content_image/303285.jpg content_image/303281.jpg content_image/303282.jpg content_image/303284.jpg
## 3 Two examples; the radial cylindrical Coulom- bic and the harmonic-oscillator 

A priori, we remove the first-order derivative in the radial cylindrical part of (19) and redefine 

to obtain 

In fact, this 1D radial cylindrical Schr¨ odinger equation provides an effective tool to study the effect of different ˜ V ( z ) settings of (18) on the spectra of two interesting models of fundamental nature. The Coulombic and the harmonic oscillator [30]. Of course, such effects could be tested for other models. 

Let us take a Coulombic radial cylindrical model ˜ V ( ρ ) = − 2 /ρ . In this case, equation (21) would read 

where ℓ = ( 1 − K 2 ϕ ) 1 / $^{2}$, K$_{z}$ = ( n$_{ρ}$ + ℓ + 1) − $^{1}$, and n$_{ρ}$ = 0 , 1 , 2 , · · · is the radial quantum number. Hence, K$_{z}$ = 1 / ( n$_{ρ}$ + √ 1 − K 2 ϕ + 1 ) and 

where K$_{z}$ is to be determined through the solution of (18) under different ˜ V ( z ) settings. 

Next, we consider the radial cylindrical harmonic oscillator model ˜ V ( ρ ) = 

<img src='content_image/303285.jpg'>

<img src='content_image/303281.jpg'>

<img src='content_image/303282.jpg'>

<img src='content_image/303284.jpg'>

