# Nuclei Single-particles Harmonic Oscillator Potential

1. Nov 20, 2008

### bluestar

I am looking into the calculations of a harmonic oscillator potential for nuclei single-particles. The information I am looking at is at:
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.

2. Nov 20, 2008

### malawi_glenn

You should not think "Level 0" means n=0 where n is the principal quantum number (same when you look at the wood-saxon + spin-orbit coupling).

level 0: 2 states (l = 0) = 2, means that this is the lowest lying energy level with L=0, i.e 1s, and it has 2states.

Then you next energy level will be:
level 1: 6 states (L = 1) = 6, i.e 1p

Now comes the fun part, solving this potential, you'll get that the 1d (the lowest lying L=2 states) and the SECOND lowest s (L=0) states have the same energy - i.e they are degenerate.

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

And so on.

So how you should read that scheme is that the levels are ordered in energy, solve the S.Equation and look for yourself ;-)

I hope this helped a bit.

3. Nov 21, 2008

### WilliamD

http://en.wikipedia.org/wiki/Image:Shells.png

Shows levels ordered by energy if you don't feel like solving any equations.

As Glenn said, it's not based upon their principal quantum number, but rather the energy level is based on the energy required to fill the subshells (which may have different principal quantum numbers).

Interestingly, notice that even levels only include even numbered subshells and odd levels only include odd numbered subshells.

4. Nov 21, 2008

### bluestar

Wow! Thanks Guys,

That clears up a lot.

I greatly appreciate the help.

5. Nov 21, 2008

### malawi_glenn

So prob, we are here for you.

The Nuclear Shell Model by Kris Heyde

is a quite good book.