In quantum mechanics, the inverse square potential is a form of a central force potential which has the unusual property of the eigenstates of the corresponding Hamiltonian operator remaining eigenstates in a scaling of all cartesian coordinates by the same constant.
[1] Apart from this curious feature, it's by far less important central force problem than that of the Keplerian inverse square force system.
The potential energy function of an inverse square potential is
is some constant and
is the Euclidean distance from some central point.
is positive, the potential is attractive and if
is negative, the potential is repulsive.
The corresponding Hamiltonian operator
is the mass of the particle moving in the potential.
The canonical commutation relation of quantum mechanics,
] = i ℏ
, has the property of being invariant in a scaling
λ
= λ
λ
is some scaling factor.
are vectors, while the components
are scalars.
In an inverse square potential system, if a wavefunction
ψ ( r )
is an eigenfunction of the Hamiltonian operator
, it is also an eigenfunction of the operator
, where the scaled operators
This also means that if a radially symmetric wave function
ψ ( r )
ψ ( λ r )
is an eigenfunction, with eigenvalue
Therefore, the energy spectrum of the system is a continuum of values.
The system with a particle in an inverse square potential with positive
(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