About Angular Momentum and Spin at Quantum level.

Math manipulation leads to L- and L+,

they are the lowering and raising operators, respectively. The raising operator “L+” increases the quantum number associated with the z-component of angular momentum (m) by one unit, and the lowering operator “L-” decreases it by one unit. To illustrate, if we have L+ “operated” on state m particle, then measure on Z: LzL+|m>

If we have L-L+

Now the result expose the fundamental discrepancies between quantum mechanics and classical mechanics: the Mm part is the residual momentum distributed on x and y even it’s full maximum aligned on z axis.

A question: why the total angular momentum measurement on a state requires using L squared but not L? answer by chatGPT 4: In quantum mechanics, the components of angular momentum do not commute, which means they cannot simultaneously possess a definite value, i.e., two components of angular momentum cannot be simultaneously measured to infinite precision. For example, if we have angular momentum L with three components, L_x, L_y, and L_z, then [L_x, L_y] = iħ L_z. This implies that we cannot simultaneously measure L_x and L_y.

However, L^2, which is total angular momentum operator (while not the magnitude, it’s related to the magnitude), commutes with any of the components (e.g., L_z). That is, [L^2, L_z] = 0. This means L^2 and any one component of angular momentum can be simultaneously measured. Hence, we often use L^2 and L_z in practical calculations.

So when you’re measuring the total angular momentum, you use the operator L^2 because it can have a definite value at the same time as any one of the components, typically L_z, which is not the case for simply Lx, Ly or Lz.

Exactly like angular momentum measuring above, spin can not be measured simultaneously on x, y, z:

now refreshing memory, spin x, y, z are no difference than sigma or Pauli matrices:

Note for spin 1 bosons the operators have to be in 3×3 matrix:

Now if i have a spin Sz operator on a vector, which returns eigen value +1, then