❙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

    ❬r❙r̂²p̂²❙Ψ❭ = r² ❬r❙p̂²❙Ψ❭

    r̂² p̂² = L̂² + (r̂⋅p̂)² - ιħ r̂⋅p̂

    ❬r❙p̂²❙Ψ❭ = 1/r² ❬r❙L̂² + (r̂⋅p̂)² - ιħ r̂⋅p̂❙Ψ❭

        ❬r❙L̂²❙Ψ❭


        ❬r❙r̂²⋅p̂²❙Ψ❭ = r ⋅ ħ/ι ∇ ❬r❙Ψ❭ = ħ/ι r ∂/∂r ❬r❙Ψ❭


        ❬r❙(r̂⋅p̂)²❙Ψ❭ = ❬r❙(r̂⋅p̂)(r̂⋅p̂)❙Ψ❭
                     = r ħ/ι ∂/∂r ❬r❙r̂⋅p̂❙Ψ❭
                     = -ħ² r ∂/∂r (r ∂/∂r) ❬r❙Ψ❭

    1/2m ❬r❙p̂²❙Ψ❭ = -ħ/2m r/r² ∂/∂r r ∂/∂r ❬r❙Ψ❭ + 1/(2mr²) ❬r❙L²❙Ψ❭
                  = -ħ²/2m (∂²/∂r² + 2/r ∂/∂r) ❬r❙Ψ❭ + 1/(2mr²) ❬r❙L̂²❙Ψ❭
                    |               ↓               |  |        ↓       |
                              linear energy             rotational energy

    Hamiltonian can now be written

        Ĥ = -ħ²/2m (∂²/∂r² + 2/r ∂/∂r) + L̂²/(2mr²) + V(│r│)

        With the eigenvalue equation

            ❬r❙E,l,mₗ❭ = E❬r❙E,l,mₗ❭