If electric charge did not exist and electrons were bound to protons by the gravitational force

step 1 We'd like to derive these expressions. Now for a real hydrogen atom, the Bohr radius is given by

If electric charge did not exist and electrons were bound to...

Key Concepts

Spectral Lines and Series Transitions

Spectral lines, such as those identified in the Balmer series, result from transitions between quantized energy levels in an atom. The specific case of the H? line corresponds to a transition from a higher energy level (n = 3) to a lower one (n = 2). In both electromagnetic and gravitationally bound systems, the frequency of emitted radiation is directly related to the energy difference between the levels. Analyzing these transitions highlights how the binding force alters the emitted light’s frequency and energy, with gravitationally bound atoms producing frequencies that are orders of magnitude different from those of real hydrogen.

Bohr Model

The Bohr model is a semiclassical approach that quantizes the allowed orbits of a particle in a central force field by enforcing quantization of angular momentum. By equating the centripetal force with the binding force, one can derive discrete energy levels and a characteristic length scale (the Bohr radius). This framework, originally developed for the Coulomb potential in hydrogen, provides a template that can be adapted to other central potentials, such as the gravitational potential in a hypothetical scenario where gravity binds the electron and proton.

Gravitational Potential in Quantum Mechanics

In classical mechanics, the gravitational potential between two masses is given by V(r) = -Gm1m2/r. Replacing the Coulomb potential with this gravitational potential in a quantum mechanical treatment leads to analogous expressions for bound states. However, since the gravitational constant G is extremely small compared to electrostatic constants, the resulting energy scales and characteristic lengths (like a modified Bohr radius) become vastly different, allowing one to compare the efficiency of binding in gravitational versus electromagnetic interactions.

Energy Level Quantization

The quantization of energy levels in a central potential arises from solving the Schrödinger equation under the condition of bound states. By applying quantization rules, one obtains expressions for discrete energy levels (En) that depend on the strength of the binding force and the reduced mass of the system. When substituting the gravitational interaction for the Coulomb interaction, the resulting energy levels are determined by Gm1m2 instead of e^2, illustrating how the binding energy and spacing between levels are affected by the nature of the force.

Reduced Mass in Two-Body Systems

For any two-body system, such as the electron-proton system in an atom, the reduced mass is introduced to simplify the problem by reducing it to an equivalent one-body problem. The reduced mass accounts for the fact that both bodies orbit around a common center of mass. This is crucial in accurately deriving expressions for the Bohr radius and energy levels, regardless of whether the binding force is electromagnetic or gravitational.

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