❙r❭=❙x,y,z❭

with eigenvalue equations

x̂❙r❭ = x❙r❭
ŷ❙r❭ = y❙r❭
ẑ❙r❭ = z❙r❭

An arbitrary state

    ❙Ψ❭ = ∫∫∫ dx dy dz ❙x,y,z❭❬x,y,z❙Ψ❭
        = ∫ d³r ❙r❭ ❬r❙Ψ❭


Understanding a System:

    measure Ĥ, L̂², L̂𝓏 → Constitutes a complete set of commuting observables (except spin)

    I.E., There is a set of eigenstates that are eigenstates of all three operators.

    Ĥ❙E,l,mₗ❭ = E❙E,l,mₗ❭

    L̂²❙E,l,mₗ❭ = l(l+1)ħ²❙E,l,mₗ❭

    L̂𝓏❙E,l,mₗ❭ = mₗħ❙E,l,mₗ❭

    Ĥ must now include angular momentum

    L̂²= (r̂×p̂)(r̂×p̂) = (geometric identity) = r̂²p̂ - (r̂⋅p̂) + ιħr̂⋅p̂