Timeline for In order to solve for the states of a spherically symmetric parabolic potential do we need to use cartesian and cylindrical coordinates?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 17, 2022 at 0:15 | comment | added | Níckolas Alves | @AndersGustafson There has to be a mistake in your calculations. While I'm not willing to open the computation in cylindrical coordinates (it will take more time than I have available), I edited the answer to include the examples of spherical and Cartesian separation of variables, since I found the computations readily available on Wikipedia and Weinberg's book. The degeneracy at level $n$ should be $\frac{(n+1)(n+2)}{2}$ regardless of the coordinate system you use when separating variables | |
| Jun 17, 2022 at 0:13 | history | edited | Níckolas Alves | CC BY-SA 4.0 |
added 2815 characters in body
|
| Jun 16, 2022 at 23:24 | comment | added | Anders Gustafson | From this I calculated the number of combinations of (n,k,m) to be greater than the number of combinations of (n,l,m) for $n\geq{1}$ | |
| Jun 16, 2022 at 23:23 | comment | added | Anders Gustafson | I found that it seems that the Energy Levels are the same for spherical and cylindrical coordinates, but in spherical coordinates using $n\geq{0}$, when n is even l can be any even number between 0 and n, and if n is odd l can be any odd number between 1 and n. I found that for cylindrical coordinates if I use k instead of l, with $k\geq{0}$, then if k is even m can be any even number between -k and k, and if k is odd m can be any odd number between -k and k. I found that k can be any integer between 0 and k. | |
| Jun 16, 2022 at 23:20 | comment | added | Anders Gustafson | I tried using separation of variables, in spherical and cylindrical coordinates, and then since I know the eigenvalues for both the Colatitude and the Azimuthal Equations, and how they relate to each other, I used the shooting method to find the eigenvalues for the Radial Equation in spherical Coordinates, and the $\rho$, and z equations in cylindrical coordinates, as well as how the eigenvalues of Radial Equation and Colatitude Equation relate to each other in, and how the eigenvalues of the Azimuthal Equation and $\rho$ equation, and the z and $\rho$ equation relate to each other. | |
| Jun 16, 2022 at 23:18 | comment | added | Anders Gustafson | I looked up degeneracy and from what I read it looks like in QM degeneracy refers to the number of numbers relating to the shape of the wave function, so for instance for a hydrogen atom when n=2 the energy degeneracy is 4 as the possible combinations of (n,l,m) are (2,0,0), (2,1,0), (2,1,1), and (2,1,-1). | |
| Jun 14, 2022 at 8:08 | comment | added | Níckolas Alves | If by "number of configurations" you mean, e.g., the degeneracy of each energy eigenvalue, then yes, all coordinate systems will yield the very same degeneracies | |
| Jun 14, 2022 at 8:07 | comment | added | Níckolas Alves | @AndersGustafson I'm not sure I understand what you mean by "same number of configurations". About the energy levels, yes, you'll find the very same energy levels, although the energy eigenstates might be a bit different. For example, in spherical coordinates they are also angular momentum eigenstates, but in Cartesian coordinates they are not. This just means that, in each energy eingenspace, you're choosing a different basis in each coordinate system | |
| Jun 14, 2022 at 4:58 | comment | added | Anders Gustafson | Does this just mean that I should find the same Energy Levels with each coordinate system or does it also mean that I should find the same number of configurations in using each coordinate system? | |
| Jun 13, 2022 at 12:37 | comment | added | Níckolas Alves | @AndersGustafson Exactly. | |
| Jun 13, 2022 at 3:51 | comment | added | Anders Gustafson | So should you find the same states from separation of variables in each coordinate system, but just expressed differently? | |
| Jun 11, 2022 at 20:44 | history | answered | Níckolas Alves | CC BY-SA 4.0 |