How is the total spin quantum number calculated?

The total spin quantum number is derived from the relative orientations of individual spins within a system. Each electron possesses an intrinsic property called spin, characterized by two possible orientations: "spin-up" (+1/2) and "spin-down" (-1/2).

To calculate the total spin quantum number ( S ), one needs to account for the individual spin states of the electrons in the system. For systems with multiple electrons, the spins are added together according to quantum mechanical rules. Specifically, the maximum value of the total spin quantum number arises when all spins are aligned in the same direction, while the minimum occurs when they are paired oppositely.

This method accurately reflects how the spins interact and combine in accordance with the principles of quantum mechanics, leading to the distinction between multiplicity (the number of possible spin states) and the overall spin state represented by ( S ).

Other methods mentioned, such as considering all particles in the system, the distance between electrons, or measuring energy loss during transitions, do not directly relate to how spins combine to yield the total spin quantum number. Consequently, understanding the contributions of individual spins as mentioned provides insight into the total spin of a quantum system.