How Can J Be a Fraction in Spectroscopic Notation?

MariusM
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Homework Statement


Consider a ^3D_X where X=3/2 state.

a) What are the possible values of S, L, J and J_z?

Homework Equations


Spectroscopic notation for this LS coupling is ^YL_J where Y=2S+1. J ranges from|L-S| to |L+S|

The Attempt at a Solution


Since L=2 and S must be equal to 1, how can J be a fraction? Shouldn't J be either 1, 2, 3?
 
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You are absolutely right. The problem is incorrect as it stands.
 
Thanks for helping me clarify this!
 
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This figure is from "Introduction to Quantum Mechanics" by Griffiths (3rd edition). It is available to download. It is from page 142. I am hoping the usual people on this site will give me a hand understanding what is going on in the figure. After the equation (4.50) it says "It is customary to introduce the principal quantum number, ##n##, which simply orders the allowed energies, starting with 1 for the ground state. (see the figure)" I still don't understand the figure :( Here is...
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The last problem I posted on QM made it into advanced homework help, that is why I am putting it here. I am sorry for any hassle imposed on the moderators by myself. Part (a) is quite easy. We get $$\sigma_1 = 2\lambda, \mathbf{v}_1 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} \sigma_2 = \lambda, \mathbf{v}_2 = \begin{pmatrix} 1/\sqrt{2} \\ 1/\sqrt{2} \\ 0 \end{pmatrix} \sigma_3 = -\lambda, \mathbf{v}_3 = \begin{pmatrix} 1/\sqrt{2} \\ -1/\sqrt{2} \\ 0 \end{pmatrix} $$ There are two ways...
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