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Question
Why is the spin quantum number of electron is + 1/2 or - 1/2 and not any other numerical term?
Why is the spin quantum number of electron is + 1/2 or - 1/2 and not any other numerical term?
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Solution
Firstly, we should understand what is an electron “spin”. An electron is an elementary particle, and is described as a unit excitation of the electron field. It is not a “billiard ball” which we associate with it in our mind, therefore it is meaningless to talk about size and dimensions of an electron; essentially electron is a point.
Having said that, it is important to realize that electrons do not spin literally. When a classical object spins, it is usually about an axis. Since electrons are not classical “billiard balls”, their spin is a quantum phenomenon, having no classical analogue. It is essentially a quantum state, which was misnomered as spin due to historical reasons (see Uhlenbleck's experiment).
The modern interpretation of electron spin comes from the Stern Gerlach experiment, in which a beam of charged particles fired through an inhomogeneous magnetic field was found to deflect from their original path. The results of the experiment show that particles possess an intrinsic angular momentum that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values.
Now, when a beam of electrons is passed through a SG apparatus, it is found that half of the electrons deflect towards the positive z-axis, while half deflect towards the negative z-axis. Therefore, by convention, those electrons deflected towards the positive z-axis are designated a positive “spin”, while those deflected towards the negative z-axis are designated a negative spin.
The ±1/2 term comes by solving the Dirac equation for electrons, which explicitly comes out to be fractional for fermions (like electrons) and integral for bosons (like photons, Higgs etc.).
Having said that, it is important to realize that electrons do not spin literally. When a classical object spins, it is usually about an axis. Since electrons are not classical “billiard balls”, their spin is a quantum phenomenon, having no classical analogue. It is essentially a quantum state, which was misnomered as spin due to historical reasons (see Uhlenbleck's experiment).
The modern interpretation of electron spin comes from the Stern Gerlach experiment, in which a beam of charged particles fired through an inhomogeneous magnetic field was found to deflect from their original path. The results of the experiment show that particles possess an intrinsic angular momentum that is closely analogous to the angular momentum of a classically spinning object, but that takes only certain quantized values.
Now, when a beam of electrons is passed through a SG apparatus, it is found that half of the electrons deflect towards the positive z-axis, while half deflect towards the negative z-axis. Therefore, by convention, those electrons deflected towards the positive z-axis are designated a positive “spin”, while those deflected towards the negative z-axis are designated a negative spin.
The ±1/2 term comes by solving the Dirac equation for electrons, which explicitly comes out to be fractional for fermions (like electrons) and integral for bosons (like photons, Higgs etc.).
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