Numerical solution of SE for cylindrical well

[aaaa202]

I am simulating electrons inside a cylindrical well like the one shown on the first figure.
My current work has been on solving the Schrodinger equation numerically for the above potential and then finding corrections to the solution such that it is consistent with Poissons equation.
To do so I need to apply the inverse Laplacian to the electron density as shown in figure 2, which will give me the correction to the potential profile (at least in the first iteration). The problem is however, that applying the inverse Laplacian to the electron density shown gives me a correction as shown on figure 3. As seen it makes sense that this is peaked around the origin, since the electron density is highest at this point. What however doesn't make sense, is that it curves upwards in one direction and doesn't go to zero in a spherically symmetric manner. Everything in my problem has circular symmetry, so I have no idea why it doesn't go to zero with circular symmetry. Does anyone have an idea why this can be? Is it a numerical problem? I doubt so since my mesh size is not very big.

 

[eljose79]

Hello there,

Thank you for sharing your work with us. It sounds like you are working on an interesting problem. Based on the information provided, it seems like the issue you are facing could be due to a few different reasons.

Firstly, it is possible that there may be some errors in your numerical implementation of the inverse Laplacian operation. This could be due to a variety of factors such as numerical precision, convergence criteria, or boundary conditions. I would suggest carefully reviewing your code and checking for any potential errors or inconsistencies.

Another possibility is that your initial assumptions or simplifications for the problem may not accurately reflect the physical system you are trying to model. It is important to carefully consider all the factors that could affect the behavior of the electrons in the cylindrical well, such as external forces, quantum effects, and boundary conditions.

Additionally, it would be helpful to compare your results with other theoretical or experimental studies on similar systems. This could provide valuable insights and help identify any potential discrepancies in your approach.

Overall, I would recommend thoroughly reviewing your methodology and assumptions, as well as seeking feedback and advice from other experts in the field. With careful analysis and consideration, I am sure you will be able to resolve the issue and continue making progress in your research.

Best of luck with your work.