Double-valued wave functions have been found, such as | 1 2 , 1 2 ⟩ = e i ϕ ϕ / 2 sin 1 2 ⁡ ⁡ θ θ and | 1 2 , − − 1 2 ⟩ = e − − i ϕ ϕ / 2 sin 1 2 ⁡ ⁡ θ θ . These do not behave well under the ladder operators, but have been found to be useful in describing rigid quantum particles