In quantum mechanics, the spin quantum number (s) and the magnetic spin quantum number (m_s) are both properties associated with the intrinsic angular momentum, or spin, of a particle. However, they have distinct meanings and describe different aspects of the particle's spin.
The spin quantum number (s) represents the magnitude of the particle's spin angular momentum. It determines the total angular momentum that a particle can have due to its spin, and it is always a positive or zero value. The spin quantum number is given by the expression s = √(s(s+1)), where s can take on values of 0, 1/2, 1, 3/2, 2, and so on. The half-integer values such as 1/2, 3/2, etc., are typically associated with fundamental particles like electrons, protons, and neutrons, which are fermions and have "spin-1/2."
On the other hand, the magnetic spin quantum number (m_s) represents the orientation or projection of the particle's spin angular momentum along a chosen axis, usually the z-axis. It describes the possible discrete values that the component of the spin angular momentum can take in a given direction. The magnetic spin quantum number can range from -s to +s in integer steps. For example, if the spin quantum number s is 1/2, then the magnetic spin quantum number m_s can be either +1/2 or -1/2.
To summarize, the spin quantum number (s) gives the total magnitude of the spin angular momentum, while the magnetic spin quantum number (m_s) specifies the specific projection or orientation of the spin along a chosen axis.