who discovered spinning electron concept? what lead to its discovery?

The concept of the spinning electron was introduced by Samuel Goudsmit and George Uhlenbeck in 1925. They proposed the idea of electron spin to explain certain anomalies in atomic spectra that couldn't be accounted for by then-existing quantum theories. The concept of electron spin is now a fundamental aspect of quantum mechanics and plays a crucial role in understanding the behavior of electrons in atoms, molecules, and solids.

Samuel Goudsmit and George Uhlenbeck arrived at the concept of electron spin through a combination of theoretical insights and the need to explain experimental anomalies, particularly in atomic spectra.

Background and Motivation:

1. Zeeman Effect: The motivation for introducing electron spin came from the study of the Zeeman effect, which describes the splitting of spectral lines in a magnetic field. The existing quantum theory, particularly the Bohr model and Sommerfeld's extension, could not fully explain the observed fine structure of the spectral lines in the hydrogen atom.

2. Fine Structure in Spectra: The observed fine structure of atomic spectral lines indicated that there were additional quantum numbers needed beyond the principal quantum number $n$ and the orbital angular momentum quantum number $l$. The anomalous Zeeman effect (a type of Zeeman effect that couldn't be explained by orbital angular momentum alone) suggested the presence of an intrinsic angular momentum associated with the electron.

Development of the Concept:

1. Electron as a Spinning Particle: Uhlenbeck and Goudsmit proposed that the electron itself has an intrinsic form of angular momentum, which they called "spin." This spin would have a quantum number $s = \frac{1}{2}$, and the associated magnetic moment would explain the fine structure and the splitting of spectral lines observed in the anomalous Zeeman effect.

2. Pauli Exclusion Principle: Around the same time, Wolfgang Pauli had formulated the exclusion principle, which required that no two electrons in an atom could occupy the same quantum state. The introduction of electron spin provided the additional quantum number needed to satisfy Pauli's principle, allowing for the explanation of the observed electron configurations in atoms.

Mathematical Formalism:

• Goudsmit and Uhlenbeck did not initially develop a full mathematical theory of spin. Instead, they proposed that the electron has an intrinsic angular momentum with a magnitude given by $\sqrt{s(s+1)}\hbar$
• • where $s = \frac{1}{2}$
  • The magnetic moment associated with the spin was assumed to be $\mu_s = g_s \cdot \mu_B \cdot$, where $g_s$ is the g-factor (later found to be approximately 2), and $\mu_B$ is the Bohr magneton.

Evidence and Confirmation:

1. Agreement with Experimental Data: The introduction of spin allowed for a much better match between theoretical predictions and experimental observations of the fine structure of spectral lines. The splitting patterns predicted by the spin hypothesis matched the patterns seen in the anomalous Zeeman effect.

2. Dirac's Theory (1928): The concept of electron spin was further solidified with the development of the Dirac equation in 1928, which naturally incorporated spin as a relativistic effect. The Dirac equation predicted the existence of spin $\frac{1}{2}$ particles and provided a theoretical foundation for the magnetic moment of the electron.

Conclusion:

Goudsmit and Uhlenbeck's proposal of electron spin was based on the need to explain specific experimental phenomena, particularly the anomalous Zeeman effect and fine structure in atomic spectra. While they did not initially provide a full mathematical derivation, the concept was quickly accepted because it provided a necessary and successful explanation for observed data. Subsequent developments in quantum mechanics, especially Dirac's work, provided the rigorous mathematical framework for understanding electron spin.