Expectation Value of Q in orthonormal basis set Psi

[Decadohedron]

Homework Statement

Suppose that { |ψ₁>, |ψ₂>,...,|ψ_n>} is an orthonormal basis set and all of the basis vectors are eigenvectors of the operator Q with Q|ψ_j> = q_j|ψ_j> for all j = 1...n.

A particle is in the state |Φ>.

Show that for this particle the expectation value of <Q> is

∑_j=1ⁿq_j |<Φ| ψ_j> |^2

Homework Equations

The Attempt at a Solution

1. |Φ> = ∑a_n| ψ_n> with a_n = <Ψ_n|Φ>

2. Q|ψ_j> = q_jΣb_n |ψ_n>, with b_n = <Ψ_n| Ψ_j>

After introducing the delta
a_m = QΣb_nΨ_n

Then, I should have

<Q> = Σ | Q |^2

= Σ | Q|Ψ> | ^2
= Σ q_j |Σ b_n | Ψ_n|^2
= Σ q_j |a_m|^2
= Σ q_j | <Ψ_m|Φ> |^2

Then by taking the inner product

= Σ q_j | < Φ|Ψ> | ^2

That's about as close as I could get but I have this general feeling of being very wrong. Not sure how else to approach this.

 

[kuruman]

Nowhere in your attempt at a solution is there the starting point for the expectation value of the operator, namely $<\Phi|Q|\Phi>.$ Start from that and then use what you already know, namely $|\Phi>=\sum <\psi_n|\Phi>|\psi_n>$ and $Q|\psi_n> = q_n|\psi_n>.$ Note that both the bra $<\Phi|$ and the ket $|\Phi>$ in the expectation value bra-ket are summations.