# Total Angular Momentum in Nuclear Shell Model

• mackabra
In summary, the allowed total angular momentum quantum numbers J for 2 protons in a nuclear shell model state j = 3/2 are 0, 1, 2, 3, 4, and 5, depending on the total spin of the system.
mackabra

## Homework Statement

Calculate the allowed total angular momentum quantum numbers J for 2 protons in a nuclear shell model state j = 3/2.

## Homework Equations

J = j1 + j2 where j are the total angular momenta of each proton. Protons are spin 1/2, orbital angular momentum L is not given.

## The Attempt at a Solution

Protons are identical, and therefore cannot occupy the same state. I need some clarification on the the addition of the angular momenta.
I know that for a completely filled state, J = 0. However does this come from the fact that all the mj's cancel out (mj = -j, -j+1, ... j) , and how is this related to J = j1+j2+j3+etc...?

Any help would be appreciated
Thank you

for your question. In the case of two identical particles, such as two protons, the allowed total angular momentum quantum numbers J will depend on the total spin of the system. Since protons have a spin of 1/2, the possible values for the total spin of the system can be 0 or 1.

If the total spin is 0, then the allowed values for J will be given by J = j1 + j2, where j1 and j2 are the individual angular momentum quantum numbers for each proton. In this case, since j = 3/2, the possible values for J will be 0, 1, 2, and 3.

If the total spin is 1, then the allowed values for J will be given by J = j1 + j2 + 1, where j1 and j2 are the individual angular momentum quantum numbers for each proton. In this case, since j = 3/2, the possible values for J will be 1, 2, 3, and 4.

In general, for N identical particles with spin j, the allowed values for J will be given by J = Nj, (N-1)j + 1, (N-2)j + 2, ... , j + (N-1), and j. Therefore, in the case of two protons with j = 3/2, the allowed values for J will be 0, 1, 2, 3, 4, and 5.

I hope this helps clarify your understanding of how total angular momentum is related to individual angular momenta in a system of identical particles. Please let me know if you have any further questions.

## What is total angular momentum in the nuclear shell model?

Total angular momentum in the nuclear shell model is a quantum mechanical property that describes the overall rotation of a nucleus. It is the sum of the individual angular momenta of all the nucleons in the nucleus.

## How is total angular momentum calculated in the nuclear shell model?

Total angular momentum is calculated by adding the individual angular momenta of all the nucleons in the nucleus, taking into account their spin and orbital angular momentum.

## What is the significance of total angular momentum in the nuclear shell model?

Total angular momentum plays a crucial role in determining the energy levels and stability of a nucleus. It also affects the nuclear reactions and decay processes of the nucleus.

## How does total angular momentum affect nuclear properties?

Total angular momentum affects the shape, magnetic properties, and stability of a nucleus. It also determines the allowed transitions between energy states in a nucleus.

## What is the relationship between total angular momentum and nuclear spin?

Total angular momentum and nuclear spin are closely related, as they both describe the rotation of a nucleus. The spin of a nucleus is one component of its total angular momentum.

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