Final answer:
In quantum mechanics, the expectation value of the number operator for a state can be calculated by taking the inner product of the state with the eigenstate of the number operator and multiplying it by the corresponding eigenvalue.
Step-by-step explanation:
In quantum mechanics, the expectation value of an operator is the average value of that operator for a given state. The number operator, denoted as N, represents the number of particles in a particular energy level. The expectation value of the number operator for a state W(f)ψ can be calculated by taking the inner product of the state with the eigenstate of the number operator and then multiplying it by the corresponding eigenvalue. Mathematically, it can be written as:
⟨N⟩ = ⟨W(f)ψ|N|W(f)ψ⟩
where |W(f)ψ⟩ is the state vector and N is the number operator.