8. DEGENERACY AND SEPARABLE POTENTIALS [Levine 3.5] a) Show that if V(r) = Vx(x) + Vy(y)...

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8. DEGENERACY AND SEPARABLE POTENTIALS [Levine 3.5] a) Show that if V(r) = Vx(x) + Vy(y) + V(2), then = Eijk E+E+ Esk Vijk (r) = Ozi (x)yi (y)zi (2). b) Give a formula for the energy levels of a 3d cubic box of width 1 in au. c) Give the lowest 10 energy levels and their degeneracies. Sketch the spectrum. d) Can you estimate the energy and degeneracy of the 10,000-th level? e) Repeat problem for an isotropic oscillator with w = 1 in au. Combine the 3 quantum numbers into 1, and deduce a general formula for degeneracy. 8. DEGENERACY AND SEPARABLE POTENTIALS [Levine 3.5] a) Show that if V(r) = Vx(x) + Vy(y) + V(2), then = Eijk E+E+ Esk Vijk (r) = Ozi (x)yi (y)zi (2). b) Give a formula for the energy levels of a 3d cubic box of width 1 in au. c) Give the lowest 10 energy levels and their degeneracies. Sketch the spectrum. d) Can you estimate the energy and degeneracy of the 10,000-th level? e) Repeat problem for an isotropic oscillator with w = 1 in au. Combine the 3 quantum numbers into 1, and deduce a general formula for degeneracy.