(a) Show that in the ground state of the hydrogen atom the...

(a) Show that in the ground state of the hydrogen atom the speed of the electron can be written as $v=\alpha c$ where $\alpha$ is the fine-structure constant. (b) From the value of $\alpha$ what can you conclude about the neglect of relativistic effects in the Bohr calculations?

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Key Concepts

Bohr Model

The Bohr model of the hydrogen atom is a semiclassical representation in which the electron moves in circular orbits around the nucleus with quantized angular momentum. This model provides formulas for the radii and energies of electron orbits by equating the centrifugal force with the Coulomb attraction, and it successfully explains the spectral lines of hydrogen despite its simplifications.

Fine-Structure Constant

The fine-structure constant, denoted by ?, is a dimensionless constant that characterizes the strength of electromagnetic interactions. With a value approximately equal to 1/137, it appears naturally in expressions for electron velocity in atomic orbitals (such as v = ?c) and plays a central role in both quantum electrodynamics and atomic physics, linking fundamental constants in the formulation of atomic structure.

Relativistic Effects in Atomic Physics

Relativistic effects become important in atomic physics when particle speeds approach the speed of light. However, in the context of the Bohr model, the electron’s speed in the ground state (v = ?c) is much lower than c due to the small value of ?. This justifies the neglect of relativistic corrections in the Bohr calculations, as the non-relativistic approximations remain valid and accurate for describing the hydrogen atom.

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