phy-4600/lecture_notes/3-23/overview

Two dimensional harmonic oscillator
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    This is an oscillator with potential V(x,y) = μ/2 ω² (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-- HARMONIC? DUBIOUS Polynomials solution

            !!! STUDY THIS !!!

    Developed the harmonic oscillator in polar coordinates

    Ĥ𝓍𝓎 = -ħ²/2μ ∇² + μ/2 ω² r²
        = -ħ²/2μ (∂²/∂r² + 1/r ∂/∂r + 1/r² ∂²/∂θ²) + 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

    Ĥ𝓍𝓎 = -ħ²/2μ (∂²/∂r² + 1/r ∂/∂r + 1/r² ∂²/∂θ²) + μ/2 ω² r².
        = -ħ²/2μ (∂²/∂r² + 1/r ∂/∂r) + -ħ²/(2μr²) L̂𝓏²/ħ² + μ ω² r²/2

    L̂𝓏² ≐ ħ²∂²/∂θ²
    L̂𝓏 ≐ -ιħ∂/∂θ