A particle of mass m moves in a circular orbit in a central potential field U(r) = 1/2 k r². If Bohr's quantization conditions are applied, radii of possible orbits and energy levels vary with quantum number n as:

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Accepted Answer

rₙ ∝ √n, Eₙ ∝ n. Central force F = kr gives mv^2/r = kr so v ~ r. Quantization mvr = n*hbar with v ~ r gives r^2 ~ n, i.e. r_n ~ sqrt(n). Energy E = (1/2)mv^2 + (1/2)k r^2 ~ r^2 ~ n, so E_n ~ n. Hence r_n ~ sqrt(n), E_n ~ n.

A particle of mass m moves in a circular orbit in a central potential field U(r) = 1/2 k r². If Bohr's quantization conditions are applied, radii of possible orbits and energy levels vary with quantum number n as:

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