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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.cite-ref-mart-nez-y-romeron-ez-y-pez2013-1-0[1] Apart from this curious feature, it's by far less important central force problem than that of the Keplerian inverse square force system.

• Description

• Properties

• See also

• References

The potential energy function of an inverse square potential is

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

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

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

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

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

p ^ ^ i ′ = p ^ ^ i / λ λ {\displaystyle {\hat {p}}_{i}'={\hat {p}}_{i}/\lambda } , and x ^ ^ i ′ = λ λ x ^ ^ i {\displaystyle {\hat {x}}_{i}'=\lambda {\hat {x}}_{i}} ,

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

This also means that if a radially symmetric wave function ψ ψ ( r ) {\displaystyle \psi (r)} is an eigenfunction of H ^ ^ {\displaystyle {\hat {H}}} with eigenvalue E {\displaystyle E} , then also ψ ψ ( λ λ r ) {\displaystyle \psi (\lambda r)} is an eigenfunction, with eigenvalue λ λ 2 E {\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 C {\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 r = 0 {\displaystyle r=0} .cite-ref-vasyutatkachuk2016-2-0[2]

• Canonical commutation relation

• Central force

• Scale invariance

cite-note-mart-nez-y-romeron-ez-y-pez2013-11. ↑ citerefmart-nez-y-romeron-ez-y-pezsalas-brito2013Martínez-y-Romero, R. P.; Núñez-Yépez, H. N.; Salas-Brito, A. L. (2013). "The two dimensional motion of a particle in an inverse square potential: Classical and quantum aspects" (PDF). Journal of Mathematical Physics. 54 (5): 053509. doi:10.1063/1.4804356. ISSN 0022-2488. Archived from the original (PDF) on 2019-02-04. Retrieved 2017-06-11.

cite-note-vasyutatkachuk2016-22. ↑ citerefvasyutatkachuk2016Vasyuta, Vasyl M.; Tkachuk, Volodymyr M. (2016). "Falling of a quantum particle in an inverse square attractive potential". The European Physical Journal D. 70 (12). arXiv:1505.04750. doi:10.1140/epjd/e2016-70463-3. ISSN 1434-6060. S2CID 118371904.