What do you mean by spin of the electron ?

 During school days all of us have learned that there is a quantum number associated to spin of the electron (spin quantum number s) which can take only two values +1/2 or -1/2, but what does this number tell us about the electron, ever wondered what is the significance of this spin quantum number in physics (or chemistry) is it related to the spinning of the Earth around it's own axis, lets go into the historical development of the concept of spin and try to understand what exactly spin means 

In Mechanics we define spin as the rotation of a body along its own axis is that significance same for the small particles like electron? before going into much details lets understand a experiment which gave rise to this concept The Stren-Gerlach Experiment

The Stern-Gerlach Experiment : 

Stern-Gerlach Experiment  experiment was performed by Otto Stern and Walther Gerlach in Frankfurt, Germany in 1922 this is the era when the quantum mechanics was introduced to the world by Scrodinger, this setup was designed to understand the angular momentum of the electron a concept introduced by Nelis Bohr in his model of hydrogen like atom This setup consist of a silver source(1) which produces the electron beam(2) to examine which is then passed through an inhomogeneous magnetic field produced (3) the angular momentum is a property of a body due to its motion around a fixed point/axis 

The Stern-Gerlach Experiment setup

 

The expected result of this experiment was that most of the electron will pass through the magnetic flied without any deflection(it was expected that most of the electrons magnetic moment will be aligned randomly) and only few will be deflected those have the magnetic moment aligned  to the magnetic filed and depending upon the alignment we might see streams of electrons, but the observed result was complexly different they observed only two streams of electron which lead to the conclusion that the electron has some intrinsic property(property of the body:: e.g. Mass) associated to angular momentum which lead to the observed result, it was strongly believed that the Quantum Mechanics could explain this result 

Further conclusion from the Experiment:

On further discussion it was concluded that the electron has some intrinsic property associated to angular momentum (Angular spin) which can take only two extreme values (+C and -C) C is just a constant which was later calculated to be (1/2) this interpretation were made with the help of the newly developed quantum Mechanics 
Now it is clear that the spin of a electron  is complexly different from the spin that we use in mechanics , also spin is termed as an intrinsic property of a element 

with the help of the stern-Gerlach experiment we partially understand spin But still the question remains,. what is this spin ?

Let's continue our journey in understanding the spin of the element 

Further Evidences of the spin of the electron:

During 1921, Compton deduced from the properties of ferromagnetic materials that the electron must posses a magnetic moment. Further Uhlenbech and Goudsmit concluded "The electron rotates about its own axis with angular momentum h/4*pi. For this value of the angular momentum, there are only two orientations of the angular momentum vector. the gyro magnetic ratio is twice as large for the rotation about its own axis as for the orbital motion". after this one can summerize by saying that the electron posesses an internal angular momentum which can assume only the values of +/-(h/4*pi) in an arbitrary chosen direction, this was the first incidence that the spin of the electron was termed 1/2.

Wolfgang Pauli gave the mathematical formulation for the spin 1/2 particles which led to the mathematical tool spinors which is than used for explaining the intrinsic property of the fundamental element spin

Spinors 

From 1920's ever since Schrodinger gave the Schrodinger equation, most of the physicists tried to use this equation to solve the unresolved problems and the prime example for this is the theoretical explanation of the Hydrogen spectrum .
When Schrodinger's equation was first introduced, Schrodinger didn't include any terms related to intrinsic property spin, then it was Wolfgang Pauli  who introduce new terms into the Schrodinger's equation to explain the spin of the electron and obtained the mathematical tool SPINOR.

SPINOR's are mathematical entities that play a crucial role in the description of intrinsic property angular spin, in quantum mechanics, which also provides a mathematical framework to understand and model the behavior of particles with angular spin  

The Mathematical representation of the spinor

The spin is the mathematical formulation which comes as the result of the Paul Dirac's formulation of the quantum mechanics but in Schrodinger's formulation of Quantum Mechanics we have to add the spinors by hand to the wave-equation 

In conclusion, the discovery and understanding of electron spin, highlighted by the Stern-Gerlach experiment, revolutionized our comprehension of quantum mechanics. Spin is an intrinsic property, distinct from classical rotation, and its description through spinors has been essential for modeling particle behavior. Importantly, spin is not just a property of fundamental particles; it also defines the behavior of celestial objects like black holes and neutron stars. This advancement underscores the significance of continuous exploration in physics, paving the way for deeper insights and technological innovations. Stay tuned for more insights about the physical meaning of spinors, which will be provided soon.