Consider the states with the quantum numbers n = l = 1 and s = 1/2
J = L + S
What is the dimension of the Hilbert space to describe all states with these
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 ms, l and ml and j and mj.
s=1/2, so ms=-1/2 , +1/2
l=1, so ml = -1, 0, +1
j= 1/2 or 3/2, so mj = -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.