Translation Operators

    Matrix mechanical expression:

    T̂(a𝓍 î) ❙x,y,z❭ = ❙x+a𝓍,y,z❭
    T̂(a𝓎 î) ❙x,y,z❭ = ❙x,y+a𝓎,z❭
    T̂(a𝓏 î) ❙x,y,z❭ = ❙x,y,z+a𝓏❭

    Position space form:

    T(a𝓍 î) = exp(-ι p𝓍 a𝓍/ħ )

    Translation operators commute since momentum operators commute.

    But, [x̂ᵢ,p̂ⱼ]=ιħδᵢⱼ

    T̂(𝐚) = exp(-ι p𝓍 a𝓍/ħ ) exp(-ι p𝓎 a𝓎/ħ ) exp(-ι p𝓏 a𝓏/ħ )
         = exp(-ι 𝐩 𝐚/ħ )

    ❬r❙p̂❙Ψ❭ = ħ/ι ∇ ❬r❙Ψ❭