phy-4600/lecture_notes/3-23/overview

Two dimensional harmonic oscillator
───────────────────────────────────

    This is an oscillator with potential V(x,y) = μ/2 ω (x² + y²)

    The hamiltonian here leaves us with a 3-dimensional differential equation

        Ĥ = ιp²/2μ + μ/2 ω² (x² + y²) = 1/2μ (p²𝓍 + p²𝓎 + p²𝓏) + μ/2 ω²(x² + y²)

        (pic) This is then split into x,y, and z parts.

            (pic) Solved Ψ(z)


        Put together solutions of Ψ(x,y) and Ψ(z).

        (pic) To find position space representation of Ψ(x,y), recall the Hermitian Polynomials solution

            !!! STUDY THIS !!!

    Developed the harmonic oscillator in polar coordinates

    Ĥ𝓍𝓎 = -ħ²\2μ ∇² + μ/2 ω² r²

    (pic) Can be solved using separation of variables.

        Ψ(r,θ) = R(r) Θ(θ)

        The problem is invariant under rotations about z, i.e. L̂𝓏 commutes with Ĥ, so the solutions must be eigenstates of both Ĥ and L̂𝓏.

        Don't use L̂𝓏 Θ = ±ιmħ Θ

        Use L̂𝓏² Θ = -m²ħ² Θ ⇒ Θ(θ) = exp(±imθ)

        (pic) further developed hamiltonian using this information