"Understanding the Spin Quantum Number -1/2 & +1/2

[i.mehrzad]

Homework Statement

Why is the spin quantum number either +1/2 or -1/2

Homework Equations

N.A.

The Attempt at a Solution

I am not sure but this is the way i thought of it.

Whatevrr quantum numbers i have known of so far: Principal, Azumithal tc. all of them have a diffrence of one between any two consecutive quantum numbers. Therefore the magnitude of the quantum numbers are +/- 1/2.

Same magnitude because the the electrons are the same in all respects.Negative because the elecrons are spinning in opposite direstions.

This explanation has no mathematical backup and is just by observation and just wanted to know wether the observation was good enough.

 

[malawi_glenn]

That is not true.

Spin is an instritinc quantity of a particle, and there are many different particles. Bosons, who has integer spin; 0,1,2,..

Fermions has half-integer spin; 1/2 ; 3/2 ; ...

Now the direction of the spin, the so called "magnetic quantum number" in many atomic physics text; is the $$m_l$$. Which obeys, precisely as the quantum number $$l$$ does; $$m_s = -s, -s +1 , ..., 0, ... , s-1, s$$.

The $$s$$ and $$l$$ is the magnitute of the spin, and the $$m_l$$ and $$m_s$$ is the direction (the magnitude of the spin/angular momentum projected on the z-axis, the z-axis is just a convention)

Now, electrons are fermions with spin 1/2, the possible values of that spin in the z-direction is -1/2 and 1/2

 

[Blueshift5]

I can provide a more comprehensive explanation for the spin quantum number being either +1/2 or -1/2. The spin quantum number is a fundamental property of particles, specifically fermions, and it represents the intrinsic angular momentum of the particle. In the case of electrons, the spin quantum number represents the spin of the electron, which can be either "up" or "down" in relation to a given axis.

The reason why the spin quantum number has a magnitude of 1/2 is due to the fact that electrons are spin-1/2 particles. This means that they have a spin angular momentum of ħ/2, where ħ is the reduced Planck's constant. In other words, the spin of an electron is half of the smallest unit of angular momentum in quantum mechanics.

The reason why the spin quantum number can only take on values of +1/2 or -1/2 is due to the spin-statistics theorem, which states that all particles with half-integer spin, such as electrons, must obey the Pauli exclusion principle. This principle states that no two electrons can occupy the same quantum state, which includes the spin state. Therefore, the two possible spin states for an electron are +1/2 and -1/2.

In summary, the spin quantum number being either +1/2 or -1/2 is a fundamental property of particles, specifically electrons, and is related to their spin angular momentum and the Pauli exclusion principle. It is not just a result of observation, but is supported by mathematical and theoretical foundations in quantum mechanics.