Arnold Sommerfeld
The Fine Structure Constant
Abstract
Arnold Sommerfeld made significant contributions to the understanding of the fine-structure constant.
In 1916, he extended the Bohr model of the atom to include elliptical orbits, which led to the prediction of slight shifts in the energy levels of electrons,
especially those moving at speeds close to the speed of light. These shifts resulted in the splitting of spectral lines, known as the "fine structure," which
is where the fine-structure constant gets its name. Physicist appear to struggle to grasp why the fine-structure constant has the value it does.
Unlike many other physical constants it is thought that cannot be derived from theoretical principles and appears to be an empirical number without a clear
theoretical derivation which leads to speculation about its origin and significance.
Introduction
Arnold Johannes Wilhelm Sommerfeld (1868-1951) was a German theoretical physicist who made significant contributions to atomic and quantum physics.
He is best known for introducing the azimuthal quantum number and the magnetic quantum number, which helped explain the fine-structure of spectral lines.
Sommerfeld also worked on wave mechanics and the theory of electrons in metals, which was valuable for understanding thermoelectricity and metallic conduction.
Arnold Sommerfeld
1868-1951
The Fine Structure Constant (denoted as \(\alpha\)) is a dimensionless physical constant that purportedly characterizes the strength of the electromagnetic
interaction between elementary charged particles, such as electrons and protons. Its approximate value is 1/137, or about 0.00729735256933.
The fine structure constant appears in the formulas that describe the splitting of spectral lines in atoms, known as fine structure splitting.
This occurs because of the interaction between the electron's spin and its orbital motion around the nucleus, which is influenced by the strength
of the electromagnetic interaction.
It will be shown that this value can be accurately established to be exactly \(7.297\ 352\ 5693 × 10^{-3}\).
This constant plays a crucial role in quantum mechanics and electromagnetism, influencing the fine structure of atomic spectra and the
behavior of charged particles. Arnold Sommerfeld found that the actual value can be determined from the velocity
of the ground state electron of hydrogen divided by the speed of light;
$$\alpha = \frac{v_e}{c} = 7.297\ 352\ 5693 × 10^{-3}$$
For some reason, the exact origin of its value remains as being one of the fundamental questions in physics.What is strange however is that the value of the constant is actually known and has already been explained by Arnold Sommerfeld, a simple fact which
seems to have been completely forgotten by modern physicists.
Overall, the fine structure constant is a central parameter in quantum mechanics and quantum electrodynamics, governing the behavior of charged particles
and the electromagnetic force. Its ubiquitous presence in these theories highlights its fundamental importance in understanding the interactions
that shape the physical universe.
Conclusion
The value of the fine-structure constant is mainly determined through experimental measurements. High-precision experiments are essential for accurately
determining its value, using techniques such as measuring the electron's magnetic moment or the fine-structure splitting in hydrogen.
This reliance on experimental data, rather than theoretical prediction, can lead to misunderstandings. Alternatively, it is suggested that the fine-structure
constant is not an arbitrary fixed scalar value but simply represents the proportionality between the orbital velocity of the ground state electron and
the speed of light, exactly as suggested by Arnold Sommerfeld.