phy-521/2D_oscillator/formalism.motes
2020-12-23 16:45:08 -05:00

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ℋ❙φ❭ = (𝓍𝓎 + 𝓏)❙φ❭ = E❙φ❭.
𝓍𝓎 = (P²𝓍 + P²𝓎)/(2μ) + ½ μω² (X² + Y²).
𝓏 = P²𝓏/2μ.
❙φ❭ = ❙φ𝓏𝓎❭ ⛒ ❙φ𝓏❭.
ℋ𝓍𝓎❙φ❭ = E𝓍𝓎❙φ❭.
E = E𝓍𝓎 + E𝓏.
❬z❙φ𝓏❭ = (2πħ)⁻¹ exp(ιp𝓏z/ħ).
E𝓏 = p𝓏²/2μ where p𝓏 is an arbitrary real constant.
= 𝓍𝓎𝓏.