Question #42480

A particle of mass $m$ moves in a circular orbit around the origin under the influence of a central potential $U(r) = \alpha r$, where $\alpha > 0$ is a constant. The Bohr quantization rule for angular momentum, $L = n\hbar$ (where $n$ is a positive integer), is applicable to this system. Let $E_1$ be the total mechanical energy of the particle in the ground state ($n=1$). An external non-conservative tangential force is applied to the particle, doing work $W$ on it. This causes the particle to slowly transition to a higher excited state where its de Broglie wavelength is exactly half of its value in the ground state. Assuming no energy is radiated during this process, what is the work done $W$ by the external force?