Question

The spin wave function of two electrons is (x ↑ x ↓ –x ↓ x ↑)/ √2. What is the eigen value of S₁.S₂? S₁ and S₂ are spin operators of 1 and 2 electrons

Answer 1

Given spin state is a singlet state, that is S = 0 

S₁ + S₂ = S 

Form scalar product by itself 

S₁ · S₁ + S₁ · S₂ + S₂ · S₁ + S₂ · S₂ = S · S 

S₁² + 2S₁ · S₂ + S₂² = S₂ = 0 

Now, S₁² = S₂² = (1/2)(1/2 + 1) = 3/4 

Therefore S₁ · S₂ = −(3/4)ℏ²