“Inverse square potential”的意思、由来-开放百科全书

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.

Description

where

is some constant and

is the Euclidean distance from some central point. If

is positive, the potential is attractive and if

is negative, the potential is repulsive. The corresponding Hamiltonian operator

Properties

The canonical commutation relation of quantum mechanics,

, has the property of being invariant in a scaling

where

is some scaling factor. It should be noted that the momentum

and the position

are vectors, while the components

and the radius

are scalars. In an inverse square potential system, if a wavefunction

is an eigenfunction of the Hamiltonian operator

, it is also an eigenfunction of the operator

, where the scaled operators

and

are defined as above.

This also means that if a radially symmetric wave function

is an eigenfunction of

with eigenvalue

, then also

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

.^[2]

See also

References

1. ^{{cite journal|last1=Martínez-y-Romero|first1=R. P.|last2=Núñez-Yépez|first2=H. N.|last3=Salas-Brito|first3=A. L.|title=The two dimensional motion of a particle in an inverse square potential: Classical and quantum aspects|journal=Journal of Mathematical Physics|volume=54|issue=5|year=2013|pages=053509|issn=0022-2488|doi=10.1063/1.4804356|url=http://dirac.fciencias.unam.mx/papers/14804356.pdf}}
2. ^{{cite journal|last1=Vasyuta|first1=Vasyl M.|last2=Tkachuk|first2=Volodymyr M.|title=Falling of a quantum particle in an inverse square attractive potential|journal=The European Physical Journal D|volume=70|issue=12|year=2016|issn=1434-6060|doi=10.1140/epjd/e2016-70463-3|url=https://epjd.epj.org/articles/epjd/abs/2016/12/d160463/d160463.html}}