A physics student is reviewing the historical development of atomic structure and encounters the idea that only certain circular orbits are permitted for an electron bound to a hydrogen nucleus. The student would like to understand, at a qualitative level, why discrete radii emerge and how this feature resolves an apparent conflict with classical electromagnetic theory.
(a) Describe the physical requirement that restricts electron orbits to a discrete set of radii in the hydrogen atom. Your explanation should refer to both the geometry of a circular path and the wave-like nature of the electron.
(b) Using qualitative reasoning, explain how the radius of an allowed orbit changes as the integer quantum number n increases. Include a brief discussion of the corresponding change in the electron’s total energy.
(c) Classical electrodynamics predicts that an accelerating charge should continuously radiate energy and spiral into the nucleus. Explain why the assumption of discrete orbits circumvents this prediction within the Bohr framework.