I am looking into the calculations of a harmonic oscillator potential for nuclei single-particles. The information I am looking at is at:(adsbygoogle = window.adsbygoogle || []).push({});

http://en.wikipedia.org/wiki/Shell_model the specific section “Deformed harmonic oscillator approximated model”

The specific question is, I don’t understand why the l quantum number oscillates from 0 to 1 as n increases. Then after the first 2 levels the next series of l quantum numbers oscillates from 2 to 3 as n increases. And then again after 2 more levels the next series of l quantum numbers oscillates from 4 to 5. And of course all states at a particular level are added together to reflect the cumulative state as shown in the table. Here is a copy of the table that I am confused about which occurs right after the first two equations in that section.

“In particular, the first six shells are:

•level 0: 2 states (l = 0) = 2.

•level 1: 6 states (l = 1) = 6.

•level 2: 2 states (l = 0) + 10 states (l = 2) = 12.

•level 3: 6 states (l = 1) + 14 states (l = 3) = 20.

•level 4: 2 states (l = 0) + 10 states (l = 2) + 18 states (l = 4) = 30.

•level 5: 6 states (l = 1) + 14 states (l = 3) + 22 states (l = 5) = 42. “

Why don’t the levels/states go like this?

•level 0: (l = 0) = 2 >> 2 states

•level 1: (l = 0) = 2 +(l = 1) = 6 >> 8 states

•level 2: (l = 0) = 2 +(l = 1) = 6+(l = 2) = 10 >> 18 states

•level 3: (l = 0) = 2 +(l = 1) = 6+(l = 2) = 10+(l = 3) = 14 >> 32 states

•level 4: (l = 0) = 2 +(l = 1) = 6+(l = 2) = 10+(l = 3) = 14+ (l = 4) = 18 >> 50 states

•level 5: (l = 0) = 2 +(l = 1) = 6+(l = 2) = 10+(l = 3) = 14+ (l = 4) = 18 + (l = 5) = 22. >> 72 states

I understand how the states are calculated because of the table presented at the very top of this section. I just don’t understand how they determine which l states goes with each level.

Your insight would be appreciated.

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# Nuclei Single-particles Harmonic Oscillator Potential

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