For the ‘particle on a ring’ system described in Exercise E7C.5(a) the expectation value of a quantity represented by the operator ˆΩ is given by

Ωₘₗ = ²π∫₀ ψ*ₘₗ (ϕ)ˆΩψₘₗ (ϕ)dϕ

where ψₘₗ (ϕ) are the normalized wavefunctions ψₘₗ (ϕ) = (1/2π)¹/² eᶦᵐˡϕ, with mₗ an integer. Compute the expectation value of the position, specified by the angle ϕ, for the case mₗ = +1, and then for the general case of integer mₗ. E7C.8(b) For the system described in Exercise E7C.8(a), evaluate the expectation value of the angular momentum represented by the operator (h/i)d/dϕ for the case mₗ = +1, and then for the general case of integer mₗ.