# Relation between spin degrees of freedom and the dimensions of Hilbert space

I came across a question which reads

"Suppose the spin degree of freedom of two particles (nonzero rest mass and nonzero spin) is described completely by a Hilbert space of dimension twenty one. Which of the following could be the spin of one of the particles?

(a) 2

(b) 3/2

(c) 1

(d) 1/2"

I am confused about the basics behind this problem; how can we relate spin degrees of freedom with dimensions of Hilbert space?

• I find it hard to believe that you could encounter this question without learning about the relation between the spin of a particle and the dimension of its spin space first. Where did you come across it? – ACuriousMind Feb 11 at 13:06
• @ACuriousMind I am an undergrad and there's an exam which is set for both undergrads and students doing their masters here in India. I found the question there. – Harshdeep Singh Feb 11 at 13:26
• I remember an exact question in the near past but I can't find it (deleted???). Anyway, in how many ways you could express 21 as product of two integers ??? – Frobenius Feb 11 at 17:50