Let the normalized eigenstates of the Hamiltonian H = be ,  and . The expectation value  and the variance of H in the state  =  are

Concept:

Normalized eigenvector is nothing but an eigenvector having unit length. It can be found by simply dividing each component of the vector by the length of the vector.

Calculation:

H = 

Eigenvalue =  = 0

(2-λ)((2-λ)² - 1) = 0

λ₁ = 2, λ₂ = 1, λ₃ = 3

|ψ> = (|ψ₁>+|ψ₂>-i|ψ₃>)    

P(E = 2) =  = ,

P(E = 1) =  ,

P(E = 3) = 

<H> = 2× +  + 3× 

= 2

<H²> = 

(Δ E)² = <E²> - <E>² =  

The correct answer is option (3).