaystyle {\hat {H}}({\hat {\mathbf {p} }}',{\hat {\mathbf {x} }}')}$, where the scaled operators ${\displaystyle {\hat {p}}_{i}'}$ and ${\displaystyle {\hat {x}}_{i}'}$ are defined as above.

This also means that if a radially symmetric wave function ${\displaystyle \psi (r)}$ is an eigenfunction of ${\displaystyle {\hat {H}}}$ with eigenvalue ${\displaystyle E}$, then also ${\displaystyle \psi (\lambda r)}$ is an eigenfunction, with eigenvalue ${\displaystyle \lambda ^{2}E}$. Therefore, the energy spectrum of the system is a continuum of values.

The system with a particle in an inverse square potential with positive ${\displaystyle C}$ (attractive potential) is an example of so-called falling-to-center problem, where there is no lowest energy wavefunction and there are eigenfunctions where the particle is arbitrarily localized in the vicinity of the central point ${\displaystyle r=0}$.^[2]

• Canonical commutation relation
• Central force
• Scale invariance