For the hypothetical logarithmic-potential atom U(r) = a ln(r ÷ r₀), Bohr quantization gives rₙ proportional to n. Which dependence therefore appears in the energy difference between two stationary levels n₁ and n₂?
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Question

For the hypothetical logarithmic-potential atom U(r) = a ln(r ÷ r₀), Bohr quantization gives rₙ proportional to n. Which dependence therefore appears in the energy difference between two stationary levels n₁ and n₂?

Options

G1 Physics · Accepted Answer

For this potential, mv² = a, so the kinetic energy K = a ÷ 2 is the same for every orbit. The orbital radius satisfies rₙ ∝ n. Potential energy is Uₙ = a ln(rₙ ÷ r₀). Since rₙ is proportional to n, the difference in potential energies between two levels is ΔU = a ln(r₂ ÷ r₁) = a ln(n₂ ÷ n₁). The kinetic-energy contribution cancels because it is the same for both levels. Therefore the level-energy difference depends on ln(n₂ ÷ n₁). Correct option: C.