Inverse square potential

The potential energy function of an inverse square potential is

${\displaystyle V(r)=-{\frac {C}{r^{2}}}}$,

where ${\displaystyle C}$ is some constant and ${\displaystyle r}$ is the Euclidean distance from some central point. If ${\displaystyle C}$ is positive, the potential is attractive and if ${\displaystyle C}$ is negative, the potential is repulsive. The corresponding Hamiltonian operator ${\displaystyle {\hat {H}}({\hat {\mathbf {p} }},{\hat {r}})}$ is

${\displaystyle {\hat {H}}={\frac {{\hat {\mathbf {p} }}^{2}}{2m}}-{\frac {C}{{\hat {r}}^{2}}}}$,

where ${\displaystyle m}$ is the mass of the particle moving in the 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