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## Homework Statement

Consider the states with the quantum numbers n = l = 1 and s = 1/2

Let

**J = L + S**

What is the dimension of the Hilbert space to describe all states with these

quantum numbers?

## Homework Equations

## The Attempt at a Solution

I believe the dimension of the Hilbert space is dependent on the number of base vectors.

If I am correct, the 'spin-angle functions' would give me the number of bases vector, and thus the dimension of the Hilbert-space.

This is dependent on the quantum numbers s and m

_{s}, l and m

_{l}and j and m

_{j}.

s=1/2, so m

_{s}=-1/2 , +1/2

l=1, so m

_{l}= -1, 0, +1

j= 1/2 or 3/2, so m

_{j}= -1/2 , 1/2 OR -3/2, -1/2, +1/2, +3/2

For j=1/2 you get 2*3*2=12 different combinations, and for j=3/2 it's 2*3*4=24 different combinations.

This gives 36 different base vectors, and therefore the dimension of Hilbert space would be 36?

I have no idea how to go from here. I have the book "introduction to quantum mechanics" by D.J. Griffiths.

If someone could help me out, it would be greatly appreciated.

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