Quantum cylindrical quadrupole

The quantum cylindrical quadrupole is a solution to the Schrödinger equation, i ℏ ∂ ∂ t ψ ( x , t ) = − ℏ 2 2 m ∂ 2 ∂ x 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , {\displaystyle \mathrm {i} \hbar {\frac {\partial }{\partial t}}\psi (x,t)=-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial x^{2}}}\psi (x,t)+V(x)\psi (x,t),} where ℏ {\displaystyle \hbar } is the reduced Planck constant, m {\displaystyle m} is the mass of the particle, i {\displaystyle \mathrm {i} } is the imaginary unit and t {\displaystyle t} is time. One peculiar potential that can be solved exactly is when the electric quadrupole moment is the dominant term of an infinitely long cylinder of charge.

Source: Wikipedia — Quantum cylindrical quadrupole (CC BY-SA 4.0)