Kvantna mehanika potenciala x^-2

V tem članku se iščejo lastna stanja potenciala x^-2, ki v treh dimenzijah predstavlja polje električnih dipolov. Primer takšnega sistema je gibanje elektrona v bližini polarne molekule. Praviloma vsak privlačni potencial dovoljuje sipalna in diskretizirana vezana stanja. Takšni so potencialna jama, funkcija delta, harmonski potencial in Coulombski potencial. Med naštetimi pa z razlogom ni x^-2 potenciala, saj krši osnovna načela kvantne mehanike. Za rešitev takšnega potenciala je potrebna uporaba bolj prefinjenih metod, kot sta renormalizacija in sebi adjungirana razširitev.

Quantum mechanics of the x^-2 potential

This article will search for eigenstates of the x^-2 potential. Three dimensional analog of this potential would represent an electric field of dipoles as in movement of an electron, which is attached to a closed-shell polar molecule. In general, every attractive potential allows scattering states and discrete bound states. This is true for potential well, delta function, harmonic oscillator and Coulomb potential. Careful readers should have noticed that x^-2 potential is missing in previous sentence, since it violates fundamental principles of quantum mechanics. To find any eigenstate of the x^-2 potential more sophisticated methods such as renormalization and self-adjoint extension need to be used.